ALGEBRA 2 NY STATE
COMMON CORE
Kingston High School
2017-2018
eMATHINSTRUCTION, RED HOOK, NY 12571, © 2015
Table of Contents U n i t 1 - Foundations of Algebra ............................................................................................................. 1
U n i t 2 - Linear Functions, Equations, and their Algebra ......................................................... 48
U n i t 3 - Exponential Functions ............................................................................................................ 68
U n i t 4 - Logarithmic Functions ............................................................................................................ 108
U n i t 5 - Sequences and Series ............................................................................................................... 141
U n i t 6 - Quadratic Functions and Their Algebra ...................................................................... 166
U n i t 7 - Graphic Characteristics of Functions ............................................................................ 210
U n i t 8 - Extension Lessons for Honors course ............................................................................. 236
U n i t 9 - Regression .................................................................................................................................... 248
U n i t 1 0 - Polynomial and Rational Functions ............................................................................ 258
U n i t 1 1 - The Circular Functions ....................................................................................................... 345
U n i t 1 2 - Probability................................................................................................................................ 392
U n i t 1 3 - Statistics..................................................................................................................................... 443
A l g e b r a 2 U n i t 1 - Algebraic Essentials Review
Unit 1 Lesson 1 – Variables, terms, and Expressions Page 1
U n i t 1 - Foundations of Algebra
LESSON 1: VARIABLES, TERMS, AND EXPRESSIONS
Math has a unique language to clarify concepts and remove ambiguity from the analysis of problems. Here are
some basic definitions so that we can all speak this language. Algebra II starts with some review of Algebra I.
Exercise #1: Consider the expression 22 3 7x x .
(c) What is the sum of this expression with the
expression 25 12 2x x ?
To add like terms, simply add the coefficients and leave the variables and powers unchanged. But, why does
this work? Below is an example of the technical steps to combine two like terms.
2 2 2
2 2
4 6 4 6
10 10
x y x y x y
x y x y
Exercise #2: Because the expression 8 2 3 5 3 1x x can be rewritten into a simpler form 31 29x , these
two expressions are equivalent. How can you test this equivalency? Show work for your test.
SOME BASIC DEFINITIONS
Variable: A quantity that is unknown, or can change within the context of a problem, represented by a letter
or symbol.
Terms: A single number or combination of numbers and variables using only multiplication or division.
Expression: A combination of terms using addition and subtraction.
(a) How many terms does this expression contain? (b) Evaluate this expression, without your
calculator, when 3x . Show your
calculations.
LIKE TERMS
Like Terms: Two or more terms that have the same variables raised to the same powers. Only the coefficients
(the multiplying numbers) can differ.
A l g e b r a 2 U n i t 1 - Algebraic Essentials Review
Unit 1 Lesson 1 – Variables, terms, and Expressions Page 2
LESSON 1: VARIABLES, TERMS, AND EXPRESSIONS
HOMEWORK
1. For each of the following expressions, state the number of terms.
(a) 23 1x (b) 2 38 7 2x x x (c) 2 2 417 2
2xy x y xy
2. Simplify each of the following expressions by combining like terms. Be careful to only combine terms that
have the same variables and powers.
(a) 2 22 8 1 5 2 8x x x x (b) 2 25 2 10 7 5x x x x
(c) 2 2 2 24 2 9x y xy xy x y (d) 3 2 2 3 2 2 37 2 4 2 9 4x x y xy y x x y y
3. Given the algebraic expression 2
12 12
1
x
x
do the following:
(c) Nina believes that this expression is equivalent to dividing 12 by one less than x. Do your results from (a)
and (b) support this assertion? Explain.
(a) Evaluate the expression for when 7x . (b) Evaluate the expression for when 4x .
A l g e b r a 2 U n i t 1 - Algebraic Essentials Review
Unit 1 Lesson 1 – Variables, terms, and Expressions Page 3
HOMEWORK (cont.)
4. Classify each of the following as either a monomial (single term), a binomial (two terms) or a trinomial
(three terms).
(a) 24x (b) 23 2 1x x (c) 216 x
(d) 2 2 25x y (e) 55
3
x (f) 216 10 4t t
5. Use the distributive property first and then combine each of the following linear expressions into a single,
equivalent binomial expression.
(a) 5 2 3 2 4 1x x (b) 2 10 1 3 4 5x x
6. Which of the following is equivalent to the expression 2 6 4 2 1 3x x ?
(1) 8 2x (3) 4 2 3x
(2) 5 2 1x (4) 10 1x
7. Simplify the expression 8 3 1 2 5 7x x . _____________________________
A l g e b r a 2 U n i t 1 - Foundations of Algebra
Unit 1 Lesson 2 – Solving Linear Equations Page 4
LESSON 2: SOLVING LINEAR EQUATIONS
You will learn many new equation solving techniques in Algebra 2, but the most basic of all equations are those
where the variable, say x, is only raised to the first power. These are called linear equations. You need to easily
solve linear equations in order to be successful in the beginning of Algebra 2. Let's practice.
Exercise #1: Solve each of the following linear equations for the value of x.
(a) 3 5 26x (b) 8 7 4 5x x
(c) 8
62
x (d) 6 4 2 1 2 20x x x
Strange things can sometimes happen when solving linear (and other) equations. Sometimes we get no solutions
at all, in which case the equation is known as inconsistent. Other times, any value of x will solve the equation, in
which case it is known as an identity.
Exercise #2: Try to solve the following equation. State whether the equation is an identity or inconsistent.
Explain.
6 2 4 3 2 5x x x x
A l g e b r a 2 U n i t 1 - Foundations of Algebra
Unit 1 Lesson 2 – Solving Linear Equations Page 5
Exercise #3: An identity is an equation that is true for all values of the substitution variable. Trying to solve
them can lead to confusing situations. Consider the equation:
2 6 1 3 3 2x x x
(a) Test the values of 5x and 3x in this equation. Show that they are both solutions.
(b) Attempt to solve the equation until you are sure this is an identity.
Exercise #4: Which of the following equations are identities, which are inconsistent, and which are neither?
(a) 8 2 3 5 1x x x x (b) 4 2
8 2 92
xx
(c) 2 8 7 2 2 3x x x (d) 16 4
2 1 2 14
xx x
A l g e b r a 2 U n i t 1 - Foundations of Algebra
Unit 1 Lesson 2 – Solving Linear Equations Page 6
LESSON 2: SOLVING LINEAR EQUATIONS
HOMEWORK
1. Solve each of the following linear equations. If the equation is inconsistent, state so. If the equation is an
identity, also state so. Reduce any non-integer answers to reduced fractions.
(a) 7 5 2 35x x (b) 7 53
x (c) 4 5 4 1x x
(d) 5 3
1 142
x (e) 3 1 2 9x x (f) 4 2 1 5 6x x x x
(g) 5 2 6 2 4 3 8 9x x x (h) 2 5
6 18
x x (Cross multiply to begin)
(i) 10 4
7 5 12
xx
(j)
8 2018 2 7 2
2
xx
A l g e b r a 2 U n i t 1 - Foundations of Algebra
Unit 1 Lesson 2 – Solving Linear Equations Page 7
HOMEWORK (cont.)
2. Laura is thinking of a number such that the sum of the number and five times two more than the number is 26
more than four times the number. Determine the number Laura is thinking of.
3. As if #2 wasn't confusing enough, Laura is now trying to come up with a number where three less than 8 times
the number is equal to half of 16 times the number after it was increased by 1. No number seems to work.
Explain why.
4. When finding the intersection of two lines, you first "set the linear equations equal" to each other. Find the
intersection point of the two lines whose equations are shown below. Be sure to find both the x and y
coordinates.
5 1 and 2 11y x y x
5. Explain why you cannot find the intersection points of the two lines shown below. Give both an algebraic
reason and a graphical reason.
4 1 and 4 10y x y x
A l g e b r a 2 U n i t 1 - Foundations of Algebra
Unit 1 Lesson 3 – Common Algebraic Expressions Page 8
LESSON 3: COMMON ALGEBRAIC EXPRESSIONS
In Algebra 2 you will spend a lot of time evaluating and simplifying algebraic expressions.
You must be able to evaluate algebraic expressions for values of the variables in them.
Exercise #1: Consider the algebraic expression 24 1x .
Exercise #2: Consider the more complex algebraic expression (known as a rational expression) 3
4 3
7
x
x
.
Expressions can contain operators such as square and cube roots and absolute value. Practice these below:
Exercise #3: Is the absolute value expression 8 2x equivalent to 10x ? How can you check this?
ALGEBRAIC EXPRESSION
Algebraic expressions are just combinations of constants and variables using addition, subtraction,
multiplication, and division along with exponents and roots (square roots, cube roots, etc.).
(a) Describe the operations occurring within this
expression and the order in which they occur.
(b) Evaluate this expression for the replacement
value 3x . Show each step in your
calculation. Do not use a calculator.
(a) Without using your calculator, find the value of
this expression when 3x . Reduce your
answer to simplest terms. Show your steps.
(b) If a student entered the following into their
calculator, it would give them the incorrect
answer. Why?
4 3 3/ 3^3 7
A l g e b r a 2 U n i t 1 - Foundations of Algebra
Unit 1 Lesson 3 – Common Algebraic Expressions Page 9
Exercise #4: Consider the algebraic expression 225 x , which contains a square root.
(c) Max thinks that the square root operation distributes over the subtraction. In other words, he thinks he can
take the square root of each TERM, and believes the following equation is an identity:
225 5x x
Show that this is not an identity.
Algebraic expressions can be complicated, but if you consider order of operations (PEMDAS) and work
generally from inside to outside then you can evaluate any expression for replacement values.
Exercise #5: Consider the rather complicated expression 2
8
5 4
x
x
.
Exercise #6: Which of the following is the value of
24 9
3
x x when 10x ?
(1) 31 (3) 18
(2) 24 (4) 84
(a) Evaluate this expression for 3x . (b) Why can you not evaluate the expression for 13x ?
(a) What operation comes last in this expression? (b) Evaluate the expression for 2x . Simplify it
completely.
A l g e b r a 2 U n i t 1 - Foundations of Algebra
Unit 1 Lesson 3 – Common Algebraic Expressions Page 10
LESSON 3: COMMON ALGEBRAIC EXPRESSIONS
HOMEWORK
1. Which of the following expressions has the greatest value when 5x ? Show how you arrived at your choice.
22 7x 3 5
3
x
10 2
3
x
x
2. A zero of an expression is a value of the input variable that results in the expression having a value of zero
(catchy and appropriate name). Is 3x a zero of the quadratic expression shown below? Justify your yes/no
answer.
24 8 12x x
3. Which of the following is the value of the rational expression 22 3
6 4
x
x
when 2x ?
(1) 12
2 (3) 14
1
(2) 5
8 (4)
2
7
4. If 5x and 2y then 2 2
x y
x y
is
(1) 1
7 (3)
3
29
(2) 13
3 (4)
7
19
A l g e b r a 2 U n i t 1 - Foundations of Algebra
Unit 1 Lesson 3 – Common Algebraic Expressions Page 11
HOMEWORK (cont.)
5. What is the value of 10 3x x if 2x ?
(1) 7 (3) 3
(2) 5 (4) 17
6. If 2x then 24 2 5
10
x x has a value of
(1) 5
2 (3)
2
5
(2) 7
5 (4)
1
2
7. The revenue, in dollars, that eMathInstruction makes off its videos depends on how many views they receive.
If x represents the number of views, in hundreds, then the profit can be found with the expression:
216 10
2x x
How much revenue would they make if their videos were viewed 600 times?
8. Sam believes that the two expressions below are equivalent. Test values and see if you can build evidence for
or against this belief.
3 8x x 2 5 24x x
A l g e b r a 2 U n i t 1 - Foundations of Algebra
Unit 1 Lesson 4 – Basic Exponent Properties Page 12
LESSON 4: BASIC EXPONENT PROPERTIES
Exponents represent repeated multiplication. The way they combine, or don't combine, is dictated by this
simple premise. The process is to multiply the numerical coefficients and add the powers
Find the product of the monomials 52x and
24x . Explain why the final exponent on the variable x is 7.
Exercise #1: Multiply the following monomials.
(a) 2 65 3x x (b) 42 6x x (c) 4 1036
2x x
(d) 2
34x
Remember, monomials (or terms) can have more than one variable, just as long as they are all combined using
multiplication and division only. Multiplying monomials that have more than one variable still just involves
application of exponent laws.
Exercise #2: Find each of the following products, which involve monomials of multiple variables.
(a) 3 2 54 5x y xy (b) 7 3 2 62 4x y x y (c) 2 51 5
2 2xy x y
A l g e b r a 2 U n i t 1 - Foundations of Algebra
Unit 1 Lesson 4 – Basic Exponent Properties Page 13
One of the key skills needed this year is factoring expressions, especially factoring out a common factor. To build
some skills, consider the following problem.
Exercise #3: Fill in the missing blank in each of the following equations involving a product such that the equation
is then an identity.
(a) 5 26 2 ______x x (b) 8 312 4 ______x x (c) 2 4 320 2 _________x y xy
The final skill we will review in this lesson is using the distributive property of multiplication (and division)
over addition (and subtraction).
Exercise #4: Use the distributive property to multiply the following monomials and polynomials.
(a) 2 5 3x x (b) 3 25 2 3 6x x x (c) 2 27 2 3x x x
(d) 2 2 2xy x y (e) 2 4 2 2 33 2 4x y x y xy y
Exercise #5: Similar to Exercise #3, fill in the missing portion of each product so that the equation is an identity.
(a) 28 12 4 ___________________x x x (b) 4 3 2 27 21 28 7 _______________x x x x
(c) 3 2 2 3 5 210 20 35 5 ________________x y x y xy xy
(d) 24 2 9 2 2 _______________x x x x
A l g e b r a 2 U n i t 1 - Foundations of Algebra
Unit 1 Lesson 4 – Basic Exponent Properties Page 14
LESSON 4: BASIC EXPONENT PROPERTIES
HOMEWORK
1. Multiply 2 53x y and 5 27x y ____________________
2. Find each of the following products of monomials.
(a) 2 43 10x x (b) 52 9x x (c) 2 5 34 8x y x y (d) 2
45x
(e) 2 54 15t t (f) 47 5x xy (g) 4212
3x x
(h) 2 42 5 6x x x
3. Fill in the missing portion of each product to make the equation an identity.
(a) 6 218 3 ________x x (b) 2 7 240 8 ________x y xy (c) 490 15 __________x y xy
(d) 6 224 3 ________x x (e) 4 10 2 248 16 _______x y x y (f) 8 6 4 349 7 ________x y x y
4. Use the distributive property to write each of the following products as polynomials.
(a) 4 5 2x x (b) 5 10x x (c) 26 4 8x x x
(d) 2 210 2 8x x x (e) 3 2 57 2 5xy x y y (f) 2 2 3 2 2 38 2 5x y x x y xy y
A l g e b r a 2 U n i t 1 - Foundations of Algebra
Unit 1 Lesson 4 – Basic Exponent Properties Page 15
HOMEWORK (cont.)
(#4 cont.: Use the distributive property to write each of the following products as polynomials)
(g) 3 27 4 2 1x x x (h) 216 2 2 3t t t (i) 2 212 2xy x xy y
5. Fill in the missing part of each product in order to make the equation into an identity.
(a) 5 3 310 35 5 ____________x x x (b) 3 2 2 38 2 10 2 ________________x y x y xy xy
(c) 2 5 218 45 9 _______________t t t (d) 4 3 2 245 30 15 15 ___________________x x x x
(e) 5 6 5 5 __________x x x x (f) 2 3 3 3 ______________x x x x
Another important exponent property occurs when a monomial with an exponent is raised to another power. See
if you can come up with a general pattern.
6. Write each of the following out as extended products and then simplify. The first is done as an example.
(a) 3
2 2 2 2 6x x x x x (b) 2
3x
(c) 4
5x (d) 3
4x
7. So, what is the pattern? For positive integers a and b: __________b
ax ?
A l g e b r a 2 U n i t 1 - Foundations of Algebra
Unit 1 Lesson 5 – Multiplying Polynomials Page 16
LESSON 5: MULTIPLYING POLYNOMIALS
Polynomials are expressions that are combinations of terms using addition and subtraction that can have only
constants and positive integer powers. For example 2x3 - 3x2 + 5 is a polynomial.
This lesson covers multiplying polynomials. The distributive property will be used, also an area model will be
shown. Let's start by multiplying binomials. How many terms does a binomial have?____
Exercise #1: Find the product of (3x2 - 4) and (x2 +1) using the distributive property.
Exercise #2: Consider the product of 3 2x with 2 5x .
Exercise #3: Find the product of the binomial 4 3x with the trinomial 22 5 3x x using the distributive
property or using the area model.
3x
2
(a) Find this product using the distributive property
twice
(b) Represent this product on the area model shown
below. 2x 5
A l g e b r a 2 U n i t 1 - Foundations of Algebra
Unit 1 Lesson 5 – Multiplying Polynomials Page 17
It is critical to understand that when two polynomials are multiplied then the result is equivalent to this product
and this equivalence can be tested.
Exercise #4: Consider the product of 2x and 2 5x .
Exercise #5: The product of three binomials, just like the product of two, can be found with repeated applications
of the distributive property.
(a) Find the product: 2 4 7x x x .
(b) For what three values of x will the cubic polynomial that you found in part (a) have a value of zero? What
famous law is this known as?
(c) Test one of the three values you found in (b) to verify that it is a zero of the cubic polynomial.
(a) Evaluate this product for 4x . Show the work
that leads to your result.
(b) Find a trinomial that represents the product of
these two binomials.
(c) Evaluate the trinomial for 4x . Is it
equivalent to the answer you found in (a)?
(d) What is the value of the trinomial when 2x ?
Can you explain why this makes sense based on
the two binomials?
A l g e b r a 2 U n i t 1 - Foundations of Algebra
Unit 1 Lesson 5 – Multiplying Polynomials Page 18
LESSON 5: MULTIPLYING POLYNOMIALS
HOMEWORK
1. Multiply the following binomials and express each product as an equivalent trinomial. Use an area model to
help find your product, if necessary.
(a) 5 8x x (b) 3 2 2 7x x (c) 5 2 2 3x x
(d) 2 24 10x x (e) 3 32 1 5 4x x (f) 2 21 9x x
2. Find each of the following products in equivalent form.
(a) 25 3 2x x x (b) 22 3 4 5 7x x x
(c) 3
2 5x
A l g e b r a 2 U n i t 1 - Foundations of Algebra
Unit 1 Lesson 5 – Multiplying Polynomials Page 19
HOMEWORK (cont.)
3. A square of an unknown side length x inches has one side length increased by 4 inches and the other increased
by 7 inches.
(a) If the original square is shown below with side lengths marked as x, label the second diagram to represent
the new rectangle constructed by increasing the sides as described above.
(b) Label each portion of the second diagram with their areas in terms of x (when applicable). State the product
of 4x and 7x as a trinomial below.
4. Expression 8 4x x .
(c) Show that this trinomial is also equal to zero at the larger value of x from part (a).
x
x
x
x
(c) If the original square had a side length of 2x
inches, then what is the area of the second
rectangle? Show how you arrived at your
answer.
(d) Verify that the trinomial you found in part (b)
has the same value as (c) for 2x .
(a) For what values of x will this expression be
equal to zero? Show how you arrived at your
answer.
(b) Write this product as an equivalent trinomial.
A l g e b r a 2 U n i t 1 - Foundations of Algebra
Unit 1 Lesson 6 – Using Tables on your Calculator Page 20
LESSON 6: USING TABLES ON YOUR CALCULATOR
The graphing calculator is an amazing device that can do many things. One helpful function is evaluating
expressions for different input values. We will be using two tools on the calculator today, STORE and TABLE.
First let's use STORE. The STORE button on your calculator is located above the ON button (lower left).
Exercise #1: Find the value of each of the following expressions by using the STORE feature on your calculator.
(a) 2 2 7x x for 5x (b)
2 6
3 5
x
x
for 10x (c) 27 20x x for 2x
Sometimes the calculator gives useful information even when it cannot evaluate an expression.
Exercise #2: Consider the expression 6 2x . What happens when you use STORE to evaluate this expression
for 5x ? Evaluate the expression by hand to explain what the calculator is telling us.
Exercise #3: Multiply the binomials 3 2x and 5x .
(c) Use STORE to evaluate the trinomial from (a) for 5x . Explain why the trinomial turns out to be this specific
value at 5x (look at the original binomials).
(a) Find their product in trinomial form. (b) Evaluate both the trinomial and the original
product for 2x . What do you notice?
A l g e b r a 2 U n i t 1 - Foundations of Algebra
Unit 1 Lesson 6 – Using Tables on your Calculator Page 21
STORE is helpful when you are determining the value of an expression at one or two input values of x. But, if
you want to know an expression's value for multiple inputs, then TABLES are a much better tool. On the
graphing calculator, the TABLE button is on the upper right, (2nd > GRAPH.)
Exercise #4: The expression 3 22 16 32x x x has an integer zero somewhere on the interval 0 10x . Use
a TABLE to find the zero on this interval. Show the table.
Table commands are good at proving two expressions are equivalent. This is helpful when you've done several
manipulations and you want to be sure you've produced an algebraically equivalent expression.
Exercise #5: Consider the more complex algebraic expression shown below:
5 8 3 2x x x x
(a) This relatively complex expression simplifies into a linear binomial expression. Determine this expression
carefully. Show your work below.
(b) Set up a table using the original expression and the one you found in (a) over the interval 0 5x . Compare
values to determine if you correctly simplified the original expression.
x 1y 2y
0
1
2
3
4
5
A l g e b r a 2 U n i t 1 - Foundations of Algebra
Unit 1 Lesson 6 – Using Tables on your Calculator Page 22
LESSON 6: USING TABLES ON YOUR CALCULATOR
HOMEWORK
1. Use the STORE feature on your calculator to evaluate each of the following. No work needs to be shown.
(a) 7 18x for 8x (b) 23 2 5x x for 3x (c)
3 25 4 20x x x for 5x
(d) 2 2 8x x for 1x (e) 2
5 3
4 5
x
x
for 2x (f)
4
9
x
x
for 5x
2. The STORE feature helps when checking to see if a value is a solution to an equation. Let's see how this works
in this problem. Consider the linear equation:
6 3 4 9x x
3. Two of the following values of x are solutions to the equation: 2 4 12 10 4x x x . Determine which they
are and provide a justification for your answer.
2x 5x 6x 8x
(a) Solve this equation for x. (b) Using STORE, determine the value of both the
left hand expression, 6 3x , and the right hand
expression, 4 9x , at the value of x you found
in (a).
(c) Why does what you found in part (b) verify that your solution is correct (or possibly incorrect if you made
a mistake in (a))?
A l g e b r a 2 U n i t 1 - Foundations of Algebra
Unit 1 Lesson 6 – Using Tables on your Calculator Page 23
HOMEWORK (cont.)
4. The quadratic expression 2 8 10x x has its smallest value for some integer value of x on the interval
0 10x . Set up a TABLE to find the smallest value of the expression and the value of x that gives this value.
Show your table below.
5. Consider the expression 7 3 1 4x x x x .
(a) Multiply the two sets of binomials and combine like terms in order to write this expression as an equivalent
trinomial in standard form. Show your work.
(b) Set up a TABLE to verify that your answer in part (a) is equivalent to the original expression. Don't hesitate
to point out that it is not equivalent (which means you either made a mistake in your algebra or in your
table set up). Show your table.
6. The product of three binomials is shown below. Write this product as a polynomial in standard form. (Its
highest power will be 3x ).
1 2 4x x x
7. Set up a table for the answer you found in #6 on the interval 5 5x . Where does this expression have
zeroes?
A l g e b r a 2 U n i t 1 - Foundations of Algebra
Unit 1 Lesson 7 – Introduction to Functions Page 24
LESSON 7: INTRODUCTION TO FUNCTIONS
Most higher level mathematics is built upon the concept of a function.
Make sure to know the following definition:
Exercise #1: An internet music service offers a plan whereby users pay a flat monthly fee of $5 and can then
download songs for 10 cents each.
(a) What are the independent and dependent variables in this scenario?
Independent: Dependent:
(b) Fill in the table below for a variety of independent values:
(c) Let the number of downloads be represented by the variable x and the amount charged be represented by the
variable y, write an equation that models y as a function of x.
(d) Based on the equation you found in part (c), produce a
graph of this function for all values of x on the interval
0 40x . Use a calculator TABLE to generate
additional coordinate pairs to the ones you found in
part (b).
DEFINITION: A function is any “rule” that assigns exactly one output value (y-value) for each input value (x-
value). These rules can be expressed in different ways, the most common being equations, graphs, and tables
of values. We call the input variable independent and output variable dependent.
Number of downloads, x 0 5 10 20
Amount Charged, y
Number of Downloads, x
Am
ou
nt
Ch
arg
ed,
y
30 40 20 10
2.50
5.00
7.50
10.00
A l g e b r a 2 U n i t 1 - Foundations of Algebra
Unit 1 Lesson 7 – Introduction to Functions Page 25
Exercise #2: One of the following graphs shows a relationship where y is a function of x and one does not.
(a) Draw the vertical line whose equation is 3x on both graphs.
(b) Give all output values for each graph at an input of 3 .
Relationship A: Relationship B:
(c) Explain which of these relationships is a function and why.
Exercise #3: The graph of the function 2 4 1y x x is shown below.
(a) State this function’s y-intercept.
(b) Between what two consecutive integers does the larger x-
intercept lie?
(c) Draw the horizontal line 2y on this graph.
(d) Using these two graphs, find all values of x that solve
the equation below:
2 4 1 2x x
(e) Verify that these values of x are solutions by using STORE on your graphing calculator.
y
x
y
x
Rel
ati
on
ship
A
y
x
Rel
ati
on
ship
B
A l g e b r a 2 U n i t 1 - Foundations of Algebra
Unit 1 Lesson 7 – Introduction to Functions Page 26
LESSON 7: INTRODUCTION TO FUNCTIONS
HOMEWORK
1. Determine for each of the following graphed relationships whether y is a function of x using the Vertical Line
Test.
(a) (b) (c)
(d) (e) (f)
2. What are the outputs for an input of 5x given functions defined by the following formulas:
(a) 3 4y x (b) 250 2y x (c) 2xy
y
x
y
x
y
x
y
x
y
x
y
x
A l g e b r a 2 U n i t 1 - Foundations of Algebra
Unit 1 Lesson 7 – Introduction to Functions Page 27
Relationship #1 Relationship #2
HOMEWORK (cont.)
3. Evan is walking home from the museum. He starts 38 blocks from home and walks 2 blocks each minute.
Evan’s distance from home is a function of the number of minutes he has been walking.
(a) Which variable is independent and which variable is dependent in this scenario?
(b) Fill in the table below for a variety of time values.
Time, t, in minutes 0 1 5 10
Distance from home, D, in blocks
4. In one of the following tables, the variable y is a function of the variable x. Explain which relationship is a
function and why the other is not.
x y
-2 11
0 7
2 11
4 23
6 23
x y
0 0
1 -1
1 1
4 -2
4 2
(c) Determine an equation relating the
distance, D, that Evan is from home as a
function of the number of minute, t, that he
has been walking.
(d) Determine the number of minutes, t, that it
takes for Evan to reach home.
A l g e b r a 2 U n i t 1 - Foundations of Algebra
Unit 1 Lesson 8 – Function Notation Page 28
LESSON 8: FUNCTION NOTATION
Functions are basic tools that convert inputs, values of the independent variable, to outputs, values of
the dependent variable. We will use function notation all year. The first exercise shows this notation in
formulas.
Exercise #1: Evaluate each of the following given the function definitions and input values.
(a)
5 2
3
2
f x x
f
f
(b)
2 4
3
0
g x x
g
g
(c)
2
3
2
xh x
h
h
Do not confuse function notation with multiplication. Function notation is summarized below.
Function rules commonly come in one of three forms: (1) equations (as in Exercise #1), (2) graphs, and
(3) tables. The next few exercises shows function notation in these three forms.
Exercise #2: Boiling water at 212 degrees Fahrenheit is left in a room that is at 65 degrees Fahrenheit and begins
to cool. Temperature readings are taken each hour and are given in the table below. In this scenario, the
temperature, T, is a function of the number of hours, h.
(a) Evaluate 2T and 6T . (b) For what value of h is 76T h ?
(c) Between what two consecutive hours will 100T h ?
FUNCTION NOTATION
Output Rule Input
h
(hours)
0 1 2 3 4 5 6 7 8
T h
F
212 141 104 85 76 70 68 66 65
A l g e b r a 2 U n i t 1 - Foundations of Algebra
Unit 1 Lesson 8 – Function Notation Page 29
Exercise #3: The function y f x is defined by the graph shown below.
Use this graph to answer the following questions
(a) Evaluate 1 , 1 , and 5f f f .
(b) Evaluate 0f . What special feature on a graph does 0f
always correspond to?
Exercise #4: For a function y g x it is known that 2 7g . Which of the following points must lie on the
graph of g x ?
(1) 7, 2 (3) 0, 7
(2) 2, 7 (4) 2, 0
Exercise #5: Physics students drop a ball from the top of a 50 foot high building and model its height as a function
of time with the equation 250 16h t t . Using TABLES on your calculator, determine, to the nearest tenth of a
second, when the ball hits the ground. Provide tabular outputs to support your answer.
(c) What values of x solve the equation 0f x .
What special features on a graph does the set of
x-values that solve 0f x correspond to?
(d) Between what two consecutive integers does
the largest solution to 3f x lie?
y
x
A l g e b r a 2 U n i t 1 - Foundations of Algebra
Unit 1 Lesson 8 – Function Notation Page 30
LESSON 8: FUNCTION NOTATION
HOMEWORK
1. Without using your calculator, evaluate each of the following given the function definitions and input
values.
(a)
3 7
4
2
f x x
f
f
(b)
23
2
3
g x x
g
g
(c)
5
41
14
h x x
h
h
2. Using STORE on your calculator, evaluate each of the following more complex functions.
(a)
23 5
4 10
5
0
xf x
x
f
f
(b)
225
4
3
xg x
x
g
g
(c)
30 1.2
3
0
xh x
h
h
3. Based on the graph of the function y g x shown below, answer the following questions.
(a) Evaluate 2 , 0 , 3 and 7g g g g .
(b) What values of x solve the equation 0g x
(c) Graph the horizontal line 2y on the grid
above and label.
(d) How many values of x solve the equation
2g x ?
y
x
A l g e b r a 2 U n i t 1 - Foundations of Algebra
Unit 1 Lesson 8 – Function Notation Page 31
HOMEWORK (cont.)
4. Ian invested $2500 in an investment vehicle that is guaranteed to earn 4% interest compounded yearly. The
amount of money, A, in his account as a function of the number of years, t, since creating the account is given
by the equation 2500 1.04t
A t .
(a) Evaluate 0 and 10A A .
5. A ball is shot from an air-cannon at an angle of 45 with the horizon. It travels along a path given by the
equation 21
50h d d d , where h represents the ball’s height above the ground and d represents the
distance the ball has traveled horizontally. Using your calculator to generate a table of values, graph this
function for all values of d on the interval 0 50d . Look at the table to properly scale the y-axis.
What is the maximum height that the ball
reaches? At what value of d does it reach this
height?
(b) What do the two values that you found in part
(a) represent?
(c) Using tables on your calculator, determine, to
the nearest whole year, the value of t that solves
the equation 5000A t . Justify your answer
with numerical evidence.
(d) What does the value of t that you found in part
(c) represent about Ian’s investment?
A l g e b r a 2 U n i t 1 - Foundations of Algebra
Unit 1 Lesson 9 –Function Composition Page 32
LESSON 9: FUNCTION COMPOSITION
Functions convert the value of an input variable into the value of an output variable. This output could then be
used as an input to a second function. This process is known as composition of functions, in other words,
combining the action or rules of two functions.
Exercise #1: A circular garden with a radius of 15 feet is to be covered with topsoil at a cost of $1.25 per square
foot of garden space.
In this exercise, we see that the output of an area function is used as the input to a cost function. This idea can be
generalized to generic functions, f and g as shown in the diagram below.
There are two notations that are used to indicate composition of two functions. These will be introduced in the
next few exercises, both with equations and graphs.
Exercise #2: Given 2 5 and 2 3f x x g x x , find values for each of the following.
(a) 1f g (b) 2g f (c) 0g g
(d) 2f g (e) 3g f (f) 1f f
(a) Determine the area of this garden to the
nearest square foot.
(b) Using your answer from (a), calculate the
cost of covering the garden with topsoil.
Input = x
Output from f
becomes the input to g Final output = y
A l g e b r a 2 U n i t 1 - Foundations of Algebra
Unit 1 Lesson 9 –Function Composition Page 33
Exercise #3: The graphs below are of the functions and y f x y g x . Evaluate each of the following
questions based on these two graphs.
(a) 2g f (b) 1f g (c) 1g g
(d) 2g f (e) 0f g (f) 0f f
y
x
y
x
A l g e b r a 2 U n i t 1 - Foundations of Algebra
Unit 1 Lesson 9 –Function Composition Page 34
LESSON 9: FUNCTION COMPOSITION
HOMEWORK
1. Given 3 4f x x and 2 7g x x evaluate:
(a) 0f g (b) 2g f (c) 3f f
(d) 6g f (e) 5f g (f) 2g g
2. Given 2 11 and 2h x x g x x evaluate:
(a) 18h g (b) 4g h (c) 11g g
(d) 0h h (e) 38h g (f) 0g h
3. The graphs of and y h x y k x are shown below. Evaluate the following based on these two graphs.
(a) 2h k (b) 0k h (c) 2h h (d) 2k k
y
x
y
x
A l g e b r a 2 U n i t 1 - Foundations of Algebra
Unit 1 Lesson 9 –Function Composition Page 35
HOMEWORK (cont.)
4. Scientists modeled the intensity of the sun, I, as a function of the number of hours since 6:00 a.m., h, using
the function 212
36
h hI h
. They then model the temperature of the soil, T, as a function of the intensity
using the function 5000T I I . Which of the following is closest to the temperature of the soil at 2:00
p.m. ?
(1) 54 (3) 67
(2) 84 (4) 38
5. Physics students are studying the effect of the temperature, T, on the speed of sound, S. They find that the
speed of sound in meters per second is a function of the temperature in degrees Kelvin, K, by 410S K K
The degrees Kelvin is a function of the temperature in Celsius given by 273.15K C C . Find the speed
of sound when the temperature is 30 degrees Celsius. Round to the nearest tenth.
6. Consider the functions 2 9f x x and 9
2
xg x
. Calculate the following.
(a) 15g f (b) 3g f (c) ))9(( gf
(d) What appears to always be true when you compose these two functions?
A l g e b r a 2 U n i t 1 - Foundations of Algebra
Unit 1 Lesson 10 –Domain & Range of Functions Page 36
LESSON 10: THE DOMAIN AND RANGE OF A FUNCTION
Because functions convert values of inputs into value of outputs, we talk about the sets that
represent these inputs and outputs. The set of inputs that result in an output is called the domain of the
function. The set of outputs is called the range.
Exercise #1: Consider a function that has as its inputs the months of the year, and as its outputs the number of
days in each month. In this case, the number of days is a function of the month of the year. Assume this function
is restricted to non-leap years.
Exercise #2: State the range of the function 2 1f n n if its domain is the set 1, 3, 5 . Show the domain
and range in the mapping diagram below.
Exercise #3: The function y g x is completely defined by the graph shown below. Answer the following
questions based on this graph.
(a) Determine the minimum and maximum x-values
represented on this graph.
(b) Determine the minimum and maximum y-values
represented on this graph.
(c) State the domain and range of this function using
set builder notation.
y
x
(a) Write, in roster form, the set that represents this
function’s domain.
(b) Write, in roster form, the set that represents this
function’s range.
Domain of f Range of f
A l g e b r a 2 U n i t 1 - Foundations of Algebra
Unit 1 Lesson 10 –Domain & Range of Functions Page 37
Some functions, defined with graphs or equations, have domains and ranges that stretch out infinitely.
Consider the following exercise in which a standard parabola is graphed.
Exercise #4: The function 2 2 1f x x x is graphed on the grid below. Which of the following represent its
domain and range written in interval notation?
(1)
Domain: 2, 4
Range: 4, 6
(3)
Domain: ,
Range: 4,
(2)
Domain: 2, 4
Range: 4,
(4)
Domain: 2, 4
Range: 4, 6
For most functions defined by an algebraic formula, the domain consists of the set of all real numbers, given the
concise symbol R. Sometimes, though, there are restrictions placed on the domain of a function by the structure
of its formula. Two basic restrictions will be illustrated in the next few exercises.
Exercise #5: The function 2 1
4
xf x
x
has outputs given by the following calculator table.
(a) Evaluate 1 and 6f f from the table.
(b) Why does the calculator give an ERROR at 4x ?
(c) Are there any values except 4x that are not in the domain of f ? Explain.
Exercise #6: Which of the following values of x would not be in the domain of the function 4y x ? Explain
your answer.
(1) 0x (3) 3x
(2) 5x (4) 8x
y
x
x f x
1 -1
2 -2.5
3 -7
4 Error
5 11
6 6.5
7 5
A l g e b r a 2 U n i t 1 - Foundations of Algebra
Unit 1 Lesson 10 –Domain & Range of Functions Page 38
LESSON 10: THE DOMAIN AND RANGE OF A FUNCTION
HOMEWORK
1. A function is given by the set of ordered pairs 2, 5 , 4, 9 , 6,13 , 8,17 . Write its domain and range
in roster form.
Domain: Range:
2. The function 2 5h x x maps the domain given by the set 2, 1, 0,1, 2 . Which of the following sets
represents the range of h x ?
(1) 0, 6,10,12 (3) 5, 6, 9
(2) 5, 6, 7 (4) 1, 4, 5, 6, 9
3. Which of the following values of x would not be in the domain of the function defined by 2
3
xf x
x
?
(1) 3x (3) 3x
(2) 2x (4) 2x
4. Determine any values of x that do not lie in the domain of the function 3 2
2 10
xf x
x
. Justify your
response.
5. Which of the following values of x does lie in the domain of the function defined by 2 7g x x ?
(1) 0x (3) 3x
(2) 2x (4) 5x
6. Which of the following would represent the domain of the function 6 2y x ?
(1) : 3x x (3) : 3x x
(2) : 3x x (4) : 3x x
A l g e b r a 2 U n i t 1 - Foundations of Algebra
Unit 1 Lesson 10 –Domain & Range of Functions Page 39
HOMEWORK (cont.)
7. The function y f x is completely defined by the graph shown below.
(a) Evaluate 4 , 3 , and 6f f f .
(b) Draw a rectangle that circumscribes (just surrounds)
the graph.
(c) State the domain and range of this function using
interval notation.
Domain: Range:
8. Which of the following represents the range of the quadratic function shown in the graph below?
(1) 4, (3) , 4
(2) , 4 (4) 4,
9. A child starts a piggy bank with $2. Each day, the child receives 25 cents at the end of the day and puts it in
the bank. If A represents the amount of money and d stands for the number of days then 2 0.25A d d
gives the amount of money in the bank as a function of days (think about this formula).
y
x
(a) Evaluate .
(c) Explain why the domain does not contain
the value .
(b) For what value of d will .
(d) Explain why the range does not include the
value .
y
x
A l g e b r a 2 U n i t 1 - Foundations of Algebra
Unit 1 Lesson 11 – One to One Functions Page 40
LESSON 11: ONE-TO-ONE FUNCTIONS
A special category of functions is called one-to-one. The following exercise shows the difference between a
function that is one-to-one and one that is not.
Exercise #1: Consider the two functions given by the equations 22 and f x x g x x .
(a) Map the domain 2, 0, 2 using each function. Fill in the range and show the mapping arrows.
(b) What is different between these two functions in terms of how the elements of this domain get mapped to the
elements of the range?
Exercise #2: Of the four tables below, one represents a relationship where y is a one-to-one function of x.
Determine which it is and explain why the others are not.
ONE-TO-ONE FUNCTIONS
A function is called one-to-one if implies that .
In other words, different inputs always give different outputs.
-2
0
2
Domain of f Range of f
-2
0
2
Domain of g Range of g
x y
4 2
4 -2
9 3
9 -3
(1)
x y
-2 1
-1 0
0 1
1 2
(2)
x y
1 2
2 4
3 8
4 16
(3)
x y
-3 10
-2 9
-1 7
-2 10
(4)
A l g e b r a 2 U n i t 1 - Foundations of Algebra
Unit 1 Lesson 11 – One to One Functions Page 41
Exercise #3: Consider the following four graphs which show a relationship between the variables y and x.
Exercise #4: Which of the following represents the graph of a one-to-one function?
Exercise #5: The distance that a number, x, lies from the number 5 on a one-dimensional number line is given
by the function 5D x x . Show by example that D x is not a one-to-one function.
y
x
(1)
y
x
(2)
y
x
(3)
y
x
(4)
(a) Circle the two graphs above that are
functions. Explain how you know they are
functions.
(b) Of the two graphs you circled, which is one-
to-one? Explain how you can tell from its
graph.
THE HORIZONTAL LINE TEST
If any horizontal line passes through the graph of a function at most one time, then that function is one-to-one.
This test works because horizontal lines represent constant y-values. If a horizontal line intersects a graph more
than once, an output has been repeated, and the function is not one-to-one.
(1)
y
x
(2)
y
x
(3)
y
x
(4)
y
x
A l g e b r a 2 U n i t 1 - Foundations of Algebra
Unit 1 Lesson 11 – One to One Functions Page 42
LESSON 11: ONE-TO-ONE FUNCTIONS
HOMEWORK 1. Which of the following graphs illustrates a one-to-one relationship?
2. Which of the following graphs does not represent that of a one-to-one function?
3. In which of the following graphs is each input not paired with a unique output?
4. In which of the following formulas is the variable y a one-to-one function of the variable x? (Hint – try
generating some values either in your head or using TABLES or graphs on your calculator.)
(1) 2y x (3) 2y x
(2) y x (4) 5y
(4)
y
x
(1)
y
x
(2)
y
x
(3)
y
x
(1)
y
x
(2)
y
x
(3)
y
x
(4)
y
x
(1)
y
x
(2)
y
x
(3)
y
x
(4)
y
x
A l g e b r a 2 U n i t 1 - Foundations of Algebra
Unit 1 Lesson 11 – One to One Functions Page 43
HOMEWORK (cont.)
5. Which of the following tables illustrates a relationship in which y is a one-to-one function of x?
(1) (2) (3) (4)
6. A recent newspaper gave temperature data for various days of the week in table format. In which of the tables
below is the reported temperature a one-to-one function of the day of the week?
(1) (2) (3) (4)
7. Physics students drop a basketball from 5 feet above the ground and its height is measured each tenth of a
second until it stops bouncing. The height of the basketball, h, is clearly a function of the time, t, since it was
dropped.
8. Consider the function ( ) round( )f x x , which rounds the input, x, to the nearest integer. Is this function
one-to-one? Explain or justify your answer.
x y
-2 -1
0 -3
2 -1
4 1
6 3
x y
-2 -8
-1 -1
0 0
1 1
2 8
x y
-2 -5
-1 -4
0 -1
-1 7
-2 5
x y
-2 11
-1 -4
0 -5
1 -4
2 11
h (ft)
t (sec)
5
(a) Sketch the general graph of what you believe
this function would look like.
(b) Is the height of the ball a one-to-one function
of time? Explain your answer.
x y
Mon 75
Tue 68
Wed 65
Thu 74
x y
Mon 75
Tue 72
Wed 68
Thu 72
x y
Mon 58
Tue 52
Mon 81
Tue 76
x y
Mon 56
Tue 58
Mon 85
Tue 85
A l g e b r a 2 U n i t 1 - Foundations of Algebra
Unit 1 Lesson 12 – Key Features of Functions Page 44
LESSON 12: KEY FEATURES OF FUNCTIONS
The graphs of functions have many key features whose terminology we will use all year.
Exercise #1: The function y f x is shown graphed to the right.
Answer the following questions based on this graph.
(a) State the y-intercept of the function.
(b) State the x-intercepts of the function. What is the alternative
name that we give the x-intercepts?
(c) Over the interval 1 2x is f x increasing or decreasing?
How can you tell?
(h) If a second function g x is defined by the formula 12
2g x f x , then what is the y-intercept of g?
y
x
(d) Give the interval over which 0f x . What is
a quick way of seeing this visually?
(e) State all the x-coordinates of the relative
maximums and relative minimums. Label each.
y f x
(f) What are the absolute maximum and minimum
values of the function? Where do they occur?
(g) State the domain and range of f x using
interval notation.
A l g e b r a 2 U n i t 1 - Foundations of Algebra
Unit 1 Lesson 12 – Key Features of Functions Page 45
Exercise #2: Consider the function 2 1 8g x x defined over
the domain 4 7x .
(a) Sketch a graph of the function to the right.
(b) State the domain interval over which this function is decreasing.
You need to be able to think about functions in all of their forms, including equations, graphs, and tables. Tables
can be quick to use, but sometimes hard to understand.
Exercise #3: A continuous function f x has a domain of 6 13x with selected values shown below. The
function has exactly two zeroes and has exactly two turning points, one at 3, 4 and one at 9, 3 .
y
x
(c) State zeroes of the function on this interval. (d) State the interval over which 0g x
(e) Evaluate 0g by using the algebraic definition
of the function. What point does this correspond
to on the graph?
(f) Are there any relative maximums or minimums
on the graph? If so, which and what are their
coordinates?
x -6 -1 0 3 5 8 9 13
f x 5 0 -2 -4 -1 0 3 1
(a) State the interval over which 0f x . (b) State the interval over which f x is
increasing.
A l g e b r a 2 U n i t 1 - Foundations of Algebra
Unit 1 Lesson 12 – Key Features of Functions Page 46
LESSON 12: KEY FEATURES OF FUNCTIONS
HOMEWORK
1. The function f x is shown to the right. Answer the following
questions based on its graph.
(a) Evaluate each of the following based on the graph:
(i) 4f (ii) 3f
(b) State the zeroes of f x .
(c) Over which of the following intervals is f x always
increasing?
(1) 7 3x (3) 5 5x
(2) 3 5x (4) 5 3x
y
x
(d) State the coordinates of the relative maximum
and the relative minimum of this function.
Relative Maximum:_________________
Relative Minimum:_________________
(e) Over which of the following intervals is
0f x ?
(1) 7 3x (3) 5 2x
(2) 2 7x (4) 5 2x
(f) A second function g x is defined using the
rule 2 5g x f x . Evaluate 0g . What
does this correspond to on the graph of g?
(g) A third function h x is defined by the formula
3 3h x x . What is the value of 2g h ?
Show how you arrived at your answer.
A l g e b r a 2 U n i t 1 - Foundations of Algebra
Unit 1 Lesson 12 – Key Features of Functions Page 47
HOMEWORK (cont.)
2. For the function 2
9 1g x x do the following.
(a) Sketch the graph of g on the axes provided.
(b) State the zeroes of g.
(c) Over what interval is g x decreasing?
(d) Over what interval is 0g x ? (e) State the range of g.
3. Draw a graph of y f x that matches the following
characteristics.
Increasing on: 8 4 and 1 5x x
Decreasing on: 4 1x
8 5f and zeroes at 6, 2, and 3x
Absolute maximum of 7 and absolute minimum of 5
4. A continuous function has a domain of 7 10x and has selected values shown in the table below. The
function has exactly two zeroes and a relative maximum at 4,12 and a relative minimum at 5, 6 .
y
x
y
x
x -7 -4 -1 0 2 5 7 10
f x 8 12 0 -2 -5 -6 0 4
(a) State the interval on which f x is decreasing.
(b) State the interval over which 0f x .
A l g e b r a 2 U n i t 2 - Linear Functions, Equations, and their Algebra
Unit 2 Lesson 1 – Average Rate of Change Page 48
U n i t 2 - Linear Functions, Equations, and their Algebra
LESSON 1: AVERAGE RATE OF CHANGE
We begin our linear unit by studying the rate of change of functions. When we model using functions, we
are very often interested in the rate that the output is changing compared to the rate of the input.
Exercise #1: The function f x is shown graphed to the right.
(a) Evaluate each of the following based on the graph:
(i) 0f (ii) 4f (iii) 7f (iv) 13f
(b) Find the change in the function, f , over each of the
following domain intervals. Find this both by subtraction
and show this on the graph.
(i) 0 4x (ii) 4 7x (iii) 7 13x
(c) Why can't you simply compare the changes in f from part (b) to determine over which interval the
function changing the fastest?
(d) Calculate the average rate of change for the function over each of the intervals and determine which interval
has the greatest rate.
(i) 0 4x (ii) 4 7x (iii) 7 13x
(e) Using a straightedge, draw in the lines whose slopes you found in part (d) by connecting the points shown on
the graph. The average rate of change gives a measurement of what property of the line?
y
x
A l g e b r a 2 U n i t 2 - Linear Functions, Equations, and their Algebra
Unit 2 Lesson 1 – Average Rate of Change Page 49
The average rate of change is a very important mathematical concept because it gives us a way to quantify
how fast a function changes, on average, over a certain domain interval. We used its formula in the last
exercise:
Exercise #2: Consider the two functions 5 7f x x and 22 1g x x .
(a) Calculate the average rate of change for both functions over the following intervals. Do your work carefully
and show the calculations that lead to your answers.
(i) 2 3x (ii) 1 5x
(b) The average rate of change for f was the same for both (i) and (ii) but was not the same for g. Why is that?
Exercise #3: The table below represents a linear function. Fill in the missing entries.
AVERAGE RATE OF CHANGE
For a function over the domain interval , the function's average rate of change is calculated by:
x 1 5 11 45
y -5 1 22
A l g e b r a 2 U n i t 2 - Linear Functions, Equations, and their Algebra
Unit 2 Lesson 1 – Average Rate of Change Page 50
LESSON 1: AVERAGE RATE OF CHANGE
HOMEWORK
1. For the function g x given in the table below, calculate the average rate of change for each of the following
intervals.
(a) 3 1x (b) 1 6x (c) 3 9x
(d) Explain how you can tell from the answers in (a) through (c) that this is not a table that represents a linear
function.
2. Consider the simple quadratic function 2f x x . Calculate the average rate of change of this function over
the following intervals:
(a) 0 2x (b) 2 4x (c) 4 6x
(d) Clearly the average rate of change is getting larger at x gets larger.
How is this reflected in the graph of f shown sketched to the right?
x 3 1 4 6 9
g x 8 2 13 12 5
A l g e b r a 2 U n i t 2 - Linear Functions, Equations, and their Algebra
Unit 2 Lesson 1 – Average Rate of Change Page 51
HOMEWORK (cont.)
3. Which has a greater average rate of change over the interval 2 4x the function 16 3g x x or the
function 22f x x . Provide justification for your answer.
4. An object travels such that its distance, d, away from its starting point is shown as a function of time, t, in
seconds, in the graph below.
(a) What is the average rate of change of d over the
interval 5 7t ? Include proper units in your
answer.
(b) The average rate of change of distance over time
(what you found in part (a)) is known as the
average speed of an object. Is the average speed
of this object greater on the interval 0 5t or
11 14t ? Justify.
5. What makes the average rate of change of a linear function different from that of any other function? What is
the special name that we give to the average rate of change of a linear function?
Time (seconds)
Dis
tan
ce (
feet)
A l g e b r a 2 U n i t 2 - Linear Functions, Equations, and their Algebra
Unit 2 Lesson 2 – Forms of a Line Page 52
LESSON 2: FORMS OF A LINE
Linear functions come in a variety of forms. The two shown below have been introduced in Common Core
Algebra I and Common Core Geometry.
Exercise #1: Consider the linear function 3 5f x x .
Exercise #2: Consider a line whose slope is 5 and which passes through the point 2, 8 .
Exercise #3: Which of the following represents an equation for the line that is parallel to 3
72
y x and which
passes through the point 6, 8 ?
(1) 2
8 63
y x (3) 3
8 62
y x
(2) 3
8 62
y x (4) 2
8 63
y x
TWO COMMON FORMS OF A LINE
Slope-Intercept: Point-Slope:
where m is the slope (or average rate of change) of the line and represents one point on the line.
(a) Determine the y-intercept of this function by
evaluating .
(b) Find its average rate of change over the interval
.
(a) Write the equation of this line in point-slope
form, .
(b) Write the equation of this line in slope-intercept
form
A l g e b r a 2 U n i t 2 - Linear Functions, Equations, and their Algebra
Unit 2 Lesson 2 – Forms of a Line Page 53
Exercise #4: A line passes through the points 5, 2 and 20, 4 .
Exercise #5: The graph of a linear function is shown below.
(a) Write the equation of this line in y mx b form.
(b) What must be the slope of a line perpendicular to the
one shown?
(c) Draw a line perpendicular to the one shown that
passes through the point 1, 3 .
(d) Write the equation of the line you just drew in point-
slope form.
(a) Determine the slope of this line in simplest
rational form.
(b) Write an equation of this line in point-slope
form.
(c) Write an equation for this line in slope-
intercept form.
(d) For what x-value will this line pass through a
y-value of 12?
y
x
(e) Does the line that you drew contain the point
30, 15 ? Justify.
A l g e b r a 2 U n i t 2 - Linear Functions, Equations, and their Algebra
Unit 2 Lesson 2 – Forms of a Line Page 54
LESSON 2: FORMS OF A LINE
HOMEWORK
1. Which of the following lines is perpendicular to 5
73
y x and has a y-intercept of 4?
(1) 5
43
y x (3) 3
45
y x
(2) 3
45
y x (4) 3
45
y x
2. Which of the following lines passes through the point 4, 8 ?
(1) 8 3 4y x (3) 8 3 4y x
(2) 8 3 4y x (4) 8 3 4y x
3. Which of the following equations could describe the graph of the linear function shown below?
(1) 2
43
y x (3) 2
43
y x
(2) 2
43
y x (4) 2
43
y x
4. For a line whose slope is 3 and which passes through the point 5, 2 :
5. For a line whose slope is 0.8 and which passes through the point 3,1 :
y
x
(a) Write the equation of this line in point-slope
form, 1 1y y m x x .
(b) Write the equation of this line in slope-
intercept form, . y mx b
(a) Write the equation of this line in point-slope
form, 1 1y y m x x .
(b) Write the equation of this line in slope-
intercept form, . y mx b
A l g e b r a 2 U n i t 2 - Linear Functions, Equations, and their Algebra
Unit 2 Lesson 2 – Forms of a Line Page 55
HOMEWORK (cont.)
6. The two points 3, 6 and 6, 0 are plotted on the grid below.
(a) Find an equation, in y mx b form, for the line passing
through these two points. Use of the grid is optional.
(b) Does the point 30, 16 lie on this line? Justify.
7. A linear function is graphed below along with the point 3,1 .
(a) Draw a line parallel to the one shown that passes through
the point 3,1 .
(b) Write an equation for the line you just drew in point-slope
form.
(c) Between what two consecutive integers
does the y-intercept of the line you drew
fall?
(d) Determine the exact value of the y-intercept
of the line you drew.
y
x
y
x
A l g e b r a 2 U n i t 2 - Linear Functions, Equations, and their Algebra
Unit 2 Lesson 3 – Inverse Functions Page 56
LESSON 3: INVERSE FUNCTIONS
The idea of inverses, or opposites, is very important in math. The word inverse is used in many different
contexts, including the additive inverse and multiplicative inverse of a number. The actions of certain
functions can be reversed. The rule of a function’s reversal can also be a function.
Exercise #1: Consider the two linear functions given by the formulas .
(a) Calculate . (b) Calculate . (c) Calculate .
(d) Calculate . (e) Without calculation, determine the value of .
The two functions seen in Exercise #1 are inverses because they literally “undo” one another. The general idea
of inverses, , is shown below in the mapping diagram.
Exercise #2: If the point lies on the graph of , then which of the following points must lie on
the graph of its inverse?
(1) (3)
(2) (4)
3 7 2 7
and 2 3
x xf x g x
5 and 11f g 7
0 and g2
f
1f g
5f g f g
and f x g x
3, 5 y f x
3, 5 5, 3
5, 31 1
,3 5
a b
Domain of f
Range of g Domain of g
Range of f
A l g e b r a 2 U n i t 2 - Linear Functions, Equations, and their Algebra
Unit 2 Lesson 3 – Inverse Functions Page 57
Inverse functions have their own special notation, as follows:
Exercise #3: The linear function is shown graphed below. Use its graph to answer the following
questions.
(a) Evaluate and .
(b) Determine the y-intercept of .
(c) On the same set of axes, draw a graph of .
(d) Write the equation of
Exercise #4: A table of values for the simple quadratic function is given below along with its graph.
(a) Graph the inverse by switching the ordered pairs.
(b) What do you notice about the graph of this function’s inverse?
LESSON 6: INVERSE FUNCTIONS
HOMEWORK
2
23
f x x
1 2f 1 4f
1f x
1y f x
)(1 xf
f x x( ) 2
INVERSE FUNCTION NOTATION
If a function has an inverse that is also a function we represent it as .
x -2 -1 0 1 2
f(x) 4 1 0 1 4
x
f x1( )
EXISTENCE OF INVERSE FUNCTIONS
A function will have an inverse that is also a function if and only if it is one-to-one. A quick way to know
if a function has an inverse that is also a function is to apply the Horizontal Line Test.
y
x
y
x
A l g e b r a 2 U n i t 2 - Linear Functions, Equations, and their Algebra
Unit 2 Lesson 3 – Inverse Functions Page 58
LESSON 3: INVERSE FUNCTIONS
HOMEWORK
1. If the point lies on the graph of , which of the following points must lie on the graph of its
inverse?
(1) (3)
(2) (4)
2. The function has an inverse function . If then which of the following must
be true?
(1) (3)
(2) (4)
3. The graph of the function is shown below. The value of is
(1) 2.5 (3) 0.4
(2) (4)
4. Which of the following functions would have an inverse that is also a function?
(1) (2) (3) (4)
5. For a one-to-one function it is known that and . Which of the following must be true
about the graph of this function’s inverse?
(1) its y-intercept = (3) its x-intercept =
(2) its y-intercept = 8 (4) its x-intercept =
7, 5 y f x
5, 7 7, 5
1 1,
7 5
1 1,
7 5
y f x 1y f x f a b
1f b a 1f b a
1 1 1f
a b
1f b a
y g x 1 2g
4 1
0 6f 8 0f
6 6
8
y
x
y
x
y
x
y
x
y
x
A l g e b r a 2 U n i t 2 - Linear Functions, Equations, and their Algebra
Unit 2 Lesson 3 – Inverse Functions Page 59
HOMEWORK (cont.)
6. The function is entirely defined by the graph shown:
(a) Sketch a graph of . Create a table of values if needed.
(b) Write the domain and range of
using interval notation.
7. The function is a one-to-one function that uses a circle’s radius as an input and gives the
circle’s area as its output. Selected values of this function are shown in the table below.
(c) The original function converted an input, the circle’s radius, to an output, the circle’s area.
What are the inputs and outputs of the inverse function?
Input: Output:
8. The domain and range of a one-to-one function, , are given below in set-builder notation. Give the
domain and range of this function’s inverse also in set-builder notation.
y h x
1y h x
1 and y h x y h x
2y A r r
y A r
y f x
Domain:
Range:
y h x
Domain:
Range:
1y h x
y
x
r 1 2 3 4 5 6
A r 4 9 16 25 36
(a) Determine the values of
from using the table.
1 19 and 36A A (b) Determine the values of
1 100 andA
1 225A
Domain: | 3 5
Range: | 2
y f x
x x
y y
1
Domain:
Range:
y f x
A l g e b r a 2 U n i t 2 - Linear Functions, Equations, and their Algebra
Unit 2 Lesson 4 – Inverses of Linear Functions Page 60
LESSON 4: INVERSES OF LINEAR FUNCTIONS
Recall that one-to-one functions have inverses that are also functions. Except for horizontal lines, all linear
functions are one-to-one and thus have inverses that are also functions. In this lesson we will investigate these
inverses and how to find their equations.
Exercise #1: On the grid below the linear function 2 4y x is graphed along with the line y x .
(a) How can you quickly tell that 2 4y x is a one-to-one
function?
(b) Graph the inverse of 2 4y x on the same grid. Recall that
this is easily done by switching the x and y coordinates of the
original line.
(c) What can be said about the graphs of 2 4y x and its inverse
with respect to the line y x ?
As we can see from part (e) in Exercise #1, inverses of linear functions include the inverse operations of the
original function but in reverse order. The simple method of finding the equation of any inverse is to simply
switch the x and y variables in the original equation and solve for y.
Exercise #2: Which of the following represents the equation of the inverse of 5 20y x ?
(1) 1
205
y x (3) 1
45
y x
(2) 1
205
y x (4) 1
45
y x
y
x
(d) Find the equation of the inverse in y mx b
form.
A l g e b r a 2 U n i t 2 - Linear Functions, Equations, and their Algebra
Unit 2 Lesson 4 – Inverses of Linear Functions Page 61
Although this is a simple procedure, common errors are often made when solving for y. Be careful with each
algebraic step.
Exercise #3: Which of the following represents the inverse of the linear function 2
83
y x ?
(1) 3
82
y x (3) 3
82
y x
(2) 3
122
y x (4) 3
122
y x
Exercise #4: What is the y-intercept of the inverse of 3
95
y x ?
(1) 15y (3) 9y
(2) 1
9y (4)
5
3y
Sometimes we are asked to work with linear functions in their point-slope form. The method of finding the
inverse and plotting it, though, do not change just because the linear equation is written in a different form.
Exercise #5: Which of the following would be an equation for the inverse of 6 4 2y x ?
(1) 1
2 64
y x (3) 6 4 2y x
(2) 1
2 64
y x (4) 2 4 6y x
Exercise #6: Which of the following points lies on the graph of the inverse of 8 5 2y x ? Explain your
choice.
(1) 8, 2 (3) 10, 40
(2) 8, 2 (4) 2, 8
Exercise #7: Which of the following linear functions would not have an inverse that is also a function? Explain
how you made your choice.
(1) y x (3) 2y
(2) 2y x (4) 5 1y x
A l g e b r a 2 U n i t 2 - Linear Functions, Equations, and their Algebra
Unit 2 Lesson 4 – Inverses of Linear Functions Page 62
LESSON 4: INVERSES OF LINEAR FUNCTIONS
HOMEWORK
1. The graph of a function and its inverse are always symmetric across which of the following lines?
(1) 0y (3) y x
(2) 0x (4) 1y
2. Which of the following represents the inverse of the linear function 3 24y x ?
(1) 1
83
y x (3) 1
243
y x
(2) 1
83
y x (4) 1 1
3 24y x
3. If the y-intercept of a linear function is 8, then we know which of the following about its inverse?
(1) Its y-intercept is 8 . (3) Its y-intercept is 1
8.
(2) Its x-intercept is 8. (4) Its x-intercept is 8 .
4. If both were plotted, which of the following linear functions would be parallel to its inverse?
Explain your thinking.
(1) 2y x (3) 5 1y x
(2) 2
43
y x (4) 6y x
5. Which of the following represents the equation of the inverse of 4
243
y x ?
(1) 4
243
y x (3) 3
184
y x
(2) 3
184
y x (4) 4
243
y x
6. Which of the following points lies on the inverse of 2 4 1y x ?
(1) 2, 1 (3) 1
,12
(2) 1, 2 (4) 2,1
A l g e b r a 2 U n i t 2 - Linear Functions, Equations, and their Algebra
Unit 2 Lesson 4 – Inverses of Linear Functions Page 63
HOMEWORK (cont.)
7. A linear function is graphed below. Answer the following questions based on this graph.
(a) Write the equation of this linear function in y mx b form.
(b) Sketch a graph of the inverse of this function on the same grid.
(c) Write the equation of the inverse in y mx b form.
(d) What is the intersection point of this line with its inverse?
8. A car traveling at a constant speed of 58 miles per hour has a distance of y-miles from Poughkeepsie, NY,
given by the equation 58 24y x , where x represents the time in hours that the car has been traveling.
(c) Give a physical interpretation of the answer you found in part (b). Consider what the input and output of
the inverse represent in order to answer this question.
9. Given the general linear function y mx b , find an equation for its inverse in terms of m and b.
(a) Find the equation of the inverse of this linear
function in y = mx = b form.
(b) Evaluate the function you found in part (a) for
an input of .
y
x
A l g e b r a 2 U n i t 2 - Linear Functions, Equations, and their Algebra
Unit 2 Lesson 5 –Systems of Linear Equations Page 64
LESSON 5: SYSTEMS OF LINEAR EQUATIONS
Systems of equations, or more than one equation, arise frequently in mathematics. To solve a system means
to find all values that simultaneously make all equations true. Of special importance are systems of linear
equations.
Exercise #1: Solve the following two-by-two (2x2) system of equations. Express answer in (x,y) form. Check.
3 2 9
2 7
x y
x y
In this lesson, we will extend to linear systems of 3 equations and 3 unknowns.
Steps to solving a system of 3 equations:
1. Label the equations A, B, and C
2. Choose a variable to eliminate (x, y or z).
3. Add/subtract 2 sets of 2 equations producing 2 equations: label them D and E.
4. Solve equations D and E as a system to get 1 variable answer.
5. Substitute into D or E to get 2nd variable answer.
6. Substitute into A, B or C to get 3rd.
7. Check in original equations.
Exercise #2: Consider the 3x3 system of linear equations shown below. Each equation is labeled A, B and C:
(A) 2x + y + z = 15
(B) 6x − 3y − z = 35
(C) −4x + 4y − z = −14
A l g e b r a 2 U n i t 2 - Linear Functions, Equations, and their Algebra
Unit 2 Lesson 5 –Systems of Linear Equations Page 65
Exercise #3: Solve the 3x3 system of equations shown below. Express answer in (x,y,z) form. Check answer.
4 3 6
2 4 2 38
5 7 19
x y z
x y z
x y z
Exercise #4: Solve the 3x3 system of equations shown below. Express answer in (x,y,z) form. Check answer.
4 2 3 23
5 3 37
2 4 27
x y z
x y z
x y z
(a) Which variable will be easiest to eliminate?
Why?
A l g e b r a 2 U n i t 2 - Linear Functions, Equations, and their Algebra
Unit 2 Lesson 5 –Systems of Linear Equations Page 66
LESSON 5: SYSTEMS OF LINEAR EQUATIONS
HOMEWORK 1. Show that 10, 4, and 7x y z (10,4,7) is a solution to the system below without solving the system.
2 25
4 5 1
2 8 32
x y z
x y z
x y z
2. Solve the following system of equations. Show all work. Express answer in (x,y,z) form. Check answer.
4 2 21
2 2 13
3 2 5 70
x y z
x y z
x y z
A l g e b r a 2 U n i t 2 - Linear Functions, Equations, and their Algebra
Unit 2 Lesson 5 –Systems of Linear Equations Page 67
HOMEWORK (cont.)
3. Solve the following system of equations. Show all work. Express answer in (x,y,z) form. Check answer.
2 5 35
3 4 31
3 2 2 23
x y z
x y z
x y z
4. Solve the following system of equations. Show all work. Express answer in (x,y,z) form. Check answer.
2 3 2 33
4 5 3 54
6 2 8 50
x y z
x y z
x y z
A l g e b r a 2 U n i t 3 - Exponential Functions
Unit 3 Lesson 1 – Exponential Properties Page 68
U n i t 3 - Exponential Functions LESSON 1: EXPONENTIAL PROPERTIES
Rules of Exponents and Exponents that are Not Positive– Let a and b be positive integers and x and y real numbers:
a. Multiplication Rule ba xx ___________ ex. a4 ∙ a2 = _____________
b. Division Rule b
a
x
x ___________ ex. _____________
c. Power Rule bax )( ___________ ex. (a4)2 = _____________
d. Power of a Product axyz)( = _____________ ex. (3abc)3 = ________________
*Note azyx )(
e. Power of a Quotient
a
y
x
= ____________ ex. = ____________
f. Zero Exponent 0x = ________ ex. 9,9990 = _______
g. Zero exponent w/coefficient a 0x = ________ ex 9,999x0 = ____________
h. Negative Exponent ax ___________ ex. 3-3 = ____________
i. Fraction raised to a negative exponent
a
y
x
= __________ ex. = ________
j. Fraction w/neg. exponent in numerator: _________ ex. _________
k. Fraction w/neg. exponent in denominator: _________ ex. ________
4
5
3
3
3
2
3
a
2
4
3
7
3x
b
x a
ax
b
3
7
x
A l g e b r a 2 U n i t 3 - Exponential Functions
Unit 3 Lesson 1 – Exponential Properties Page 69
Examples: Simplify
1. 32
34
)(y
yy ____________ 2.
4
5
)( x
x
__________ 3.
3
5)(
xy
xy ______________
4. (x2y)k _____________ 5.
3
2
2
1
x ___________ 6. 2x ∙24 ______________
7. 2
1
xy
yx ____________ 8.
1
2
6
2xy
yx __________ 9. (4x2y-4)(4xy-3) _______________
A l g e b r a 2 U n i t 3 - Exponential Functions
Unit 3 Lesson 1 – Exponential Properties Page 70
LESSON 1: EXPONENTIAL PROPERTIES
HOMEWORK
1. Express each of the following expressions in "expanded" form, i.e., do all of the multiplication and/or
division possible and combine as many exponents as possible.
(a) 3 12x x (b) 3 54 5x x (c) 2 7 33 5x y x y (d) 3 6 44 7x y x
(e) 9
3
x
x (f)
3 7
2
5
15
x y
xy (g)
3
10
x
x (h)
4 3
8
10
25
x y
x
(i) 8
5x (j) 0
310x (k) 3
54x (l) 4
2x
(p)
(u) (s) (t)
(r)
(m) (n) (o)
(q)
A l g e b r a 2 U n i t 3 - Exponential Functions
Unit 3 Lesson 1 – Exponential Properties Page 71
HOMEWORK (cont.)
2. 3. 4.
5. 6. 7.
8. 9. 10.
A l g e b r a 2 U n i t 3 - Exponential Functions
Unit 3 Lesson 2 – Exponential Properties Practice Page 72
LESSON 2: Exponential Properties Practice Multiplication:
1. __________________ 2. ___________________
3. __________________ 4. xxx 5237 ___________________
More examples: Write the expression without a denominator.
1. 4
24
a
ba= 2.
4
24a
ba =
Write the expression using only positive exponents.
3. 3
33
y
x
= 4. 36
625
xy
zyx =
5. 733
72
2
4
cba
cba = 6.
32
55
15
30
yx
yx
=
Describe the mistake:
7. (-2)²(-2)³ = 45 8. 4
2
8
xx
x 9. 1234 xxx
y
x
xy
yx
12
42
3
y
y
y 6
3
12
8 4
3
x
yx
x
y 614
3
10 20
2
A l g e b r a 2 U n i t 3 - Exponential Functions
Unit 3 Lesson 2 – Exponential Properties Practice Page 73
Evaluate each function for the given value. Don’t use a calculator.
1. f(x) = 3x 1x f(2) = ______________
2. f(x) = 2
x
3
2 + 4
x
3
2; f(1) = _____________
3. f(x) = (3 2x + 6 2x )3 ; f(–3) = _____________
4. f(x) = 1
202
x
xx ; f(2) = _______________
Simplify:
5.
3
71
124
2
4
yx
zyx _____________ 6.
212
25
5
cba
A l g e b r a 2 U n i t 3 - Exponential Functions
Unit 3 Lesson 2 – Exponential Properties Practice Page 74
LESSON 3: EXPONENTIAL PROPERTIES PRACTICE
HOMEWORK
7. If 0 35 4f x x x then f a
(1) 12 5a (3) 3
15
4a
(2) 3
45
a (4) 12 1a
8. Which of the following is equivalent to
38
25
4
6
x
x for all 0x ? Show the manipulations that
lead to your final answer.
(1) 1416
9x (3) 142
3x
(2) 416
9x (4) 42
3x
1. 2. 3.
4. 5. 6.
A l g e b r a 2 U n i t 3 - Exponential Functions
Unit 3 Lesson 2 – Exponential Properties Practice Page 75
HOMEWORK (cont.)
In 39-50, write each expression with only positive exponents. Express the answer in simplest form.
In 67-72 write each quotient as a product without a denominator.
A l g e b r a 2 U n i t 3 - Exponential Functions
Unit 3 Lesson 3 – Rational Exponents Page 76
LESSON 3: RATIONAL Exponents
Today we will introduce rational ( ___________________ ) exponents and extend your exponential knowledge that much further.
If n is a positive integer then n
m
x = mn x or n mx
𝑥12
= 𝑥13
= 𝑥14
= 𝑥23 =
𝑥32
= 𝑦53 = 𝑦
25
= 𝑎34 =
To evaluate an expression containing a fractional exponent: 1. Rewrite the expression in radical form. 2. Evaluate the root. 3. Evaluate the power. Practice: Evaluate and/or simplify without a calculator.
1. 25 2
1
= 2. (27x) 3
1
=
3. (16x) 2
1
= 4. 125 3
1
=
5. (-125) 3
1
= 6. 81 4
1
=
7. ( 27x) 3
2
= 8. (16x) 4
3
=
9. (–8) 3
2
= 10. (27x) 3
2
=
A l g e b r a 2 U n i t 3 - Exponential Functions
Unit 3 Lesson 3 – Rational Exponents Page 77
11. 3
2
27
8
= 12.
3
2
27
8
13. (x² y³) 3
2
14.
Rewrite using rational exponents. Then simplify your answer.
64ab
3 12627 zyx √√𝑥4
Evaluate each function for the given value.
1. f(x) = 2
3
x + 2
1
x f(9) = ______________
2. f(x) = 4x 3
2
– x 3
4
; f(8) = _____________
Note: If n (the index of the radical) is an even number, the base, x, cannot be negative. However, if n is an odd number, the base, x, can be negative. That’s because if n is even, an imaginary number would result.
For example: 3
1
)8( = 3 8 = –2 (a real number)
However, 2
1
)16( = 2 16 = 4i (an imaginary number)
4 6x
3
2
1
4
yx
A l g e b r a 2 U n i t 3 - Exponential Functions
Unit 3 Lesson 3 – Rational Exponents Page 78
LESSON 3: RATIONAL EXPONENTS
HOMEWORK
1. Rewrite the following as equivalent roots and then evaluate as many as possible without your calculator.
(a) 1
236 (b) 1
327 (c) 1
532 (d) 1
2100
(e) 1
4625 (f) 1
249 (g) 1
481
(h) 1
3343
2. Evaluate each of the following by considering the root and power indicated by the exponent. Do as many
as possible without your calculator.
(a) 2
38 (b) 3
24 (c) 3
416
(d) 5
481
(e) 5
24
(f) 3
7128 (g) 3
4625 (h) 3
5243
3. Given the function 3
25 4f x x , which of the following represents its y-intercept?
(1) 40 (3) 4
(2) 20 (4) 30
A l g e b r a 2 U n i t 3 - Exponential Functions
Unit 3 Lesson 3 – Rational Exponents Page 79
HOMEWORK (cont.)
4. Which of the following is equivalent to 1
2x
?
(1) 1
2x (3)
1
x
(2) x (4) 1
2x
5. Written without fractional or negative exponents, 3
2x
is equal to
(1) 3
2
x (3)
3
1
x
(2) 3 2
1
x (4)
1
x
6. Which of the following is not equivalent to 3
216 ?
(1) 4096 (3) 64
(2) 38 (4) 316
7. Marlene claims that the square root of a cube root is a sixth root? Is she correct? To start, try rewriting
the expression below in terms of fractional exponents. Then apply the Product Property of Exponents.
3 a
8. f(x) = (3x) 3
1
+ 2x 0 ; f(9) = _____________
A l g e b r a 2 U n i t 3 - Exponential Functions
Unit 3 Lesson 4 – More practice with Rational Exponents Page 80
LESSON 4: MORE PRACTICE WITH RATIONAL Exponents
All rules of exponents apply to fractional (rational) exponents. These rules justify many standard manipulations with square roots (and other radicals). For example, simplifying roots:
It’s easy to simplify these: 29x _____________ 22249 zyx _____________
However, if the radicand is not a perfect square, there is another step to take. We only consider a square root "simplified" when all of its perfect square factors have had their square roots evaluated.
To Simplify a Radical: 1. Express the radicand as the product of factors - a) the largest perfect (square or cube or nth) factors, followed by: b) the non-perfect (square or cube or nth) factors 2. Find the root of the “perfect” factors from (a). This is the coefficient of the radical answer. The (b) part is the radicand of the radical answer. Perfect Squares - 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, ... 400, ... Perfect Cubes - 8, 27, 64, 125, 216, 343, 512, 729, 1000, … Other powers to know - 24= 34= 25= 26= Simplify each of the following square roots. Show the manipulations that lead to your answers.
1. 50 = 2. 700 = 3. 472xy =
4. 532x = 5. 418x = 6. 5 3200x y =
7. 9 4147x y = 8. 1016128 yx = 7. 1001275 ba
A l g e b r a 2 U n i t 3 - Exponential Functions
Unit 3 Lesson 4 – More practice with Rational Exponents Page 81
We can extend the simplifying process to include cube roots and higher-order roots. 1. Simplify each of the following higher order roots:
(a) 3 16 (b)
3 108 (c) 3 250 (d)
3 8128x
(e) 4 162 (f) 4 816x (g) 10 54 48x y (h) 12 155 64x y
2. Simplify:
(a) 72 18x x (b) 2 5 83 98x y x y
(c) 12 143 64x y (d) 7 113 375x y
A l g e b r a 2 U n i t 3 - Exponential Functions
Unit 3 Lesson 4 – More practice with Rational Exponents Page 82
LESSON 4: MORE PRACTICE WITH RATIONAL EXPONENTS
HOMEWORK
1. Which of the following is not equivalent to 9x ?
(1) 3x (2)
92x (3)
9
x (4) 4x x
2. The radical expression 5 350x y can be rewritten equivalently as
(1) 25 2xy xy (2) 25 2x y xy (3) 5xy xy (4) 210 5x y xy
3. If the function 312y x was placed in the form by ax then which of the following is the value
of a b ?
(1) 36 (2) 36 (3) 4 (4) 4
4. Rewrite each of the following expressions without roots by using fractional exponents.
(a) x (b) 3 x (c) 7 x (d) 5x
(e) 3 11x (f) 4
1
x (g)
3 2
1
x (h)
9
1
x
5. Rewrite each of the following without the use of fractional or negative exponents, using radicals.
(a) 1
6x (b) 110x (c)
13x
(d)
15x
A l g e b r a 2 U n i t 3 - Exponential Functions
Unit 3 Lesson 4 – More practice with Rational Exponents Page 83
HOMEWORK (cont.)
6. Simplify each of the following square roots that contain variables in the radicand.
(a) 98x (b) 16 1175x y
7. Express each of the following roots in simplest radical form.
(a) 3 816x (b) 5 103 108x y
8. Mikayla was trying to rewrite the expression 1
225x in an equivalent form that is more
convenient to use. She incorrectly rewrote it as 5 x . Explain Mikalya's error.
9. If the expression 1
x was placed in
ax form, then which of the following would be the value of a?
(1) 2 (2) 1
2 (3) 2 (4)
1
2
A l g e b r a 2 U n i t 3 - Exponential Functions
Unit 3 Lesson 5 – More Exponent practice Page 84
LESSON 5: MORE EXPONENT PRACTICE
It is important to be able to manipulate expressions involving exponents, whether those
exponents are positive, negative, or fractional. The basic laws of exponents are shown below.
They apply regardless of the nature of the exponent (i.e. positive, negative, or fractional).
Although these problems can be challenging, the key will be to carefully apply these exponent
laws systematically.
Example 1: Simplify each of the following expressions. Leave no negative exponents in your answers.
(a)
3 4
25
x x
x
(b)
4
2
5 7
x y
x y (c)
2 4
6
x y
x y
(d)
23 4
43
x y
xy
In the last example, all powers were integers. The next example introduces fractional powers.
Remember that they still follow the exponent rules above. If needed, use your calculator to help
add and subtract the powers.
Example 2: Simplify each of the following expressions. Write each without the use of negative exponents.
(a)
1 13 2
16
x x
x
(b)
51
2
332
x
x x (c)
3
23
8
4
32
x
x
EXPONENT LAWS
1. 𝑥𝑎 • 𝑥𝑏 = 5. 𝑥−𝑎 = 𝑎𝑛𝑑 1
𝑥−𝑎=
2. 𝑥𝑎
𝑥𝑏 = 6. (𝑥 • 𝑦)𝑎 =
3. (𝑥𝑎)𝑏 = 7. 𝑥0 =
4. 𝑥𝑚
𝑛Τ = 8. 𝑎𝑛
𝑏𝑛 =
(For integers m and n)
A l g e b r a 2 U n i t 3 - Exponential Functions
Unit 3 Lesson 5 – More Exponent practice Page 85
Don’t forget that fractional exponents have an equivalent interpretation as radicals. You should
be able to move from one representation to another.
Example 3: Rewrite each expression below in both its simplest form and using radical expressions.
(a) 5
3x (b)
52
43
x
x (c)
32
1
x
(d) 3x
x (e)
25 38x (f)
1
327
6
x
x
Example 4: Which of the following is equivalent to 3 78x ?
(1) 7
38x (2) 3
72x (3) 7
32x (4) 3
78x
Example 5: The expression 1
4x is the same as
(1) 1
21
2x
(2) 1
24x (3) 1
22x
(4) 1
21
2x
A l g e b r a 2 U n i t 3 - Exponential Functions
Unit 3 Lesson 5 – More Exponent practice Page 86
LESSON 5: MORE EXPONENT PRACTICE
HOMEWORK
1. Rewrite each of the following expressions in simplest form and without negative exponents.
(a)
3 7
32
x x
x (b)
4
10
5
25
x
x (c)
23 4
33
x y
x y (d)
5
3
3
2
8
x
x
2. Which of the following represents the value of 4
2
a
b
when 3 and 2a b ?
(1) 4
9 (2)
1
36 (3)
4
81 (4)
1
3
3. Simplify each expression below so that it contains no negative exponents. Do not write the expressions
using radicals.
(a)
7 12 2
324
x y
x y (b)
4
13
23
x
x
(c) 2 1
2 33 25 2x y x y
4. Which of the following represents the expression
12
52
24
6
x
x
written in simplest form?
(1) 3
4
x (2)
2
4
x (3)
34x (4) 24x
A l g e b r a 2 U n i t 3 - Exponential Functions
Unit 3 Lesson 5 – More Exponent practice Page 87
HOMEWORK, CONT.
5. Rewrite each of the following expressions using radicals. Express your answers in simplest form.
(a) 3
24x (b) 2
3x
(c) 3
4 5x
(d) 3 x
x (e)
2
53
x x
x
(f)
3
2
24
x
x
6. Which of the following is equivalent to 3
5
20
x
x?
(1) 3
1
4 x (2)
5 2
1
4 x (3)
5
4
x (4)
5
1
4 x
7. When written in terms of a fractional exponent the expression 2
x x
x
is
(1) 7
2x (2) 1
2x
(3) 5
2x (4) 3
2x
8. Expressed as a radical expression, the fraction
1 13 2
1
x x
x is
(1) 6
1
x (2)
11 6x (3) 11 6
1
x (4)
6 11x
A l g e b r a 2 U n i t 3 - Exponential Functions
Unit 3 Lesson 6 –Exponential Function Basics Page 88
LESSON 6: EXPONENTIAL FUNCTION BASICS
This lesson reviews many of the basic components of exponential graphs and behavior. Exponential functions, those whose exponents are variable, are extremely important in math, science, and engineering.
Exercise #1: Consider the function 2xy . Fill in the table below without using your calculator and then
sketch the graph on the grid provided.
Exercise #2: Now consider the function 12
x
y . Using your calculator to help you, fill out the table
below and sketch the graph on the axes provided.
BASIC EXPONENTIAL FUNCTIONS
where
x 2xy
3
2
1
0
1
2
3
y
x
x 12
x
y
3
2
1
0
1
2
3
y
x
A l g e b r a 2 U n i t 3 - Exponential Functions
Unit 3 Lesson 6 –Exponential Function Basics Page 89
Exercise #3: Based on the graphs and behavior you saw in Exercises #1 and #2, state the domain and range
for an exponential function of the form xy b .
Domain (input set): Range (output set):
Exercise #4: Are exponential functions one-to-one? How can you tell? What does this tell you about their
inverses?
Exercise #5: Now consider the function 7 3x
y .
(a) Determine the y-intercept of this function
algebraically. Justify your answer.
(b) Does the exponential function increase or decrease?
Explain your choice.
(c) Create a rough sketch of this function, labeling its y-
intercept.
Exercise #6: Consider the function 1 43
x
y .
(a) How does this function’s graph compare to that of
13
x
y ? What does adding 4 do to a function's
graph?
(b) Determine this graph’s y-intercept algebraically.
Justify your answer.
(c) Create a rough sketch of this function, labeling its y-
intercept.
y
x
y
x
A l g e b r a 2 U n i t 3 - Exponential Functions
Unit 3 Lesson 6 –Exponential Function Basics Page 90
LESSON 6: EXPONENTIAL FUNCTION BASICS
HOMEWORK
1. Which of the following represents an exponential function?
(1) 3 7y x (3) 3 7x
y
(2) 37y x (4)
23 7y x
2. If 6 9x
f x then 12
f ? (Remember what we just learned about fractional exponents and do
withou a calculator.)
(1) 7
2 (3) 27
(2) 18 (4) 15
2
3. If 3xh x and 5 7g x x then 2h g
(1) 18 (3) 38
(2) 12 (4) 27
4. Which of the following equations could describe the graph shown below?
(1) 2 1y x (3) 2 1y x
(2) 23
x
y (4) 4xy
5. Which of the following equations represents the graph shown?
(1) 5xy (3) 1 22
x
y
(2) 4 1xy (4) 3 2xy
y
x
1 2 3 -1 -2 -3
5
10
A l g e b r a 2 U n i t 3 - Exponential Functions
Unit 3 Lesson 6 –Exponential Function Basics Page 91
HOMEWORK (cont.)
6. Sketch graphs of the equations shown below on the axes given. Label the y-intercepts of each graph.
(a) 1183
x
y (b) 25 4x
y
7. The Fahrenheit temperature of a cup of coffee, F, starts at a temperature of 185 F . It cools down according
to the exponential function 201
113 722
m
F m
, where m is the number minutes it has been cooling.
8. The graph below shows two exponential functions, with real number constants a, b, c, and d. Given the
graphs, only one pair of the constants shown below could be equal in value. Determine which pair could
be equal and explain your reasoning.
b and d a and b a and c
9. Explain why the equation below can have no real solutions. If you need to, graph both sides of the equation
using your calculator to visualize the reason.
3 5 2x
(a) How do you interpret the statement that
60 86F ?
(b) Determine the temperature of the coffee after
one day using your calculator. What do you
think this temperature represents about the
physical situation?
y
x
y
x
x
y a b
x
y c d
A l g e b r a 2 U n i t 3 - Exponential Functions
Unit 3 Lesson 7 –Finding Equations of Exponential Functions Page 92
LESSON 7: FINDING EQUATIONS OF EXPONENTIAL FUNCTIONS
In this lesson, you will learn how to write equations of exponential functions when you have information about the starting value and base (multiplier or growth constant). Let's review a basic problem.
Exercise #1: An exponential function of the form x
f x a b is presented in the table below. Determine
the values of a and b and explain your reasoning.
a _________
b _________
Final Equation: _____________________ Explanation:
Finding an exponential equation becomes more challenging if we do not have output values for inputs that are increasing by 1 unit at a time. Just like with lines, any two points will determine the equation of an exponential function. Steps: 1. Write 2 equations 2. Divide the 2 equations 3. Solve for b 4. Solve for a
Exercise #2: An exponential function of the form x
y a b passes through 2, 36 and 5,121.5
Exercise #3: For an exponential function of the form x
f x a b , it is known that 0 8f and
3 1000f . Find the values of a and b, and write the equation.
x 0 1 2 3
f x 5 15 45 135
(1) By substituting these two points into the general
form of the exponential, create a system of
equations in the constants a and b.
(2) Divide these two equations to eliminate the
constant a. Recall that when dividing two like
bases, you subtract their exponents.
(3) Solve the resulting equation from (2) for the
base, b. (4) Use your value from (3) to determine the value
of a. State the final equation.
A l g e b r a 2 U n i t 3 - Exponential Functions
Unit 3 Lesson 7 –Finding Equations of Exponential Functions Page 93
Steps: 1. Write 2 equations 2. Divide the 2 equations 3. Solve for b 4. Solve for a
Exercise #4: An exponential function exists such that 4 3 and 6 48f f , which of the following must
be the value of its base? Explain or illustrate your thinking.
(1) 16b (3) 6b
(2) 2b (4) 4b
Now let's practice this with a decreasing exponential function.
Exercise #5: Find the equation of the exponential function shown graphed below. Be careful in terms of your
exponent manipulation. State your final answer in the form x
y a b .
Exercise #6: A bacterial colony is growing at an exponential rate. It is known that after 4 hours, its population
is at 98 bacteria and after 9 hours it is 189 bacteria. Determine an equation in x
y a b form that models the
population, y, as a function of the number of hours, x. (Round to the nearest hundredths.)
At what percent rate is the population growing per hour?
y
x
A l g e b r a 2 U n i t 3 - Exponential Functions
Unit 3 Lesson 7 –Finding Equations of Exponential Functions Page 94
LESSON 7: FINDING EQUATIONS OF EXPONENTIAL FUNCTIONS
HOMEWORK
1. For each of the following coordinate pairs, find the equation of the exponential function, in the form
x
y a b that passes through the pair. Show the work that you use to arrive at your answer.
(a) 0,10 and 3, 80 (b) 0,180 and 2, 80
2. For each of the following coordinate pairs, find the equation of the exponential function, in the form
x
y a b that passes through the pair. Show the work that you use to arrive at your answer.
(a) 2,192 and 5,12288 (b) 1,192 and 5, 60.75
3. Each of the previous problems had values of a and b that were rational numbers. They do not need not
be. Find the equation for an exponential function that passes through the points 2,14 and 7, 205 in
x
y a b form. When you find the value of b do not round your answer before you find a. Then, find
both to the nearest hundredth and give the final equation. Check to see if the points fall on the curve.
A l g e b r a 2 U n i t 3 - Exponential Functions
Unit 3 Lesson 7 –Finding Equations of Exponential Functions Page 95
HOMEWORK (cont.)
4. A population of koi goldfish in a pond was measured over time. In the year 2002, the population was
recorded as 380 and in 2006 it was 517. Given that y is the population of fish and x is the number of
years since 2000, do the following:
5. Engineers are draining a water reservoir until its depth is only 10 feet. The depth decreases exponentially
as shown in the graph below. The engineers measure the depth after 1 hour to be 64 feet and after 4 hours
to be 28 feet. Develop an exponential equation in x
y a b to predict the depth as a function of hours
draining. Round a to the nearest integer and b to the nearest hundredth. Then, graph the horizontal line
10y and find its intersection to determine the time, to the nearest tenth of an hour, when the reservoir
will reach a depth of 10 feet.
(a) Represent the information in this problem as
two coordinate points.
(b) Determine a linear function in the form
y mx b that passes through these two
points. Don't round the linear parameters (m
and b).
(c) Determine an exponential function of the form
x
y a b that passes through these two points.
Round b to the nearest hundredth and a to the
nearest tenth.
(d) Which model predicts a larger population of
fish in the year 2000? Justify your work.
Wa
ter D
epth
(ft
)
Time (hrs)
A l g e b r a 2 U n i t 3 - Exponential Functions
Unit 3 Lesson 8 – The method of common bases Page 96
LESSON 8: THE METHOD OF COMMON BASES
In this lesson we will look at solving exponential equations using a method known as The Method of
Common Bases. If bx = by then ____________
Exercise #1: Solve each of the following simple exponential equations by writing each side of the equation
using a common base.
(a) 2 16x (b) 3 27x (c) 1
525
x (d) 16 4x
In each of these cases, even the last, more challenging one, we could manipulate the right-hand side of the equation so that it shared a common base with the left-hand side of the equation. We can exploit this fact by manipulating both sides so that they have a common base. First, though, we need to review an exponent law.
Exercise #2: Simplify each of the following exponential expressions.
(a) 32x
(b) 4
23x
(c) 3 7
15x
(d) 21
34x
Exercise #3: Solve each of the following equations by finding a common base for each side.
(a) 8 32x (b) 2 19 27x (c) 4
112525
xx
Exercise #4: Which of the following represents the solution set to the equation 2 32 64x ?
(1) 3 (3) 11
(2) 0, 3 (4) 35
A l g e b r a 2 U n i t 3 - Exponential Functions
Unit 3 Lesson 8 – The method of common bases Page 97
This technique can be used in any situation where all bases involved can be written with a common base. In a practical sense, this is rather rare. Yet, these types of algebraic manipulations help us see the structure in exponential expressions. Try to tackle the next, more challenging, problem.
Exercise #5: Two exponential curves,
5
24x
y
and
2 11
2
x
y
are shown below. They intersect at point
A. A rectangle has one vertex at the origin and the other at A as shown. We want to find its area.
(a) Fundamentally, what do we need to know
about a rectangle to find its area?
(b) How would knowing the coordinates of
point A help us find the area?
(c) Find the area of the rectangle algebraically using the Method of Common Bases. Show your work
carefully.
Exercise #6: At what x coordinate will the graph of 25x ay intersect the graph of
3 11
125
x
y
? Show
the work that leads to your choice.
(1) 5 1
3
ax
(3)
2 1
5
ax
(2) 2 3
11
ax
(4)
5 3
2
ax
y
x
A
A l g e b r a 2 U n i t 3 - Exponential Functions
Unit 3 Lesson 8 – The method of common bases Page 98
LESSON 8: THE METHOD OF COMMON BASES
HOMEWORK
1. Solve each of the following exponential equations using the Method of Common Bases. Check your
answers.
(a) 2 53 9x (b) 3 72 16x (c) 4 5 15125
x
(d) 2 18 4x x (e) 3 2
2 12161296
xx
(f)
315 151 3125
25
x x
2. Algebraically determine the intersection point of the two exponential functions shown below.
1 2 38 and 4x xy y
3. Algebraically determine the zeroes of the exponential function 2 92 32xf x . Recall that the reason
it is known as a zero is because the output is zero.
A l g e b r a 2 U n i t 3 - Exponential Functions
Unit 3 Lesson 8 – The method of common bases Page 99
HOMEWORK (cont.)
4. One hundred must be raised to what power in order to be equal to a million cubed? Solve this problem
using the Method of Common Bases. Show the algebra you do to find your solution.
5. The exponential function
2
5110
25
x
y
is shown graphed along with the horizontal line 115y . Their
intersection point is ,115a . Use the Method of Common Bases to find the value of a. Show your work.
6. The Method of Common Bases works because exponential functions are one-to-one, i.e. if the outputs are
the same, then the inputs must also be the same. This is what allows us to say that if 32 2x , then x must
be equal to 3. But it doesn't always work out so easily.
If 2 25x , can we say that x must be 5? Could it be anything else? Why does this not work out as easily
as the exponential case?
y
x
2
5110
25
x
y
115y , 115a
A l g e b r a 2 U n i t 3 - Exponential Functions
Unit 3 Lesson 9 – Exponential Modeling with % Growth and Decay Page 100
LESSON 9: EXPONENTIAL MODELING WITH PERCENT
RATE GROWTH AND DECAY Exponential functions are very important in modeling a variety of real world phenomena because certain things either increase or decrease by fixed percentages or rates over given units of time.
Exercise #1: Suppose that you deposit money into a savings account that receives 5% interest per year on the
amount of money that is in the account for that year. Assume that you deposit $400 into the account initially.
The thinking process from Exercise #1 can be generalized to any situation where a quantity is increased or
decreased by a fixed percentage over a fixed interval of time. This pattern is summarized as follows
Exercise #2: Which of the following gives the savings S in an account if $250 was invested at an interest rate
of 3% per year?
(1) 250 4t
S (3) 1.03 250t
S
(2) 250 1.03t
S (4) 250 1.3t
S
EXPONENTIAL MODELS
Some real-life quantities increase or decrease by a fixed percent, r, in decimal form. The amount A of
such a quantity after t time periods (e.g. years, minutes, etc.) can be modeled by
Exponential Growth Exponential Decay
𝐴 = 𝑎(1 + 𝑟)𝑡 𝐴 = 𝑎(1 − 𝑟)𝑡
where a represents the initial amount (amount at ) and t represents time.
(a) How much will the savings account increase by
over the course of the year?
(b) How much money is in the account at the end of
the year?
(c) By what single number could you have
multiplied the $400 by in order to calculate your
answer in part (b)?
(d) Using your answer from part (c), determine the
amount of money in the account after 2 and 10
years. Round all answers to the nearest cent
when needed.
(e) Give an equation for the amount in the savings
account as a function of the number of years
since the $400 was invested.
(f) Using a table on your calculator determine, to the
nearest year, how long it will take for the initial
investment of $400 to double. Provide evidence
to support your answer.
A l g e b r a 2 U n i t 3 - Exponential Functions
Unit 3 Lesson 9 – Exponential Modeling with % Growth and Decay Page 101
Exercise #3 In 2000 the world population was about 6.09 billion. During the next several years, the world
population increased by about 1.18% each year.
a. Write an exponential growth model giving the population A (in billions) t years after 2000.
Then, estimate the world population in 2005.
b. Estimate the year when the world population was 7 billion.
Exercise #4: If the population of a town is decreasing by 4% per year and started with 12,500 residents, which
of the following is its projected population in 10 years? Show the exponential model you use to solve this
problem.
(1) 9,230 (3) 18,503
(2) 76 (4) 8,310
Exercise #5: The stock price of WindpowerInc is increasing at a rate of 4% per week. Its initial value was
$20 per share. On the other hand, the stock price in GerbilEnergy is crashing (losing value) at a rate of 11%
per week. If its price was $120 per share when Windpower was at $20, after how many weeks will the stock
prices be the same? Model both stock prices using exponential functions. Then, graphically find when the
stock prices will be equal. Draw a well labeled graph to justify your solution.
Exercise #6: State the multiplier (base) you would need to multiply by in order to decrease a quantity by the
given percent listed.
(a) 10% (b) 2% (c) 25% (d) 0.5%
A l g e b r a 2 U n i t 3 - Exponential Functions
Unit 3 Lesson 9 – Exponential Modeling with % Growth and Decay Page 102
LESSON 9: EXPONENTIAL MODELING WITH PERCENT GROWTH AND DECAY
HOMEWORK
1. If $130 is invested in a savings account that earns 4% interest per year, which of the following is closest
to the amount in the account at the end of 10 years?
(1) $218 (3) $168
(2) $192 (4) $324
2. A population of 50 fruit flies is increasing at a rate of 6% per day. Which of the following is closest to the
number of days it will take for the fruit fly population to double?
(1) 18 (3) 12
(2) 6 (4) 28
3. If a radioactive substance is quickly decaying at a rate of 13% per hour approximately how much of a 200
pound sample remains after one day?
(1) 7.1 pounds (3) 25.6 pounds
(2) 2.3 pounds (4) 15.6 pounds
4. A population of llamas stranded on a desert island is decreasing due to a food shortage by 6% per year. If
the population of llamas started out at 350, how many are left on the island 10 years later?
(1) 257 (3) 102
(2) 58 (4) 189
5. Which of the following equations would model a population with an initial size of 625 that is growing at
an annual rate of 8.5%?
(1) 625 8.5t
P (3) 1.085 625tP
(2) 625 1.085t
P (4) 28.5 625P t
6. The acceleration of an object falling through the air will decrease at a rate of 15% per second due to air
resistance. If the initial acceleration due to gravity is 9.8 meters per second, which of the following
equations best models the acceleration t seconds after the object begins falling?
(1) 215 9.8a t (3) 9.8 1.15t
a
(2) 9.8
15a
t (4) 9.8 0.85
ta
A l g e b r a 2 U n i t 3 - Exponential Functions
Unit 3 Lesson 9 – Exponential Modeling with % Growth and Decay Page 103
HOMEWORK (cont.)
7. Red Hook has a population of 6,200 people and is growing at a rate of 8% per year. Rhinebeck has a
population of 8,750 and is growing at a rate of 6% per year. In how many years, to the nearest year, will
Red Hook have a greater population than Rhinebeck? Show the equation or inequality you are solving and
solve it graphically.
8. A warm glass of water, initially at 120 degrees Fahrenheit, is placed in a refrigerator at 34 degrees
Fahrenheit and its temperature is seen to decrease according to the exponential function
86 0.83 34h
T h
9. Percents combine in strange ways that don't seem to make sense at first. It would seem that if a population
grows by 5% per year for 10 years, then it should grow in total by 50% over a decade. But this isn't true.
Start with a population of 100. If it grows at 5% per year for 10 years, what is its population after 10 years?
What percent growth does this represent?
(a) Verify that the temperature starts at 120 degrees
Fahrenheit by evaluating 0T .
(b) Using your calculator, sketch a graph of T
below for all values of h on the interval
0 24h . Be sure to label your y-axis and y-
intercept.
(c) After how many hours will the temperature be
at 50 degrees Fahrenheit? State your answer to
the nearest hundredth of an hour. Illustrate
your answer on the graph your drew in (b).
A l g e b r a 2 U n i t 3 - Exponential Functions
Unit 3 Lesson 10 – r-values for multiple or fractional time periods Page 104
LESSON 10: r-VALUES FOR MULTIPLE OR FRACTIONAL TIME PERIODS
Given the exponential growth formula traA )1( , when t = 1 (e.g. 1 year), solve for r.
This is an annual growth rate. (Multiply by 100 if you want the annual percent growth rate.)
Finding the annual rate of growth (find r when t=1)
Exercise #1: A population of wombats is growing at a constant percent rate. If the population on January 1st
is 1027 and a year later is 1079, what is its annual percent growth rate to the nearest tenth of a percent?
Finding the percent growth over multiple years (when t>1)
Exercise #2: Now let's determine the percent growth in wombat population over a decade. Assume the
rounded annual percent increase found in Exercise #1 continues for the next decade.
Finding the growth rate for a fraction of the year (when 0<t<1, in exercises 3 and 4)
Exercise #3: Use the wombats from Exercise #1. Assuming their annual growth rate is constant, what is the
monthly growth rate to the nearest tenth of a percent? Assume a constant sized month.
Exercise #4: If a population grows at a constant rate of 22% every 5 years, what is its percent growth rate
over a 2 year time span? Round to the nearest tenth of a percent.
(a) After 10 years, what will the original
population be multiplied by, rounded to the
nearest hundredth? Show the calculation.
(b) Using your answer from (a), what is the decade
percent growth rate?
(a) First, give an expression that will calculate the
yearly percent growth rate based on the fact that
the population grew 22% in 5 years.
(b) Now use this expression to calculate the percent
growth over 2 years.
A l g e b r a 2 U n i t 3 - Exponential Functions
Unit 3 Lesson 10 – r-values for multiple or fractional time periods Page 105
Exercise #5: World oil reserves (the amount of oil unused in the ground) are depleting at a constant 2% per
year. Determine the percent decline over the next 20 years based on this 2% yearly decline.
Exercise #6: The population of squirrels in Ulster County is growing at an annual rate of 1.8%. Find the
percent rate of growth (a) every 2 years (b) every 5 years (c) every Decade (d) Monthly Show the
calculations that lead to each answer. Round each to the nearest tenth of a percent.
Exercise #7: A radioactive substance’s half-life is the amount of time needed for half (or 50%) of the substance
to decay. A certain radioactive substance has a half-life of 15 years.
(a) First, give an expression that will calculate the yearly percent decay rate based on the fact that the substance
decayed 50% in 15 years.
(b) What percent of the substance would be radioactive after 60 years?
(c) What percent of the substance would be radioactive after 5 years? Round to the nearest tenth of a percent.
Exercise #8: Rewrite the following in the form A = a(1 ± r)t. State the growth or decay rate r.
a) A = a(2)t/3 b) A = a(4)t/6 c) A = a(.5)t/12 d) A = a(.25)t/9
(a) Write and evaluate an expression for what we
would multiply the initial amount of oil by after
20 years.
(b) Use your answer to (a) to determine the percent
decline, r, after 20 years. Be careful! Round to
the nearest percent.
A l g e b r a 2 U n i t 3 - Exponential Functions
Unit 3 Lesson 10 – r-values for multiple or fractional time periods Page 106
LESSON 10: r-VALUES FOR MULTIPLE OR FRACTIONAL TIME PERIODS
HOMEWORK
1. A quantity is growing at a constant 3% yearly rate. Which of the following would be its percent growth after
15 years?
(1) 45% (3) 56%
(2) 52% (4) 63%
2. If a credit card company charges 13.5% yearly interest, which of the following calculations would be used in
the process of calculating the monthly interest rate?
(1) 0.135
12 (3)
121.135
(2) 1.135
12 (4)
1121.135
3. The county debt is growing at an annual rate of 3.5%. What percent rate is it growing every 2 years?
Every 5 years? Every decade? Show the calculations that lead to each answer. Round each to the nearest tenth
of a percent.
4. A population of llamas is growing at a constant yearly rate of 6%. At what rate is the llama population
growing per month? Assume all months are equally sized and there are 12 of these per year. Round to the
nearest tenth of a percent.
A l g e b r a 2 U n i t 3 - Exponential Functions
Unit 3 Lesson 10 – r-values for multiple or fractional time periods Page 107
HOMEWORK (cont.)
5. Shana is trying to increase the number of calories she burns by 5% per day. By what percent is she
trying to increase per week? Round to the nearest tenth of a percent.
6. If a bank account doubles in size every 5 years, then by what percent does it grow after only 3 years? Round
to the nearest tenth of a percent. Hint: First write an expression that would calculate its growth rate after a
single year.
7. An object’s speed decreases by 5% for each minute that it is slowing down. Which of the following is closest
to the percent that its speed will decrease over half-an hour?
(1) 21% (3) 48%
(2) 79% (4) 150%
8. Over the last 10 years, the price of corn has decreased by 25% per bushel.
(a) Assuming a steady percent decrease, by what percent does it decrease each year? Round to the nearest
tenth of a percent.
(b) Assuming this percent continues, by what percent will the price of corn decrease by after 50 years? Show
the calculation that leads to your answer. Round to the nearest percent.
A l g e b r a 2 U n i t 4 - Logarithmic Functions
Unit 4 Lesson 1 – Introduction to Logarithms Page 108
U n i t 4 - Logarithmic Functions
LESSON 1 - INTRODUCTION TO LOGARITHMS
Exponential functions are of such importance that their inverses, functions that “reverse” their action, are important themselves. These functions, known as logarithms, will be introduced in this lesson.
Exercise #1: The function 2xf x is shown graphed on the
axes below along with its table of values.
(a) Is this function one-to-one? Explain your answer.
(b) Based on your answer from part (a), what must
be true about the inverse of this function?
(c) Create a table of values below for the inverse of
2xf x and plot this graph on the axes given.
(d) What would be the first step to find an equation for this inverse algebraically? Write this step down and
then stop.
Defining Logarithmic Functions – The function logby x is the name we give the inverse of xy b . For
example, ________________________ is the inverse of 2xy . Based on Exercise #1(d), we can write an
equivalent exponential equation for each logarithm as follows:
log is the same as y
by x b x
Based on this, we see that a logarithm gives as its output (y-value) the exponent we must raise b to in order to produce its input (x-value).
x 3 2 1 0 1 2 3
2xf x 18
14
12
1 2 4 8
x
1f x
y
x
Notice that, as always, the graphs
of and are
symmetric across
A l g e b r a 2 U n i t 4 - Logarithmic Functions
Unit 4 Lesson 1 – Introduction to Logarithms Page 109
It is critically important to understand that logarithms give exponents as their outputs.
Exercise #2: Evaluate the following logarithms. If needed, write an equivalent exponential equation. Do as
many as possible without the use of your calculator.
(a) 2log 8 (b) 4log 16 (c) 5log 625 (d) 10log 100,000
(e) 61log
36 (f) 2
1log16
(g) 5log 5 (h) 5
3log 9
Exercise #3: If the function 2log 8 9y x was graphed in the coordinate plane, which of the following
would represent its y-intercept?
(1) 12 (3) 8
(2) 13 (4) 9
Exercise #4: Between which two consecutive integers must 3log 40 lie?
(1) 1 and 2 (3) 3 and 4
(2) 2 and 3 (4) 4 and 5
Calculator Use and Logarithms – Most calculators only have two logarithms that they can evaluate directly.
One of them, 10log x , is so common that it is actually called the common log and typically is written without
the base 10.
10log logx x (The Common Log)
Exercise #5: Evaluate each of the following using your calculator.
(a) log100 (b) 1log1000
(c) log 10
Switching between exponential form and log form. Exponential Form Log Form Exponential Form Log Form
2³ = 8 38log 2 236log 6
25 = 32 3729log 9
62554 phm log
21663 2
112log 144
327 3
1
3001.0log10
tr s 2
19log b
364log 4 49
17 2
A l g e b r a 2 U n i t 4 - Logarithmic Functions
Unit 4 Lesson 1 – Introduction to Logarithms Page 110
LESSON 1 - INTRODUCTION TO LOGARITHMS
HOMEWORK
1. Which of the following is equivalent to 7logy x ?
(1) 7y x (3) 7 yx
(2) 7x y (4)
17y x
2. If the graph of 6xy is reflected across the line y x then the resulting curve has an equation of
(1) 6xy (3) 6logx y
(2) 6logy x (4) 6x y
3. The value of 5log 167 is closest to which of the following? Hint – guess and check the answers.
(1) 2.67 (3) 4.58
(2) 1.98 (4) 3.18
4. Which of the following represents the y-intercept of the function log 1000 8y x ?
(1) 8 (3) 3
(2) 5 (4) 5
5. Determine the value for each of the following logarithms. (Easy)
(a) 2log 32 (b) 7log 49 (c) 3log 6561 (d) 4log 1024
6. Determine the value for each of the following logarithms. (Medium)
(a) 21log
64 (b) 3log 1 (c) 5
1log25
(d) 71log
343
A l g e b r a 2 U n i t 4 - Logarithmic Functions
Unit 4 Lesson 1 – Introduction to Logarithms Page 111
HOMEWORK (cont.)
7. Determine the value for each of the following logarithms. Each of these will have non-integer, fractional
answers. (Difficult)
(a) 4log 2 (b) 4log 8 (c) 3
5log 5 (d) 5
2log 4
8. Between what two consecutive integers must the value of 4log 7342 lie? Justify your answer.
9. Between what two consecutive integers must the value of 51log
500 lie? Justify your answer.
10. In chemistry, the pH of a solution is defined by the equation pH log H
where H represents the concentration of hydrogen ions in the solution. Any solution with a pH less than 7 is considered acidic and any solution with a pH greater than 7 is considered basic. Fill in the table below. Round your pH’s to the nearest tenth of a unit.
11. Can the value of 2log 4 be found? What about the value of 2log 0 ? Why or why not? What does this
tell you about the domain of logb x ?
Substance Concentration
of Hydrogen pH Basic or Acidic?
Milk 71.6 10
Coffee 51.3 10
Bleach 132.5 10
Lemon Juice 27.9 10
Rain 61.6 10
A l g e b r a 2 U n i t 4 - Logarithmic Functions
Unit 4 Lesson 2 – Graphs of Logarithms Page 112
LESSON 2 - GRAPHS OF LOGARITHMS
Most logarithms have bases greater than one; the pH scale that we saw on the last homework assignment is a good example. In this lesson, we will further explore graphs of these logarithms, including their construction, transformations, and domains and ranges.
Exercise #1: Consider the logarithmic function 3logy x and its inverse 3xy .
(a) Construct a table of values for 3xy and then use this to
construct a table of values for the function 3logy x .
(b) Graph 3xy and 3logy x on the grid given. Label with equations. Label the asymptote and its equation.
(c) State the domain and range of 33 and logxy y x . Write the equation of the asymptote for each.
Exercise #2: Using your calculator, sketch the graph of 10logy x on the axes below. Label the x-intercept.
State the domain and range of 10logy x .
Domain:
Range:
Equation of asymptote:
x 2 1 0 1 2
3xy
x
3logy x
3xy
Domain:
Range:
Asymptote:
3log xy
Domain:
Range:
Asymptote:
y
x
2
10
y
x
A l g e b r a 2 U n i t 4 - Logarithmic Functions
Unit 4 Lesson 2 – Graphs of Logarithms Page 113
Exercise #3: Which of the following equations describes the graph shown below? Show or explain how you
made your choice.
(1) 3log 2 1y x
(2) 2log 3 1y x
(3) 2log 3 1y x
(4) 3log 3 1y x
The fact that finding the logarithm of a non-positive number (negative or zero) is not possible in the real number system allows us to find the domains of a variety of logarithmic functions.
Exercise #4: Determine the domain of the following functions. State your answer in set-builder notation.
(a) 2log 3 4y x (b) 𝑦 = 𝑙𝑜𝑔3(𝑥 + 4) (c) 𝑦 = 𝑙𝑜𝑔5(7 − 𝑥)
All logarithms with bases larger than 1 are always increasing. This increasing nature can be seen by calculating their average rate of change.
Exercise #5: Consider the common log, or log base 10, logf x x .
(a) Set up and evaluate an expression for the average rate of
change of f x over the interval 1 10x
(b) Set up and evaluate an expression for the average rate of
change of f x over the interval 1 100x .
y
x
y
x
A l g e b r a 2 U n i t 4 - Logarithmic Functions
Unit 4 Lesson 2 – Graphs of Logarithms Page 114
LESSON 2 - GRAPHS OF LOGARITHMS
HOMEWORK 1. The domain of 3log 5y x in the real numbers is
(1) | 0x x (3) | 5x x
(2) | 5x x (4) | 4x x
2. Which of the following equations describes the graph shown below?
(1) 5logy x (3) 3logy x
(2) 2logy x (4) 4logy x
3. Which of the following represents the y-intercept of the function 2log 32 1y x ?
(1) 8 (3) 1
(2) 4 (4) 4
4. Which of the following values of x is not in the domain of 5log 10 2f x x ?
(1) 3 (3) 5
(2) 0 (4) 4
5. Which of the following is true about the function 4log 16 1y x ?
(1) It has an x-intercept of 4 and a y-intercept of 1 .
(2) It has x-intercept of 12 and a y-intercept of 1.
(3) It has an x-intercept of 16 and a y-intercept of 1.
(4) It has an x-intercept of 16 and a y-intercept of 1 .
6. The graph of the function 𝑦 = 𝑙𝑜𝑔5𝑥 appears in which quadrants?
(a) I and II (b) I and IV (c) II and III (d) III and IV _________
y
x
A l g e b r a 2 U n i t 4 - Logarithmic Functions
Unit 4 Lesson 2 – Graphs of Logarithms Page 115
HOMEWORK (cont.)
7. Determine the domains of each of the following logarithmic functions. State your answers using any
accepted notation. Be sure to show the inequality that you are solving to find the domain and the work you use to solve the inequality.
(a) 5log 2 1y x (b) log 6y x
8. Graph the logarithmic function 4logy x on the graph paper given. For a method, see Exercise #1.
9. Logarithmic functions whose bases are larger than 1 tend to increase very slowly as x increases. Let's
investigate this for 2logf x x .
(a) Find the value of 1 , 2 , 4 , and 8f f f f without your calculator.
(b) For what value of x will 2log 10x ? For what value of x will 2log 20x ?
10. If the graph of 6xy is reflected across the line y x then the resulting curve has an equation of
(1) 6xy (2) 6logy x (3)
6logx y (4) 6x y __________
y
x
A l g e b r a 2 U n i t 4 - Logarithmic Functions
Unit 4 Lesson 3 –Solving Exponential Equations using Logs Page 116
LESSON 3 - SOLVING EXPONENTIAL EQUATIONS
USING LOGARITHMS
Earlier in this unit, we used the Method of Common Bases to solve exponential equations. This technique is quite limited, however, because it requires the two sides of the equation to be expressed using the same base. A more general method utilizes our calculators.
Exercise #1: Solve: 4 8x using (a) common bases and (b) the logarithm law shown above.
(a) Method of Common Bases (b) Logarithm Approach
The beauty of using logarithms is that it removes the variable from the exponent. We can solve almost any exponential equation using a TI-84 calculator as follows: alpha window #5.
Exercise #2: Solve each of the following equations for the value of x. Round your answers to the
nearest hundredth. **You must isolate the base before switching to log form!**
(a) 5 18x (b) 4x +12 = 112 (c) 5 2 1560x
These equations can become more complicated, but each and every time we will use the logarithm law to transform an exponential equation into one that is more familiar (linear only for now)
Exercise #3: Solve each of the following equations for x. Round your answers to the nearest hundredth.
(a) 36 50x (b) 5
21.03 2x
A l g e b r a 2 U n i t 4 - Logarithmic Functions
Unit 4 Lesson 3 –Solving Exponential Equations using Logs Page 117
Now that we are familiar with this method, we can revisit some of our exponential models from earlier in the unit. Recall that for an exponential function that is growing:
Exercise #4: A biologist is modeling the population of bats on a tropical island. When he first starts observing
them, there are 104 bats. The biologist believes that the bat population is growing at a rate of 3% per year.
Exercise #5: A stock has been declining in price at a steady pace of 5% per week. If the stock started at a
price of $22.50 per share, determine algebraically the number of weeks it will take for the price to reach $10.00.
Round your answer to the nearest week.
Exercise #6: Solve each of the following exponential equations to the nearest hundredth.
If quantity A is known to increase by a fixed percentage r, in decimal form, then A can be modeled by
𝐴(𝑡) = 𝑃(1 + 𝑟)𝑡
where P represents the amount of A present at and t represents time.
(a) Write an equation for the number of bats, ,
as a function of the number of years, t, since the
biologist started observing them.
(b) Using your equation from part (a), algebraically
determine the number of years it will take for the
bat population to reach 200. Round your answer
to the nearest year.
(a) 4 2 3 17x (b) 317 5 4
x
A l g e b r a 2 U n i t 4 - Logarithmic Functions
Unit 4 Lesson 3 –Solving Exponential Equations using Logs Page 118
LESSON 3 - SOLVING EXPONENTIAL EQUATIONS USING LOGARITHMS
HOMEWORK 1. Solve for x. When necessary, round your answer to the nearest ten thousandth. (4 decimal places)
(a) 837 x (b) x4log 3
(c) 214 x (d) 104log 5x
2. Solve each of the following exponential equations. Round each of your answers to the nearest hundredth.
(a) 39 250x (b) 50 2 1000x (c) 105 35
x
3. Solve each of the following exponential equations. Be careful with your use of parentheses. Express each
answer to the nearest hundredth.
(a) 2 56 300x (b) 1
31 12 6
x
(c) 12500 1.02 2300x
(d) 5(10)x ─ 6 = 100 (e) 13(2)4x=117
A l g e b r a 2 U n i t 4 - Logarithmic Functions
Unit 4 Lesson 3 –Solving Exponential Equations using Logs Page 119
HOMEWORK (cont.)
4. The population of Red Hook is growing at a rate of 3.5% per year. If its current population is 12,500, in
how many years will the population exceed 20,000? Round your answer to the nearest year. Only an
algebraic solution is acceptable.
5. A radioactive substance is decaying such that 2% of its mass is lost every year. Originally there were 50
kilograms of the substance present.
6. If a population doubles every 5 years, how many years will it take for the population to increase by 10
times its original amount?
First: If the population gets multiplied by 2 every 5 years, what does it get multiplied by each year? Use
this to help you answer the question.
7. Find the solution to the general exponential equation cx
a b d , in terms of the constants a, c, d
and the logarithm of base b. Think about reversing the order of operations in order to solve for x.
(a) Write an equation for the amount, A, of the
substance left after t-years.
(b) Find the amount of time that it takes for only
half of the initial amount to remain. Round
your answer to the nearest tenth of a year.
A l g e b r a 2 U n i t 4 - Logarithmic Functions
Unit 4 Lesson 4 – The number e and the Natural Logarithm Page 120
LESSON 4 - THE NUMBER e AND THE NATURAL LOGARITHM
There are many numbers that are more important than others because they find so many uses in either math or science. Good examples of important numbers are 0, 1, i, and . In this lesson you will be introduced to
an important number given the letter e for its “inventor” Leonhard Euler (1707-1783). This number plays a crucial role in Calculus and more generally in modeling exponential phenomena.
Exercise #1: Which of the graphs below shows xy e ? Explain your choice. Check on your calculator.
(1) (2) (3) (4)
Explanation:
Very often e is involved in exponential modeling of both increasing and decreasing quantities.
Exercise #2: A population of llamas on a tropical island can be modeled by the equation 0.035500 tP e , where
t represents the number of years since the llamas were first introduced to the island.
THE NUMBER e
1. Like , e is irrational. 2. e 3. Used in Exponential Modeling
y
x
y
x
y
x
y
x
(a) How many llamas were initially introduced at
0t ? Show the calculation that leads to your
answer.
(b) Algebraically determine the number of years for
the population to reach 600. Round your answer
to the nearest tenth of a year.
A l g e b r a 2 U n i t 4 - Logarithmic Functions
Unit 4 Lesson 4 – The number e and the Natural Logarithm Page 121
Because of the importance of xy e , its inverse, known as the natural logarithm, is also important.
The natural logarithm, like all logarithms, gives an exponent as its output. In fact, it gives the power that we must raise e to in order to get the input.
Exercise #3: Without the use of your calculator, determine the values of each of the following.
(a) ln e (b) ln 1 (c) 5ln e (d) ln e
Solve for x to the nearest hundredth:
(e) ln(x) = .5787 (f) 35xe
The natural logarithm follows the basic logarithm laws that all logarithms follow. The following problems give additional practice with these laws.
Exercise #4: A hot liquid is cooling in a room whose temperature is constant. Its temperature can be modeled
using the exponential function shown below. The temperature, T, is in degrees Fahrenheit and is a function of
the number of minutes, m, it has been cooling.
0.03101 67mT m e
THE NATURAL LOGARITHM
The inverse of :
(a) What was the initial temperature of the water at
0m . Do without using your calculator.
(b) How do you interpret the statement that
60 83.7T ?
(c) Using the natural logarithm, determine
algebraically when the temperature of the liquid
will reach 100 F . Show the steps in your
solution. Round to the nearest tenth of a minute.
(d) On average, how many degrees are lost per
minute over the interval 10 30m ? Round to
the nearest tenth of a degree.
A l g e b r a 2 U n i t 4 - Logarithmic Functions
Unit 4 Lesson 4 – The number e and the Natural Logarithm Page 122
LESSON 4 - THE NUMBER e AND THE NATURAL LOGARITHM
HOMEWORK 1. Which of the following is closest to the y-intercept of the function whose equation is
110 xy e ?
(1) 10 (3) 27
(2) 18 (4) 52
2. On the grid below, the solid curve represents xy e . Which of the following exponential functions could
describe the dashed curve? Explain your choice.
(1) 12
x
y (3) 2xy
(2) xy e (4) 4xy
3. Which of the following values of t solves the equation 25 15te ?
(1) ln15
10 (3) 2ln3
(2) 1
2 ln 5 (4)
ln 3
2
4. At which of the following values of x does 22 32xf x e have a zero?
(1) 5
ln2
(3) ln8
(2) ln 4 (4) 2
ln5
y
5. For the equation ctae d , solve for the variable t in terms of a, c, and d. Express your answer in terms of
the natural logarithm.
y
x
A l g e b r a 2 U n i t 4 - Logarithmic Functions
Unit 4 Lesson 4 – The number e and the Natural Logarithm Page 123
HOMEWORK (cont.)
6. Flu is spreading exponentially at a school. The number of new flu patients can be modeled using the
equation 0.1210 dF e , where d represents the number of days since 10 students had the flu.
(a) How many days will it take for the number of new flu patients to equal 50? Determine your answer
algebraically using the natural logarithm. Round your answer to the nearest day.
(b) Find the average rate of change of F over the first three weeks, i.e. 0 21d . Show the calculation
that leads to your answer. Give proper units and round your answer to the nearest tenth. What is the
physical interpretation of your answer?
7. The savings in a bank account can be modeled using .0451250 tS e , where t is the number of years the
money has been in the account. Determine, to the nearest tenth of a year, how long it will take for the
amount of savings to double from the initial amount deposited of $1250.
8. Solve for x to the nearest thousandth:
(a) 802 x (b) ln (3x) = 3.525
A l g e b r a 2 U n i t 4 - Logarithmic Functions
Unit 4 Lesson 5 – Compound Interest Page 124
LESSON 5 - COMPOUND INTEREST Compound interest is interest paid on an initial investment, called the principal, and on previously earned interest. Interest earned is often expressed as an annual percent, but the interest is usually compounded
more than once per year. So, the exponential growth model traA )1( must be modified for compound
interest problems.
Compound Interest Given P = amount initially invested (Principal)
r = annual interest rate expressed as a decimal
n = number of compounds per year
t = time, or number of years
The amount A in the account after t years is given by the formula
nt
n
rPA
1
Exercise #1: A person invests $500 in an account that earns a nominal yearly interest rate of 4%. Calculate
the amount of money in the account if the interest was compounded as follows:
(a) Quarterly (b) Monthly (c) Daily
Exercise #2: How much would $1000 invested at a nominal 2% yearly rate, compounded monthly, be worth
in 20 years? Show the calculations that lead to your answer.
(1) $1485.95 (3) $1033.87
(2) $1491.33 (4) $1045.32
A l g e b r a 2 U n i t 4 - Logarithmic Functions
Unit 4 Lesson 5 – Compound Interest Page 125
Exercise #3: If $1500 is invested at 2.5% interest compounded weekly, (a) how much will be in the
account after 10 years to the nearest dollar, (b) how many years will it take for the account to reach $2500?
Round to nearest tenth of a year.
Exercise #4: If $100 is invested at 8% interest compounded monthly, after how many years will the
amount in the account double? Round to the nearest tenth of a year.
The rate in Exercise #1 was referred to as nominal (in name only). It's known as this, because you effectively earn more than this rate if the compounding period is more than once per year. Because of this, bankers refer to the effective rate, or the rate you would receive if compounded just once per year. Let's investigate this.
Exercise #5: An investment with a nominal rate of 5% is compounded at different frequencies. Give the
effective yearly rate, accurate to two decimal places, for each of the following compounding frequencies.
Show your calculation.
(a) Quarterly (b) Monthly (c) Daily
A l g e b r a 2 U n i t 4 - Logarithmic Functions
Unit 4 Lesson 5 – Compound Interest Page 126
(a) Write a function to model the number of seals on
the island after t-years. (b) Algebraically determine the number of years for
the population to reach 455. Round your answer
to the nearest tenth of a year.
We could compound at smaller and smaller frequency intervals, eventually compounding all moments of
time. This gives rise to continuous compounding and the use of the natural base e in the continuous
compound interest formula.
Exercise #6: Tom invests $350 in a bank account that pays 2% annual interest compounded
continuously.
Exercise #7: You invest $2,500 in an account to save for college. Account 1 pays 6% annual interest
compounded quarterly. Account 2 pays 4% annual interest compounded continuously. Which account
should you choose to obtain the greater amount in 10 years? Justify your answer.
The above formula can be applied to any situation that grows at a continuous rate.
Exercise #8: A population of 314 seals on a tropical island is growing continuously at a rate
of 1.5% per year.
CONTINUOUS COMPOUND INTEREST
For an initial principal, P, compounded continuously at a nominal yearly rate of r, the investment
would be worth an amount A given by: A = Pert
(a) Write an equation for the amount this investment
would be worth after t-years.
(b) How much would the investment be worth
after 20 years?
(c) Algebraically determine the time it will take for
the investment to reach $400. Round to the nearest
tenth of a year.
(d) What is the effective annual rate for this
investment? Round to the nearest hundredth of a
percent.
A l g e b r a 2 U n i t 4 - Logarithmic Functions
Unit 4 Lesson 5 – Compound Interest Page 127
LESSON 5 - COMPOUND INTEREST
HOMEWORK
1. The value of an initial investment of $400 at 3% nominal interest compounded quarterly can be modeled
using which of the following equations, where t is the number of years since the investment was made?
(1) 4
400 1.0075t
A (3) 4
400 1.03t
A
(2) 400 1.0075t
A (4) 4
400 1.0303t
A
2. Which of the following represents the value of an investment with a principal of $1500 with a nominal
interest rate of 2.5% compounded monthly after 5 years?
(1) $1,697.11 (3) $4,178.22
(2) $1,699.50 (4) $5,168.71
3. Franco invests $4,500 in an account that earns a 3.8% nominal interest rate compounded continuously. If
he withdraws the profit from the investment after 5 years, how much has he earned on his investment?
(1) $858.92 (3) $922.50
(2) $912.59 (4) $941.62
4. An investment that returns a nominal 4.2% yearly rate, but is compounded quarterly, has an effective yearly
rate closest to
(1) 4.21% (3) 4.27%
(2) 4.24% (4) 4.32%
5. If an investment's value can be modeled with
12.027
325 112
t
A
then which of the following describes
the investment?
(1) The investment has a nominal rate of 27% compounded every 12 years.
(2) The investment has a nominal rate of 2.7% compounded every 12 years.
(3) The investment has a nominal rate of 27% compounded 12 times per year.
(4) The investment has a nominal rate of 2.7% compounded 12 times per year.
A l g e b r a 2 U n i t 4 - Logarithmic Functions
Unit 4 Lesson 5 – Compound Interest Page 128
HOMEWORK (cont.) 6. An investment of $500 is made at 2.8% nominal interest compounded quarterly.
7. An investment of $300 is made at 3.6% nominal interest compounded continuously.
8. The formula rtA Pe calculates the amount an investment earning a nominal rate of r compounded
continuously is worth. Show that the amount of time it takes for the investment to double in value is given
by the expression ln 2
r.
(a) Write an equation that models the amount A the
investment is worth t-years after the principal
has been invested.
(b) How much is the investment worth after 10
years?
(c) Algebraically determine the number of years it
will take for the investment to reach a worth of
$800. Round to the nearest hundredth.
(d) Why does it make more sense to round your
answer in (c) to the nearest quarter? State the
final answer rounded to the nearest quarter.
(a) Write an equation that models the amount A the
investment is worth t-years after the principal
has been invested.
(b) How much is the investment worth after 10
years?
(c) Algebraically determine the number of years it will take for the investment to be reach a worth of $800.
Round to the nearest hundredth.
A l g e b r a 2 U n i t 4 - Logarithmic Functions
Unit 4 Lesson 6 - More Exponential and Log Modeling Page 129
LESSON 6 - MORE EXPONENTIAL AND LOGARITHMIC MODELING
The Half-life, h, is the amount of time required for the amount of something to decrease to half its
initial value. Any exponential decay function can be rewritten as a half-life function.
We will begin with an example from the first lesson of this unit:
Exercise #1: The population of a town is decreasing by 4% per year and started with 12,500 residents.
(a) Write a function to model this situation.
(b) Algebraically determine when the population of the town will be half of the initial population.
Round to the nearest tenth of a year. The time you calculate is called the half-life, or h.
To re-write (a) as a half-life formula, start with the exponential decay formula 𝐴 = 𝑎(1 − 𝑟)𝑡 . Since the
amount decreases to half its amount, the r value is 50%, i.e., r = 1
2 over the timespan of the half-life h.
Thus the annual decay rate is (1 −1
2)
1
ℎ , or (
1
2)
1
ℎ . To calculate the amount A remaining after t years, raise
this annual decay rate to the t power: ((1
2)
1
ℎ)
𝑡
. Use the power rule of exponents to obtain the following:
(c) Write the half-life formula for the population of the town for Exercise #1.
Exercise #2: The decay of a sample of 5000 grams of carbon can be modeled by the equation,
57301( ) 5000
2
t
C t , where t is measured in years.
(a) What is the half-life of carbon? (b) How can you tell this is a half-life equation?
HALF LIFE FORMULA
For an initial quantity, a, that is decreasing at an exponential rate with a half life, h, the amount, A, left
after t “time units” is given by the formula
A = a(𝟏
𝟐)
𝒕
𝒉
A l g e b r a 2 U n i t 4 - Logarithmic Functions
Unit 4 Lesson 6 - More Exponential and Log Modeling Page 130
Exercise #3: One of the medical uses of I-131, a radioactive isotope of Iodine, is to enhance x-ray images.
The half-life of I-31 is 8.02 days. A patient is injected with 20 milligrams of I-131. Determine, to the
nearest day, the amount of time needed before the amount of I-131 in the patient’s body is 7 milligrams.
Log functions can be used to model real world phenomena.
Example #4: The slope, s, of a beach is related to the average diameter d (in millimeters) of the sand
particles on the beach by this equation: 0.159 0.118logs d .
Example #5: Two methods were used to teach athletes how to shoot a basketball. The methods were
assessed by assigning students into 2 groups, one group taught with method A, and one group taught with
method B. The students in each group took 30 foul shots after ten sessions. The average number of shots
made in each of the x sessions using method A can be modeled by ( ) 11.90 4.3lnA x x . The average
number of shots made in each of the x sessions using method B can be modeled by ( ) 9.17(1.109)xB x .
(a) In which of the 10 sessions, to the nearest whole number, will the two methods produce the same number
of made baskets? Explain how you found your answer.
(b) Find the average range of change for each method between sessions 3 and 8 for each of the methods.
Give proper units and round your answers to the nearest tenth.
(c) Explain why B(x) would not be an appropriate model for this situation if there were 15 sessions.
(b) If the average diameter of the sand particles is 0.25mm,
find the slope of the beach (to the nearest hundredth).
(c) Given a slope of 0.14, find the average diameter (to the
nearest hundredth) of the sand particles on the beach.
d
s
(a) Sand particles typically have a maximum diameter of 1mm.
Using this information, sketch a graph of the function.
A l g e b r a 2 U n i t 4 - Logarithmic Functions
Unit 4 Lesson 6 - More Exponential and Log Modeling Page 131
LESSON 6 - MORE EXPONENTIAL AND LOGARITHMIC MODELING
HOMEWORK
1. The decay of a sample of 800 grams of hydrogen can be modeled by the equation, 12.321
( ) 8002
t
H t
after
t years.
(a) What is the half-life of hydrogen? (b) How can you tell this is a half-life equation?
2. The $2,500 in your bank account is decreasing continuously at a rate of 5% per year.
(a) Use the Continuous Compound Interest Formula to write a function that models the amount of
money in your bank account after t years. (Don’t forget the r value should be negative because it’s
decreasing).
(b) When will only half of your initial deposit be left in your bank account (to the nearest tenth of a year)?
(c) Write the half-life formula for your bank account.
3. Flu is spreading exponentially at a school. The number of new flu patients can be modeled using
the equation 0.1210 dF e , where d represents the number of days since 10 students had the flu.
(a) How many days will it take for the number of new flu patients to equal 50? Round your answer
to the nearest day.
(b) Find the average rate of change of F over the first three weeks, i.e. 0 21d . Show the
calculation that leads to your answer. Give proper units and round your answer to the
nearest tenth. What is the physical interpretation of your answer?
A l g e b r a 2 U n i t 4 - Logarithmic Functions
Unit 4 Lesson 6 - More Exponential and Log Modeling Page 132
HOMEWORK (cont.)
4. Jessica keeps track of the height of a tree she planted over the first ten years. It can be modeled by
the equation 4.25 10.16ln( 1)y x where x is the number of years since she planted the tree.
(a) On average, how many feet did the tree grow each year over the time interval 0£t £10, to the nearest
hundredth.
(b) How tall was the tree when she planted it?
5. Most tornadoes last less than an hour and travel less than 20 miles.
The wind speed s (in miles per hour) near the center of a tornado is related to
the distance d (in miles) the tornado travels by this model: s = 93logd +65 .
(a) Sketch a graph of this function.
(b) On March 18, 1925, a tornado whose wind speed was about 180 miles per hour struck the Midwest. Use
your graph to determine how far the tornado traveled to the nearest mile.
x
y
A l g e b r a 2 U n i t 4 - Logarithmic Functions
Unit 4 Lesson 7 - Newton’s Law of Cooling and Exp. Formula Review Page 133
LESSON 7: NEWTON'S LAW OF COOLING AND EXPONENTIAL FORMULA REVIEW
Exercise #1: A detective is called to the scene of a crime where a dead body has just been found. He
arrives at the scene and measures the temperature of the dead body at 9:30 p.m to be 𝟕𝟐°𝐅. After
investigating the scene, he declares that the person died 𝟏𝟎 hours prior, at approximately 11:30 a.m. A
crime scene investigator arrives a little later and declares that the detective is wrong. She says that the
person died at approximately 6:00 a.m., 𝟏𝟓. 𝟓 hours prior to the measurement of the body temperature.
She claims she can prove it by using Newton’s law of cooling.
Using the data collected at the scene, decide who is correct, the detective or the crime scene investigator.
𝑇𝑎 = 68°F (the temperature of the room)
𝑇0 = 98.6°F (the initial temperature of the body)
𝑘 = 0.1335 (13.35 % per hour―calculated by the investigator from the data collected)
Recall, the temperature of the body at 9:30 p.m. is 72°F.
Exercise 2: A detective is called to the scene of a crime where a dead body has just been found. She arrives on
the scene at 10:23 pm and begins her investigation. Immediately, the temperature of the body is taken and is
found to be 80o F. The detective checks the programmable thermostat and finds that the room has been kept at a
constant 68o F. Assuming that the victim’s body temperature was normal (98.6o F) prior to death and that the
temperature of the victim’s body decreases continuously at a rate of 13.35% per hour, use Newton’s Law of
Cooling to determine the time when the victim died.
NEWTON’S LAW OF COOLING
where:
𝑻(𝒕) is the temperature of the object t “time units” has elapsed
𝑻𝒂 is the ambient temperature (the temperature of the surroundings), assumed to be constant
𝑻𝟎 is the initial temperature of the object
𝒌 is the decay constant per “time unit” (the r value where r is negative).
A l g e b r a 2 U n i t 4 - Logarithmic Functions
Unit 4 Lesson 7 - Newton’s Law of Cooling and Exp. Formula Review Page 134
Exercise #3: Two cups of coffee are poured from the same pot. The initial temperature of the coffee is 180°F
and 𝑘 is 0.2337 (for time in minutes).
Suppose both cups are poured at the same time. Cup 1 is left sitting in the room that is 75°F, and
cup 2 is taken outside where it is 42°F.
i. Use Newton’s law of cooling to write equations for the temperature of each cup of coffee
after 𝑡 minutes have elapsed.
ii. Graph and label both on the same coordinate plane and compare and contrast the end behavior
of the two graphs.
iii. Coffee is safe to drink when its temperature is below 140°F. How much time elapses before each
cup is safe to drink, to the nearest tenth of a minute. Use a graph to answer the question.
A l g e b r a 2 U n i t 4 - Logarithmic Functions
Unit 4 Lesson 7 - Newton’s Law of Cooling and Exp. Formula Review Page 135
Throughout the unit, you have learned many different exponential formulas. We will now practice writing a
few of them and converting between the different forms.
Exercise #4: Tritium has a half-life of 12.32 years.
a. Write a half-life formula, A(t), for the amount of tritium left in a 500 milligram sample after t years.
b. Write an equivalent function, B(t), it terms of the yearly rate of decay of tritium. Round all values to
four decimal places.
c. Wrtie an equivalent function, C(t), it terms of the monthly rate of decay of tritium. Round all values to
four decimal places.
Exercise #5: A deposit of $300 is made into a bank account that gets 4.3% interest compounded continuously.
a. Write a function, A(t), to model the amount of money in the account after t years.
b. Write an equivalent function, B(t), it terms of the yearly rate of interest for the account. Round all
values to four decimal places.
c. Write an equivalent function, C(t), it terms of the quarterly rate of interest for the account.
Round all values to four decimal places.
A l g e b r a 2 U n i t 4 - Logarithmic Functions
Unit 4 Lesson 7 - Newton’s Law of Cooling and Exp. Formula Review Page 136
LESSON 7: NEWTON'S LAW OF COOLING AND EXPONENTIAL FORMULA REVIEW
HOMEWORK
1. Hot soup is poured from a pot and allowed to cool in a room. The temperature in degrees Fahrenheit
of the soup after t minutes, can be modeled by the function, T(t)= 65+(212-65)e-.054t . What was the
initial temperature of the soup? What is the temperature of the room? At what rate is the temperature
of the soup decreasing?
2. Two cups of coffee are poured from the same pot. The initial temperature of the coffee is 190°F
and 𝑘 is 0.1450 (for time in minutes). Both are left sitting in the room that is 75°F, but milk is immediately
poured into cup 2 cooling it to an initial temperature of 162°F.
a. Use Newton’s law of cooling to write equations for the temperature of each cup of coffee after 𝑡
minutes have elapsed.
b. Graph and label both functions on the coordinate plane and compare and contrast the
end behavior of the two graphs.
c. Coffee is safe to drink when its temperature is below 140°F. Based on your graph,
how much time elapses before each cup is safe to drink to the nearest tenth of a minute?
A l g e b r a 2 U n i t 4 - Logarithmic Functions
Unit 4 Lesson 7 - Newton’s Law of Cooling and Exp. Formula Review Page 137
HOMEWORK (cont.)
3. A cooling liquid starts at a temperature of 200 F and cools down in a room that is held at a constant
temperature of 70 F . (Note – time is measured in minutes on this problem).
(b) Using the value of k you found in part (a), algebraically determine, to the nearest tenth of a minute, when
the temperature reaches 100 F .
4. A deposit of $1200 is made into a bank account that gets 4.3% interest compounded weekly.
a. Write a function, A(t), to model the amount of money in the account after t years.
b. Write an equivalent function, B(t), it terms of the yearly rate of interest for the account.
Round all values to four decimal places.
c. Write an equivalent function, C(t), it terms of the monthly rate of interest for the account.
Round all values to four decimal places.
(a) Use Newton’s Law of Cooling to determine the value of k if the temperature after 5 minutes is .
Round to four decimal places. (Hint: Write out the equation, plug in (5,153), and solve for k.)
A l g e b r a 2 U n i t 4 - Logarithmic Functions
Unit 4 Lesson 7 - Newton’s Law of Cooling and Exp. Formula Review Page 138
HOMEWORK (cont.)
5. A small town has a population of 12,600. The population is decreasing continuously at a rate
of 3% per year.
a. Write a function, A(t), to model the population of the town after t years.
b. Write an equivalent function, B(t), it terms of the yearly rate of decrease for the town.
Round all values to four decimal places.
c. Write an equivalent function, C(t), it terms of the daily rate of decrease for the town.
Round all values to four decimal places.
A l g e b r a 2 U n i t 4 - Logarithmic Functions
Unit 4 Lesson 7 - Newton’s Law of Cooling and Exp. Formula Review Page 139
Do now:
1. Which of the following values, to the nearest hundredth, solves: 7 500x ?
(1) 3.19 (3) 2.74
(2) 3.83 (4) 2.17
2. The solution to 32 52x
, to the nearest tenth, is which of the following?
(1) 7.3 (3) 11.4
(2) 9.1 (4) 17.1
3. To the nearest hundredth, the value of x that solves 45 275x is
(1) 6.73 (3) 8.17
(2) 5.74 (4) 7.49
4. Growth of a certain strain of bacteria is modeled by the equation 𝐺 = 𝐴(2.7)0.584𝑡, where G = final number
of bacteria, A=initial number of bacteria, and t = time (in hours). In approximately how many hours will 4
bacteria first increase to 2,500 bacteria? Round your answer to the nearest hour.
5. The rate of bacteria growth on a piece of moldy bread is represented by the equation 𝑟 = 24(2.5)𝑡, t
representing time in minutes and R representing the amount of bacteria in millions. If there was originally 24
million specimens of bacteria, how many minutes will it take for there to be triple that amount? Round your
answer to the nearest tenth of a minute.
A l g e b r a 2 U n i t 4 - Logarithmic Functions
Unit 4 Lesson 7 - Newton’s Law of Cooling and Exp. Formula Review Page 140
6. Hannah invests $3,850 at an annual rate of 6% compounded compounded continuously.
(a) Determine, to the nearest dollar, the amount of money she will have after 5 years.
(b) Determine how many years, to the nearest year, it will take for her investment to have a value of $10,000.
7. The decay of a sample of radioactive iodine can be modeled by the function 𝑓(𝑡) = 80(. 5)𝑡
60, where f grams
of the radioactive element remain after t days. In approximately how many days will 15% of the original mass
be present?
8. The Franklins inherited $3,500, which they want to invest for their child’s future college expenses. If they
invest it at 8.25% with interest compounded monthly, determine the value of the account, in dollars, after 5
years
9. The Matthews family would like to invest $2,000 for their child’s future college expenses. If they invest it at
6.75% with interest compounded monthly, determine the amount of time, to the nearest year, for the investment
to double in value.
A l g e b r a 2 U n i t 5 - Sequences and Series
Unit 5 Lesson 1 - Sequences Page 141
U n i t 5 - Sequences and Series
LESSON 1 - SEQUENCES
Sequences are ordered lists of numbers and are extremely important in math. A sequence is a function whose
domain the set of positive integers, i.e. 1, 2, 3, ..., n .
Exercise #1: Given the following explicit sequence definition: an = 2 1a n n .
(e) With explicit sequence formulas, when you are looking for a specific term in the sequence, what do you need
to do?
Sequences can also be described by using recursive definitions. When a sequence is defined recursively, terms
are found by operations on previous terms. A recursive definition always contains 2 parts – the first term and the
“formula”.
Exercise #2: A sequence is defined by the recursive formula: 1 5f n f n with 1 2f .
Exercise #3: A sequence is defined recursively as 1 12; 3 1n na a a .
(c) When you are looking for a specific term in a sequence defined recursively, what must you find first?
Generate the first five terms of this sequence. Label each term with proper function notation.
(a) Find the first three terms of this sequence, written
1 2 3, , and a a a .
(c) Which term has a value of 53? (d) Explain why there will not be a term that has a value of 70.
(b) Find the value of the 40th term.
(a) What is the value of the second term in the
sequence?
(b) What is the value of the fourth term in the
sequence?
A l g e b r a 2 U n i t 5 - Sequences and Series
Unit 5 Lesson 1 - Sequences Page 142
Exercise #4: Determine a recursive definition, in terms of f n , for the sequence shown below. Be sure to
include a starting value. Remember a recursive definition has ______ parts.
5, 10, 20, 40, 80, 160, …
Exercise #5: For the recursively defined sequence 2
1 2n nt t and 1 2t , the value of
4t is
(1) 18 (3) 456
(2) 38 (4) 1446
Exercise #6: Find an algebraic formula (explicit), a n , similar to that in Exercise #1, for each of the following
sequences. Recall that the domain that you map from will be the set 1, 2, 3, ..., n .
(a) 4, 5, 6, 7, ... (b) 2, 4, 8,16, ... (c) 5 5 5
5, , , , ...2 3 4
d) 1 1 1
1, , , , ...4 9 16
Exercise #7: Which of the following would represent the graph of the sequence 2 1na n ? Explain your choice.
(1) (2) (3) (4)
Explanation:
y
n
y
n
y
n
y
n
A l g e b r a 2 U n i t 5 - Sequences and Series
Unit 5 Lesson 1 - Sequences Page 143
3 3,
2 41 1
16,
2n na a a
Exercise #8: Match each of the explicit and recursive formulas with its sequence of numbers.
Explicit Formula Recursive Formula Sequence
1.
A.
W. -1, 2, 5, 8, …
2. 3n – 4
B.
X. 6, 3, …
3. 6n + 1
C.
Y. 7, 13, 19, 25, …
4.
D.
Z. 2, -6, 18, -54, …
Exercise #9: One of the most well-known sequences is the Fibonacci, which is defined recursively using two
previous terms. Its definition is:
and
Generate values for (in other words, the next four terms of this sequence).
1 2f n f n f n 1 1 and 2 1f f
3 , 4 , 5 , and 6f f f f
1
2 3n
11
32
n
(1) 2, ( 1) 3 ( )f f n f n
1 11, 3n na a a
(1) 7, ( ) ( 1) 6f f n f n
A l g e b r a 2 U n i t 5 - Sequences and Series
Unit 5 Lesson 1 - Sequences Page 144
UNIT 5 LESSON 1 - SEQUENCES
HOMEWORK
1. Given each of the following explicit sequence definitions, write the first four terms. A variety of notations is
used.
(a) 7 2f n n (b) 2 5na n (c) 2
3
n
t n (d) 1
1nt
n
f(1)=
f(2)=
f(3)=
f(4)=
2. Sequences below are defined recursively. Determine and label the next three terms of the sequence.
(a) 1 4 and 1 8f f n f n (b) 11 and 1 242
a n a n a
(c) 1 2n nb b n with
1 5b (d) 22 1f n f n n and 1 4f
3. Given the sequence 7, 11, 15, 19, ..., which of the following represents an explicit formula that will generate
it?
(1) 4 7a n n (3) 3 7a n n
(2) 3 4a n n (4) 4 3a n n
4. Which of the following formulas would represent the sequence 10, 20, 40, 80, 160, …
(1) 10n
na (3) 5 2n
na
(2) 10 2n
na (4) 2 10na n
A l g e b r a 2 U n i t 5 - Sequences and Series
Unit 5 Lesson 1 - Sequences Page 145
HOMEWORK (cont.)
5. For each of the following sequences, determine an algebraic formula (explicit), similar to Exercise #6, that
defines the sequence.
(a) 5, 10, 15, 20, … (b) 3, 9, 27, 81, … (c) 1 2 3 4
, , , , ...2 3 4 5
(d) 10, 20, 30, 40, … (e) -2, -4, -6, -8, …. (f) 10, 100, 1000, 10000, … 6. List the first 5 terms of the following recursive sequences:
(a) a1 = 2 an+1 = 2an (b) a1 = 81 an = 3
1an-1
7. For each of the following sequences, state a recursive definition. Be sure to include a starting value.
(a) 8, 6, 4, 2, … (b) 2, 6, 18, 54, … (c) 2, 2, 2, 2, ...
A l g e b r a 2 U n i t 5 - Sequences and Series
Unit 5 Lesson 2 - Arithmetic and Geometric Sequences Page 146
UNIT 5 LESSON 2 -ARITHMETIC AND GEOMETRIC SEQUENCES
In this lesson, we will review the basics of two particular sequences known as arithmetic (based on
constant addition to get the next term) and geometric (based on constant multiplying to get the next term).
Exercise #1: Generate the next three terms of the given __________arithmetic sequence 1 1
1 3 and
2 2n na a a
Exercise #2: Find the first four terms of the given ___________arithmetic sequence 2 6( 1)f n n .
Exercise #3: Consider ( ) ( 1) 3 with f(1) 5f n f n . Is this a recursive or explicit definition?
Exercise #4: Given that 1 46 and 18a a are members of an arithmetic sequence, determine the value of 20a .
To find d, the common difference, think of average rate of change, use d =
(a) Determine the value of (2), (3), and (4)f f f . (b) Write an explicit formula for the thn term of an
arithmetic sequence, ( )f n , based on the first
term, (1)f , d and n.
(c) Using your answer to (b), find f(1), f(2), f(3), and f(4) to make sure you found the correct formula.
ARITHMETIC SEQUENCE EXPLICIT FORMULA
where d is called the common difference and can be positive or negative.
d =
A l g e b r a 2 U n i t 5 - Sequences and Series
Unit 5 Lesson 2 - Arithmetic and Geometric Sequences Page 147
GEOMETRIC SEQUENCE EXPLICIT FORMULA
or
where r is called the common ratio and can be positive or negative and is sometimes fractional.
Geometric sequences are defined very similarly to arithmetic, but with a multiplicative constant instead of an
additive one.
Exercise #5: Generate the next three terms of the geometric sequences given below.
(a) 1 4 and 2a r (b) 11
3f n f n with 1 9f (c)
1 12 with 3 2n nt t t
Exercise #6:Determine if the sequence is arithmetic or geometric, write an explicit formula and recursive formula.
A l g e b r a 2 U n i t 5 - Sequences and Series
Unit 5 Lesson 2 - Arithmetic and Geometric Sequences Page 148
UNIT 5 LESSON 2 - ARITHMETIC AND GEOMETRIC SEQUENCES
HOMEWORK
Use the given information to fill in the other three rows in the table. Hint: If the terms are not given, find the
first few terms before completing the rest of the row.
Terms
Arithmetic or
Geometric?
Explicit Formula Recursive Formula
1.
10, 14, 18, 22. . .
2.
30, 15, 7.5, 3.75, . . .
3.
f (n) = 2 × f (n -1)
with f(1) = 6
4.
an
= an-1
- 6
with a1= 20
5.
f (n) = 5+
1
2(n -1)
6.
a
n= 3(-4)n-1
7. Consider f (n) = f (n-1) -10 with f(1) = 24 .
(a) Determine the value of (2), (3), and (4)f f f . (b) Write an explicit formula for the thn term of the
sequence, ( )f n .
(c) Using your answer to (b), find f(1), f(2), f(3), and f(4) to make sure you found the correct formula.
A l g e b r a 2 U n i t 5 - Sequences and Series
Unit 5 Lesson 2 - Arithmetic and Geometric Sequences Page 149
HOMEWORK (cont.)
8. Generate the next three terms of each arithmetic sequence shown below.
(a) 1 2 and 4a d (b)
1 23, 1a a
9. In an arithmetic sequence of numbers 1 64 and 46a a . Which of the following is the value of
12a ?
(1) 120 (3) 92
(2) 146 (4) 106
10. The first term of an arithmetic sequence whose common difference is 7 and whose 22nd term is
given by 22 143a is which of the following?
(1) 25 (3) 7
(2) 4 (4) 28
11. Generate the next three terms of each geometric sequence defined below.
(a) 1 8 with 1a r (b) 1
32n na a and
1 16a
12. In a geometric sequence, it is known that 1 1a and 4 64a . The value of
10a is
(1) 65,536 (3) 512
(2) 262,144 (4) 4096
A l g e b r a 2 U n i t 5 - Sequences and Series
Unit 5 Lesson 3 - Summation Notation Page 150
UNIT 5 LESSON 3 - SUMMATION NOTATION
Much of our work in this unit will concern adding the terms of a sequence. In order to specify this
addition or summarize it, we introduce a new notation, known as summation or sigma notation that will
represent these sums. This notation will also be used later in the course when we write formulas used in statistics.
Exercise #1: Evaluate each of the following sums.
(a)
5
3
2i
i
(b)
32
1k
k
(c) 5
1
1
3 2 j
j
(d) 5
1
1i
i
(e) 4
1
1 2( 1)k
k
(f) 3
1
1i
i i
Exercise #2: Which of represents the value of
4
1
1
i i
?
(1) 110
(3) 25
12
(2) 94
(4) 31
24
SUMMATION (SIGMA) NOTATION
where i is called the index variable, which starts at a value of a, ends at a value of n, and moves by unit
increments (increase by 1 each time).
A l g e b r a 2 U n i t 5 - Sequences and Series
Unit 5 Lesson 3 - Summation Notation Page 151
Exercise #3: Consider the sequence defined recursively by 1 2 1 22 and 0 and 1n n na a a a a . Find the value
of
7
4
i
i
a
Exercise #4: It is also good to be able to place sums into sigma notation. The values that are being summed in
the next problems form either an arithmetic or geometric sequence. Look back at Exercise #1 on the previous
page. Which problem represented the sum of the terms in an arithmetic sequence? A geometric sequence?
Exercise #5: Express each sum using sigma notation. Use n as your index variable. First, consider any patterns
you notice amongst the terms involved in the sum. Then, work to put these patterns into a formula and sum.
(a) 40 28 16 ... 20 (b) 1 1 1 5 625
25 5
(c) 6 3 0 3 15 (d) 2 6 18... 1458
Exercise #6: Some sums are more interesting than others. Determine the value of 99
1
1 1
1i i i
. Show your
reasoning. This is known as a telescoping series (or sum).
A l g e b r a 2 U n i t 5 - Sequences and Series
Unit 5 Lesson 3 - Summation Notation Page 152
UNIT 5 LESSON 3 - SUMMATION NOTATION
HOMEWORK
1. Evaluate each of the following. Place any non-integer answer in simplest rational form.
(a) 5
2
4i
i
(b) 3
2
0
1k
k
(c)
(d) 2
2 1
0
1k
k
(e) 3
1
log 10i
i
(f) 3
0
1
2256k
k
2. Which of the following is the value of 4
0
4 1k
k
?
(1) 53 (3) 37
(2) 45 (4) 80
3. The sum 7
7
4
2i
i
is equal to
(1) 158
(3) 34
(2) 32
(4) 78
4. Which of the following represents the sum ?
(1) (3)
(2) (4)
4
1 1n
n
n
2 5 10 82 101
6
1
4 3j
j
10
2
1
1j
j
103
3
2j
j
11
0
4 1j
j
A l g e b r a 2 U n i t 5 - Sequences and Series
Unit 5 Lesson 3 - Summation Notation Page 153
HOMEWORK (cont.)
5. Express each sum using sigma notation. Use n as your index variable. First, consider any patterns you notice
amongst the terms involved in the sum. Then, work to put these patterns into a formula and sum.
(a) -3 + 6 - 12 + 24 - 48 + ... 768 (b) 1
27+ 1
9+ 1
3+ ×××729
(c) 8.3+8.1+7.9 +7.7 + ... for 20 terms (d) 4+9+14+ ...44 + 49
6. A sequence is defined recursively by the formula 1 2 1 24 2 with 1 and 3n n nb b b b b . What is the value
of 5
3
i
i
b
? Show the work that leads to your answer.
7. A curious pattern occurs when we look at the behavior of the sum 1
2 1n
k
k
.
(a) Find the value of this sum for a variety of values of n below:
2
1
2 : 2 1k
n k
4
1
4 : 2 1k
n k
3
1
3: 2 1k
n k
5
1
5 : 2 1k
n k
(b) What types of numbers are you summing?
What types of numbers are the sums?
(c) Find the value of n such that .
A l g e b r a 2 U n i t 5 - Sequences and Series
Unit 5 Lesson 4 - Arithmetic Series Page 154
UNIT 5 LESSON 4 - ARITHMETIC SERIES
A series is simply the sum of the terms of a sequence.
You have already worked with series in previous lessons almost anytime you evaluated a summation problem.
Exercise #1: Given the arithmetic sequence defined by 1 12 and 5n na a a , which of the following is the
value of 5
5
1
i
i
S a
?
(1) 32 (3) 25
(2) 40 (4) 27
The sums associated with arithmetic sequences, known as arithmetic series, have interesting properties, many
applications and values that can be predicted with what is commonly known as rainbow addition.
Exercise #2: Consider the arithmetic sequence defined by 1 13 and 2n na a a . The series, based on the
first eight terms of this sequence, is shown below. Terms have been paired off as shown.
(a) What does each of the paired off sums equal?
(b) How many of these pairs are there?
(d) Generalize this now and create a formula for an arithmetic series sum based only on its first term, 1a , its last
term, na , and the number of terms, n.
THE DEFINITION OF A SERIES
If the set represent the elements of a sequence then the series, , is defined by:
3 5 7 9 11 13 15 17
(c) Using your answers to (a) and (b) find the value
of the sum using a multiplicative process.
A l g e b r a 2 U n i t 5 - Sequences and Series
Unit 5 Lesson 4 - Arithmetic Series Page 155
Exercise #3: Which of the following is the sum of the first 100 natural numbers? Show the process that leads to
your choice.
(1) 5,000 (3) 10,000
(2) 5,100 (4) 5,050
Exercise #4: Find the sum of each arithmetic series described or shown below.
Exercise #5: Kirk has set up a college savings account for his son, Maxwell. If Kirk deposits $100 per month in
an account, increasing the amount he deposits by $10 per month each month, then how much will be in the account
after 10 years?
SUM OF AN ARITHMETIC SERIES
Given an arithmetic series with n terms, , then its sum is given by:
(a) The sum of the sixteen terms given by:
.
(b) The first term is , the common difference, d,
is 6 and there are 20 terms
(c) The last term is and the common
difference, d, is .
(d) The sum .
A l g e b r a 2 U n i t 5 - Sequences and Series
Unit 5 Lesson 4 - Arithmetic Series Page 156
UNIT 5 LESSON 4 - ARITHMETIC SERIES
HOMEWORK
1. Which of the following represents the sum of 3 10 87 94 if the arithmetic series has 14 terms?
(1) 1,358 (3) 679
(2) 658 (4) 1,276
2. The sum of the first 50 natural numbers is
(1) 1,275 (3) 1,250
(2) 1,875 (4) 950
3. If the first and last terms of an arithmetic series are 5 and 27, respectively, and the series has a sum 192, then
the number of terms in the series is
(1) 18 (3) 14
(2) 11 (4) 12
4. Find the sum of each arithmetic series described or shown below.
(a) The sum of the first 100 even, natural numbers. (b) The sum of multiples of five from 10 to 75,
inclusive.
(c) A series whose first two terms are
and whose last term is 124.
(d) A series of 20 terms whose last term is equal to
97 and whose common difference is five.
A l g e b r a 2 U n i t 5 - Sequences and Series
Unit 5 Lesson 4 - Arithmetic Series Page 157
HOMEWORK (cont.)
5. For an arithmetic series that sums to 1,485, it is known that the first term equals 6 and the last term
equals 93. Algebraically determine the number of terms summed in this series.
6. Arlington High School recently installed a new black-box theatre for local productions. They only had room
for 14 rows of seats, where the number of seats in each row constitutes an arithmetic sequence starting with
eight seats and increasing by two seats per row thereafter. How many seats are in the new black-box theatre?
Show the calculations that lead to your answer.
7. Simon starts a retirement account where he will place $50 into the account on the first month and increasing
his deposit by $5 per month each month after. If he saves this way for the next 20 years, how much will the
account contain?
8. The distance an object falls per second while only under the influence of gravity forms an arithmetic sequence
with it falling 16 feet in the first second, 48 feet in the second, 80 feet in the third, etcetera. What is the total
distance an object will fall in 10 seconds? Show the work that leads to your answer.
9. A large grandfather clock strikes its bell once at 1:00, twice at 2:00, three times at 3:00, etcetera. What is the
total number of times the bell will be struck in a day? Use an arithmetic series to help solve the problem and
show how you arrived at your answer.
A l g e b r a 2 U n i t 5 - Sequences and Series
Unit 5 Lesson 5 - Geometric Series Page 158
UNIT 5 LESSON 5 - GEOMETRIC SERIES
A series is simply the sum of the terms of a sequence.
In truth, you have already worked extensively with series in previous lessons almost anytime you evaluated a
summation problem.
Exercise #1: Given a geometric series defined by the recursive formula 1 13 and 2n na a a , which of the
following is the value of 5
5
1
i
i
S a
?
(1) 106 (3) 93
(2) 75 (4) 35
Exercise #2: There is a formula for the sum of the first n terms of a geometric series, S
n. The following is steps
for deriving the formula.
Steps Work
1. Write the explicit formula for a
geometric sequence (a
n form).
a
n=
2. Write out “n” terms of the sequence
by plugging in 1 through “n”.
3. Write an equation, S
n, which gives
the sum of these terms.
S
n=
4. Multiply both sides of the equation
by r.
5. Find, in simplest form, the value of
S
n- r ×S
n (step 3 minus step 4)
n nS r S
6. Write both sides of the equation in
their factored form.
7. From the equation in step 6, find a
formula for nS in terms of
1, , and a r n by dividing by (1-r).
THE DEFINITION OF A SERIES
If the set represent the elements of a sequence then the series, , is defined by:
A l g e b r a 2 U n i t 5 - Sequences and Series
Unit 5 Lesson 5 - Geometric Series Page 159
Exercise #3: Which of the following represents the sum of a geometric series with 8 terms whose first term is 3
and whose common ratio is 4?
(1) 32,756 (3) 42,560
(2) 28,765 (4) 65,535
Exercise #4: Find the value of the geometric series shown below. Show the calculations that lead to your final
answer. 6 12 24 768
Exercise #5: A person places 1 penny in a piggy bank on the first day of the month, 2 pennies on the second
day, 4 pennies on the third, and so on. Will this person be a millionaire at the end of a 31 day month?
Exercise #6: You are offered a job that pays a salary of $51,000 the first year and a 2% increase in each
successive year. You decide to accept the job.
(a) What will your salary be during your tenth year of employment?
(b) You worked ten years for the company. What are your total earnings?
Exercise #7: Maria places $500 at the beginning of each year into an account that earns 5% interest compounded
annually. Maria would like to determine how much money is in her account after she has made her $500 deposit
at the beginning of the 11th year (this amount would not get any interest).
SUM OF A FINITE GEOMETRIC SERIES
For a geometric series defined by its first term, , and its common ratio, r, the sum of n terms is given by:
or (don’t memorize - it’s on the formula sheet)
A l g e b r a 2 U n i t 5 - Sequences and Series
Unit 5 Lesson 5 - Geometric Series Page 160
UNIT 5 LESSON 5-GEOMETRIC SERIES
HOMEWORK
1. Find the sums of geometric series with the following properties:
(a) 1 6, 3 and 8a r n (b) 1
120, , and 62
a r n (c) 1 5, 2, and 10a r n
2. If the geometric series 128
54 3627
has seven terms in its sum then the value of the sum is
(1) 4118
27 (3)
1370
9
(2) 1274
3 (4)
8241
54
3. Which of the following represents the value of the geometric series 256 + 384 + 576 +…+ 6561?
(1) 19,171 (3) 22,341
(2) 12,610 (4) 8,956
4. A geometric series has a first term of 32 and a final term of and a common ratio of . The value of
this series is
(1) 19.75 (3) 22.5
(2) 16.25 (4) 21.25
5. A geometric series whose first term is 3 and whose common ratio is 4 sums to 4095. The number of terms in
this sum is
(1) 8 (3) 6
(2) 5 (4) 4
1
4
1
2
A l g e b r a 2 U n i t 5 - Sequences and Series
Unit 5 Lesson 5 - Geometric Series Page 161
HOMEWORK (cont.)
6. Find the sum of the geometric series shown below. Show the work that leads to your answer.
127 9 3
729
7. Alex earns $52,000 in his first year of teaching and earns a 2% increase in each successive year. Write a
geometric series formula, Sn , for Alex’s total earnings over n years. Use this formula to find Alex’s total
earnings for his first 8 years of teaching, to the nearest cent.
8. A college savings account is constructed so that $1000 is placed the account on January 1st of each year
with a guaranteed 3% yearly return in interest, applied at the end of each year to the balance in the account.
If this is repeatedly done, how much money is in the account after the $1000 is deposited at the beginning
of the 19th year? Show the sum that leads to your answer as well as relevant calculations.
9. A ball is dropped from 16 feet above a hard surface. After each time it hits the surface, it rebounds to a height
that is 34
of its previous maximum height. What is the total vertical distance, to the nearest foot, the ball has
traveled when it strikes the ground for the 10th time? Write out the first five terms of this sum to help visualize.
A l g e b r a 2 U n i t 5 - Sequences and Series
Unit 5 Lesson 6 - Mortgage Payments Page 162
UNIT 5 LESSON 6 - MORTGAGE PAYMENTS
Mortgages are large amounts of money borrowed from a bank. Monthly mortgage payments can be found
using the formula below. This formula comes from a geometric series, but we will just be learning how to work
with the formula and solve for the different variables.
The most basic way to use the formula is to calculate monthly payments. 1. You took out a 30-year mortgage for $220,000 to buy a house. The interest rate on the mortgage is 5.2%. What
are your monthly payments?
When you are taking out a mortgage, you often know how much you can afford each month, and you want to
determine what size mortgage you can afford. 2. Based on your current income, you can afford mortgage payments of $900 a month. You also want to take out a
15 year mortgage to pay off the loan sooner. If the average interest rate at this time is 3.375%, what size
mortgage can you afford?
The last way we will learn to use the mortgage payment formula involves determining the length of the loan
that you can afford given the cost of a house, the amount you can spend per month, and the interest rate. 3. Imagine you have found the house of your dreams for $325,000. You know you can afford monthly mortgage
payments of $1500. You qualified for a mortgage with an interest rate of 4.75%. Algebraically determine the
number of payments you would need to make to pay off the loan at this rate to the nearest whole number. How
many years would it take you to the nearest tenth of a year?
112 12
1 112
n
n
r rP
Mr
M = monthly payment
P = amount borrowed
r = annual interest rate
n = number of monthly payments
A l g e b r a 2 U n i t 5 - Sequences and Series
Unit 5 Lesson 6 - Mortgage Payments Page 163
Exercise #1: Andy has a $200,000 mortgage at 4% yearly interest. He’s paying off his mortgage with $1,600
monthly payments (much of which are initially going to interest). How much he still owes after n-payments is
calculated with the following formula, where m is the monthly payment, r is the monthly interest rate, P is the
principal, and n is the number of payments made. Find the amount owed on this loan after 1 year and then after
10 years.
Exercise #2: I paid $297,000 for my house in December of 2008. The interest rate was 4.25% for 30 years.
Calculate the monthly payment (m) using the formula below. Recall that r is the monthly interest rate, P is the
principal, and n is the total number of payments. 1 1
n
P rm
r
Exercise #3: Using the formula from above:
a) Determine the number of months it would take to pay off a $150,000 loan at a monthly 0.5% interest rate,
with $1,000 payments.
b) How much money will it cost to pay off the loan when $1,000 is payed each month?
The amount owed after n payments, an, is:
A l g e b r a 2 U n i t 5 - Sequences and Series
Unit 5 Lesson 6 - Mortgage Payments Page 164
UNIT 5 LESSON 6 - MORTGAGE PAYMENTS
HOMEWORK
1. You took out a 15-year mortgage for $160,000 to buy a house. The interest rate on the mortgage is
5.2%.
a. What are your monthly payments?
2. Based on your current income, you can afford mortgage payments of $1250 a month. You also want to
take out a 25-year mortgage to spread the payments out over time. If the average interest rate at this
time is 3.375%, what size mortgage can you afford?
3. You have chosen a home in the perfect location for $250,000. You know you can afford monthly
mortgage payments of $1,400. You qualified for a mortgage with an interest rate of 4.75%.
Algebraically determine the number of payments you would need to make to pay off the loan at this rate.
How many years would it take you?
A l g e b r a 2 U n i t 5 - Sequences and Series
Unit 5 Lesson 6 - Mortgage Payments Page 165
HOMEWORK (cont.)
Use this formula for all problems on this page 1 1
n
P rm
r
4. Calculate the monthly payment needed to pay off a $200,000 mortgage at 5.15% yearly interest over 20 years.
Recall that r is the monthly rate. Show your work and carefully evaluate the above formula for m.
5. Do the same calculation as in the previous exercise but now make the pay off period 30 years instead of 20.
How much less is the monthly payment?
6. How many months would it take to pay off a $300,000 mortgage at a yearly interest rate of 4.75% making
monthly payments of $2,500?
7. How much money will it cost to pay off the loan if $2,500 is payed each month?
A l g e b r a 2 U n i t 6 - Quadratic Functions and Their Algebra
Unit 6 Lesson 1 - Factoring Page 166
U n i t 6 - Quadratic Functions and Their Algebra
LESSON 1 FACTORING The definition of factor, in two forms, is given below.
Always keep in mind that when we factor (verb) a quantity, we are rewriting it in a different form that is equal
to the original quantity.
Exercise #1: Rewrite each of the following binomials as a product of an integer and a binomial.
a) 5 10x b) 2 6x c) 6 15x d) 6 14x
The above type of factoring is called “factoring out” the greatest common factor (gcf). This greatest common
factor can be numbers, variables, or both.
Exercise #2: Write each of the following binomials as the product of the binomial’s gcf and another binomial.
a) 23 6x x b) 220 5x x c) 210 25x x d) 230 20x
Exercise #3: Rewrite each of the following trinomials as the product of its gcf and another trinomial.
(a) 22 8 10x x (b) 210 20 5x x (c) 3 28 12 20x x x (d) 3 26 15 21x x x
The ____________________ of a binomial is identical to it, but with the opposite sign in the middle.
Ex. Consider the binomial x + 5, this binomial has a conjugate: ____________________
If a binomial is of the “special’ form x2 – a2 , known as the difference of perfect squares, then its factors
are (x + a) and (x – a).
Exercise #4: Write each of the following binomials as the product of a conjugate pair.
a) 2 9x b) 168 494 yx c) 24 25x d) 216 81x
Exercise #5: Write each of the following binomials as the product of a conjugate pair.
a) 2 1
4x b)
10036
1006 b
a c) 24 49
81 9x d) 2 236 49x y
FACTOR – TWO IMPORTANT MEANINGS
(1) Factor (verb) – To rewrite a quantity as an equivalent product.
(2) Factor (noun) – Any individual component of a product.
CONJUGATE MULTIPLICATION PATTERN
A l g e b r a 2 U n i t 6 - Quadratic Functions and Their Algebra
Unit 6 Lesson 1 - Factoring Page 167
Factoring an expression until it cannot be factored any more is known as complete factoring. In general,
when completely factoring an expression, the first type of factoring always to consider is factoring out the gcf.
Exercise #6: Using a combination of gcf and difference of perfect squares factoring, write
each of the following in its completely factored form.
a) 25 20x b) 228 7x c) 240 250x d) 33 48x x
Just as there is a pattern for factoring the difference of
perfect squares, there are formulas for factoring the sum
AND difference of perfect cubes.
Exercise #7: To verify the sum of perfect cubes formula,
simplify the following product: 2 2( )( )a b a ab b
Perfect Cubes: 1, 8, ____, 64, ____, …
Exercise #8: Factor each of the following expressions:
a) x3 + 8 b) 1 + 64x3 c) x3 − 1000 d) x3y3 - 125 e) 27x3 + 8y3 f) 8x3 – 125y3
g) 250x4 – 2x h) 3x5 – 24x²
SUM OF PERFECT CUBES
DIFFERENCE OF PERFECT CUBES
A l g e b r a 2 U n i t 6 - Quadratic Functions and Their Algebra
Unit 6 Lesson 1 - Factoring Page 168
UNIT 6 LESSON 1 FACTORING
HOMEWORK
1. Rewrite each of the following binomials as the product of an integer and a different binomial.
a) 10 55x b) 24 40x c) 6 45x d) 18 9x
2. Rewrite each of the following binomials as the product of its gcf along with another binomial.
a) 22 8x x b) 6 27x c) 230 35x x d) 3 224 20x x
3. Rewrite each of the following binomials as the product of a conjugate pair.
a) 2 121x b) 264 x c) 24 1x d) 2 125
9x
4. Rewrite each of the following trinomials as the product of its gcf and another trinomial.
a) 24 12 28x x b) 26 4 10x x c) 3 214 35 7x x x d) 3 220 5 15x x x
5. Completely factor each of the following binomials using a combination of gcf factoring and conjugate
pairs.
a) 26 150x b) 236 4x c) 228 7x d) 327 12x x
e) 280 125x f) 32 200x x g) 28 512x h) 344 99x x
A l g e b r a 2 U n i t 6 - Quadratic Functions and Their Algebra
Unit 6 Lesson 1 - Factoring Page 169
HOMEWORK (cont.)
13. The area of any rectangular shape is given by the product of its width and length. If the area of a particular
rectangular garden is given by 215 35A x x and its width is given by 5x , then find an expression for
the garden’s length. Justify your response.
14. A projectile is fired from ground level such that its height, h, as a function of time, t, is given by 216 80h t t . Written in factored form this equation is equivalent to
(1) 16 4h t t (3) 16 5h t t
(2) 8 2 7h t t (4) 8 5h t t
A l g e b r a 2 U n i t 6 - Quadratic Functions and Their Algebra
Unit 6 Lesson 2 - Factor By Grouping Page 170
UNIT 6 LESSON 2 - FACTORING BY GROUPING
Today we will introduce a new type of factoring known as factoring by grouping. This technique requires
you to see structure in expressions. Remember, whenever we factor we also look for a gcf FIRST!
Warm up. Factor: xp + 7p = ________________
Exercise #1: Factor a binomial common factor out of each of the following expressions. Write your final
expression as the product of two binomials.
a) 2 1 7 2 1x x x b) 5 2 4 2x x x
c) 5 7 7 1x x x x d) 2 8 4 2 4x x x x
When a polynomial has 4 terms, you will factor by grouping, follow the steps below. 1. Split the polynomial into the 1st two terms, then the last two terms (draw line before the + or – operator) 2. Factor out the GCF of each set of two terms; what remains in both parentheses should be equal 3. Write what remains in parentheses, followed by the 2 GCFs in their own set of parentheses Examples: a) x3 – 4x² + 2x – 8 b) 2x3 – 3x² – 4x + 6
c) 5x3 – x² – 5x + 1 d) 3x3 + 18x² – 3x – 18
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Unit 6 Lesson 2 - Factor By Grouping Page 171
Exercise #2: Use the method of factoring by grouping to completely factor the following expressions.
(a) 6x2 - 4x +15x -10 (b) 5 3 24 2 8x x x
(c) 5x3 - 2x2 +5x - 2 (d) 18x3 + 9x2 - 2x -1
(e) 3 23 2 27 18x x x (f) x
2 y + 3xy - 25x - 75
(g) 2x2 - 30x - x +15 (h) 3 25 10 20 40x x x
Exercise #3:
Consider the expression: 3 25 9 45x x x . Enter this expression in y= on your calculator and find its
zeroes (x-intercepts). Use the following window: Xmin: -10, Xmax:10, Ymin: -10, Ymax: 50. Draw a
rough sketch. Then, factor the expression completely. Do you see the relationship between the factors and
the zeroes?
y
x
A l g e b r a 2 U n i t 6 - Quadratic Functions and Their Algebra
Unit 6 Lesson 2 - Factor By Grouping Page 172
UNIT 6 LESSON 2 - FACTORING BY GROUPING
HOMEWORK
1. Rewrite each of the following as the product of binomials. Be especially careful on the manipulations that
involve subtraction.
a) 5 7 5x x x b) 4 2 3 2x x x
c) 10 3 5 3x x x x d) 2 7 4 4 2x x x x
e) 4 3 2 1 2 2 1x x x x f) 3 7 5 5 2 4x x x x
2. Max tries to simplify the expression 5 2 3 2 3 3x x x x as follows:
5 2 3 2 3 3
3 5 2 2 3
3 3 1
x x x x
x x x
x x
3. Factor each of the following quadratic expressions completely using the method of grouping:
a) 210 6 35 21x x x b)
212 3 20 5x x x
Show using 2x that this simplification is incorrect.
Then, give the correct simplification.
A l g e b r a 2 U n i t 6 - Quadratic Functions and Their Algebra
Unit 6 Lesson 2 - Factor By Grouping Page 173
HOMEWORK (cont.)
4. Factor each of the following cubic expressions completely.
a) 3 25 2 20 8x x x b)
3 218 27 2 3x x x
c) 3 22 25 50x x x d)
3 28 10 12 15x x x
5. Factor each of the following expressions. Rearrange the expressions as needed to produce binomial
pairs with common factors.
a) xy ab ay bx b) 2x ac cx ax
Be careful when you use factoring by grouping. Don't force the method when it does not apply. This can lead
to errors.
6. Consider the expression 3 22 10 7 21x x x . Explain the error made in factoring it. How can you tell that
the factoring is incorrect?
3 2 2
2
2
2 10 7 21 2 5 7 3
2 7 5 3
2 7 2 8
x x x x x x
x x x
x x
A l g e b r a 2 U n i t 6 - Quadratic Functions and Their Algebra
Unit 6 Lesson 3 - Factoring Trinomials Page 174
UNIT 6 LESSON 3 – FACTORING TRINOMIALS
The ability to factor trinomials, expressions of the form 2ax bx c , is an important skill. Trinomials can be
factored if they are the product of two binomials.
Exercise #1: Warm up. Write each of the following products in simplest 2ax bx c form.
(a) 3 2 5 7x x (b) 5 4 2x x (c) 4 3 3 8x x
Factoring Trinomials – 2 types
Factoring trinomials, expressions of the form 2ax bx c , when a = 1 - Always look for a gcf, first. - Find factors of the constant, c, that sum to the coefficient of the linear term, b. a) x² + 12x + 27 b) x² + 10x + 24 c) x² + 8x − 20 d) x² - 2x - 15 e) y² + 13y + 12 f) x² + 7x − 30 g) x² + 8x +16 h) 2x² - 28x – 30 i) 10y² − 30y − 400 j) 3x3 – 21x2 – 54x k) 5x² + 5x - 280 l) 3y² - 33y + 90
A l g e b r a 2 U n i t 6 - Quadratic Functions and Their Algebra
Unit 6 Lesson 3 - Factoring Trinomials Page 175
Factoring trinomials, expressions of the form 2ax bx c , when a > 1
- Look for a gcf - Multiply the leading coefficient, a, and the constant, c. Write this product, a∙c. - Find the factors of a∙c that sum to the coefficient of the linear term, b. - Express the linear term as a sum and proceed using factor by grouping. a) 2x² + 7x + 6 b) 4x² − 12x + 5 c) 6x² − x − 2
d) 6x² - 17x + 12 e) 2x² - 7x + 6 f) 6x² - 17x + 5 g) 6x² − 11x + 3 h) 3x² + 7x + 2 i) 2x² + 13x + 5
A l g e b r a 2 U n i t 6 - Quadratic Functions and Their Algebra
Unit 6 Lesson 3 - Factoring Trinomials Page 176
UNIT 6 LESSON 3 FACTORING TRINOMIALS
HOMEWORK
1. Write each of the following trinomials in its factored form. (a = 1) Reminder, always look for gcf, first.
a) 2 7 18x x b)
2 14 24x x c) 2 17 30x x d)
2 5 6x x
e) 2 5 6x x f)
2 15 44x x g) 2 21 20x x h)
2 6 16x x
i) x² + 6x – 7 j) 2x² - 14x + 24 k) 3x² - 3x – 18 l) x² + 5x – 6
m) x² - x – 42 n) x² + 27x + 72 o) x² - 3x – 21 p) x² + 2x – 48
2. Write each of the following in its completely factored form.
a) 24 12 40x x b)
26 24x c) 22 20 50x x d)
275 3x
A l g e b r a 2 U n i t 6 - Quadratic Functions and Their Algebra
Unit 6 Lesson 3 - Factoring Trinomials Page 177
HOMEWORK (cont.)
3. Write each of the following trinomials in its factored form. (a > 1)
a) 25 41 8x x b)
23 4 20x x c) 22 29 15x x
d) 23 19 40x x e)
22 15 18x x f) 15x² – 19x + 6
g) 9x² – 12x + 4 h) 2x² – 7x – 4 i) 3x² + 10x – 25
.
4. Completely factor each of the following.
a) 210 55 105x x b)
212 57 15x x
A l g e b r a 2 U n i t 6 - Quadratic Functions and Their Algebra
Unit 6 Lesson 4 - Factoring completely Page 178
UNIT 6 LESSON 4 FACTORING COMPLETELY
Factoring can produce more than just two factors. In Exercise #1, we first warm-up by multiplying three
factors together.
Exercise #1: Write each of these in their simplest form. The second question should take little time to do.
(a) 2 4 7x x (b) 4 3 2 3 2x x x
To factor completely means you have factored until you can factor no more. Always look for a gcf,
first.
Factor completely
a) 2x² + 22x + 36 b) 5x³ − 10x² − 40x c) 2x² – 50 d) 18x³ − 98x e) 2x³ + 16x² – 210x f) x4 + 8x³ − 20x² g) 6x² + 26x + 20 h) 10x³ − 25x² − 35x i) 12u² − 28u − 24
A l g e b r a 2 U n i t 6 - Quadratic Functions and Their Algebra
Unit 6 Lesson 4 - Factoring completely Page 179
j) x7 − 3x5 − 4x³ k) 15x4 + 2x³ − x² l) 12x³ + 36x² + 27x
m) 26 13 6x x n) 2x4 − 128x o)
2 3 2 6a ab a b
p) 24x5 + 3x2 q) 3 23 2 12 8x x x
A l g e b r a 2 U n i t 6 - Quadratic Functions and Their Algebra
Unit 6 Lesson 4 - Factoring completely Page 180
UNIT 6 LESSON 4 – FACTORING COMPLETELY
HOMEWORK
1. Factor completely:
a) 22 14 36x x b)
25 70 245x x c) 23 192x
d) 3 26 36 96x x x e)
328 7x x f) 28 12 8x x
2. Write each of the following in completely factored form.
a) 215 110 120x x b)
3 210 26 12x x x
3. More Practice – Write each of the following expressions in its completely factored form.
a) 218 39 15x x b)
345 20x x
A l g e b r a 2 U n i t 6 - Quadratic Functions and Their Algebra
Unit 6 Lesson 4 - Factoring completely Page 181
HOMEWORK (cont.)
Note: there is one problem on this page of homework that is not factorable, which letter is it?
c) 28 30 28x x d)
3 290 90 20x x x
e) 227 3x f)
220 112 48x x
g) 5x3 + 4x2 - 45x + 36 h) 3 27 49 7x x x
i) 16x4 + 54x j) 40x3 +5
A l g e b r a 2 U n i t 6 - Quadratic Functions and Their Algebra
Unit 6 Lesson 5 – The Zero Product Law Page 182
UNIT 6 LESSON 5 - THE ZERO PRODUCT LAW
One of the most important equation solving technique stems from a fact about the number zero that is not true
of any other number:
The law can immediately be put to use in the first exercise. In this exercise, quadratic equations are given
already in factored form.
Exercise #1: Solve each of the following equations for all value(s) of x.
a) 7 3 0x x b) 2 5 4 0x x c) 4 3 2 4 3 0x x
Exercise #2: In Exercise #1c), why does the factor of 4 have no effect on the solution set of the equation?
The Zero Product Law can be used to solve any quadratic equation that is factorable (not prime). To utilize
this technique the problem solver must first set the equation equal to zero and then factor the non-zero side.
Exercise #3: Solve each of the following quadratic equations using the Zero Product Law.
a) 2 3 14 2 10x x x b)
2 23 12 7 3 2x x x x
THE ZERO PRODUCT LAW
If the product of multiple factors is equal to zero then at least one of the factors must be equal to zero.
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Unit 6 Lesson 5 – The Zero Product Law Page 183
Exercise #4: Consider the system of equations shown below consisting of a parabola and a line.
23 8 5 and 4 5y x x y x
a) Find the intersection points of these curves algebraically.
b) Using your calculator, sketch a graph of this system on the
axes to the right. Be sure to label the curves with
equations, the intersection points, and the window.
c) Verify your answers to part a) by using the INTERSECT command on your calculator.
The Zero Product Law is extremely important in finding the zero’s or x-intercepts (zeroes) of a parabola.
Exercise #5: The parabola shown at the right has the equation 2 2 3y x x
.
a) Write the coordinates of the two x-intercepts of the graph.
b) Find the x-intercepts of this parabola algebraically.
Exercise #6: Algebraically find the set of x-intercepts (zeroes) for each parabola given below.
a) 24 1y x b) 23 13 10y x x c) 25 10y x x
y
x
y
x
A l g e b r a 2 U n i t 6 - Quadratic Functions and Their Algebra
Unit 6 Lesson 5 – The Zero Product Law Page 184
UNIT 6 LESSON 5 - THE ZERO PRODUCT LAW
HOMEWORK
1. Solve each of the following equations for all value(s) of x.
a) 2 5 0x x b) 7 1 2 5 0x x c) 3 1 3 1 0x x
2. Solve each of the following quadratic equations which have already been set equal to zero.
a) 2 10 16 0x x b)
23 11 4 0x x c) 212 8 0x x
3. Solve each of the following quadratic equations by first manipulating them so that one side of the equation
is set equal to zero.
a) 2 4 40 10 15x x x b)
24 3 11 3 2x x x
c) 2 26 15 2 2 10 4x x x x d)
216 76 5 12 5t t t
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Unit 6 Lesson 5 – The Zero Product Law Page 185
HOMEWORK (cont.)
4. Consider the system of equations shown below consisting of one linear and one quadratic equation.
24 5 and 2 5 10y x y x x
a) Find the intersection points of this system algebraically.
b) Using your calculator, sketch a graph of this system
to the right. Be sure to label the curves with
equations, the intersection points, and the window.
c) Use the INTERSECT command on your calculator to verify the results you found in part a).
5. Algebraically, find the zeroes (x-intercepts) of each quadratic function given below.
a) 2 81y x b) 212 18y x x c) 22 6 8y x x
6. A quadratic function of the form 2y x bx c .
a) What are the x-intercepts of this parabola?
b) Based on your answer to part a), write the equation of this
quadratic function first in factored form and then in trinomial
form.
y
x
y
x
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Unit 6 Lesson 6 – Complete the Square Page 186
UNIT 6 LESSON 6 - SOLVING INCOMPLETE QUADRATICS AND COMPLETING THE SQUARE
Quadratics in the form ax2 + c = 0are known as incomplete. Because these equations lack a linear (b) term
they can be solved without the use of factoring and the Zero Product Law.
Exercise #1: Solve each of the following incomplete quadratics for all values of x.
(a) x2 -16 = 0 (b) 5x2 -8 =12 (c)
2
9x2 + 4 = 22
Unlike those quadratics that we factored and used the Zero Product Rule to solve, incomplete quadratics can
have irrational answers as solutions.
Exercise #2: Solve each of the following incomplete quadratics for all values of x. Place all answers in
simplest radical form.
(a) 3x2 -5 = 19 (b) 10x2 +1= 6 (c) 4x2 +5 = 8 (d) 3(x – 5)2 = 54
Any quadratic equation can be rewritten in a form where the method of
Square roots can be used. This process is known as completing the square.
Example: Solve the equation, x2 - 6x -8 = 0 .
1. Move the constant term to the other side of the equation.
2. Find 2
b and
2
2
b
3. Complete the square by adding
2
2
b
to both sides of the equation.
4. Factor the left side of the equation into a perfect square binomial.
5. Take the square root of both sides. Do not forget the plus or minus.
6. Add or subtract to solve for x.
You can use this process to solve any quadratic, whose lead coefficient, a, is 1, but it is especially useful
when the quadratic cannot be factored.
2
2
2
2
2
6 8 0
1. 6 8
2. 3 92 2
3. 6 9 8 9
4. ( 3)( 3) 17
( 3) 17
5. 3 17
6. 3 17
x x
x x
b b
x x
x x
x
x
x
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Unit 6 Lesson 6 – Complete the Square Page 187
Exercise #3: Solve each of the following prime quadratic equations by first completing the square. Express
your answers in simplest radical form.
(a) x2 -10x + 23= 0 (b) x
2 +12x +18 = 0
Exercise #4: For each of the following quadratics, express your answers to the nearest hundredth. Graph the
quadratic to verify that you have found the correct answer for the zeroes.
(a) x2 + 2x -12 = 0 (b) x
2 -14x + 7 = 0
Quadratic equations where b is even and a=1 are the easiest to solve by completing the square. When b is
odd, fractions are involved in the process.
Exercise #5: Solve each of the following quadratic equations by completing the square.
(a) 2 5 3 0x x (b) 2 9 12 0x x
You cannot complete the square when a is greater than one. In those cases, divide the entire equation by “a”
first and then complete the square.
c) 24 8 20 0x x
d) 3x2 + 9x + 3= 0
A l g e b r a 2 U n i t 6 - Quadratic Functions and Their Algebra
Unit 6 Lesson 6 – Complete the Square Page 188
UNIT 6 LESSON 6 - SOLVING INCOMPLETE QUADRATICS AND COMPLETING THE SQUARE
HOMEWORK
1. Solve each of the following incomplete quadratics. Express your answers in simplest radical form when
necessary.
a) 3x2 =108 b)
1
2x2 - 7 = 25 c) 500x2 -5 = 0
d) 5x2 =100 e) 2x2 - 20 = 70 f) 6x2 +10 =12
2. Solve each of the following quadratic equations by completing the square. Express your answer in simplest
radical form.
a) x2 - 2x - 2 = 0 b) x
2 + 6x + 4 = 0
c) x2 -5x +1= 0 d) 2x2 -8x + 2 = 0
A l g e b r a 2 U n i t 6 - Quadratic Functions and Their Algebra
Unit 6 Lesson 6 – Complete the Square Page 189
HOMEWORK (cont.)
e) 4x2 - 20x +8 = 0 f) x2 - 7x + 2 = 0
3. Rounded to the nearest hundredth the larger root of 2 22 108 0x x is
(1) 18.21 (3) 6.74
(2) 13.25 (4) 14.61
A l g e b r a 2 U n i t 6 - Quadratic Functions and Their Algebra
Unit 6 Lesson 7 – The Quadratic Formula Page 190
UNIT 6 LESSON 7 - THE QUADRATIC FORMULA
Simplify each of the following expressions:
3 11
6
=
3 50
6
=
3 72
6
=
3 25
6
=
In the last lesson, you solved the following quadratic equation, x2 -10x + 23= 0 by completing the square.
The solutions were x = 5- 2 and x = 5+ 2 .
Since any quadratic can be rearranged through the process of Completing the Square, a formula can be
developed that will solve for the roots of any quadratic equation. This famous formula, known as the
Quadratic Formula, is shown below. You worked with this as well in Algebra I.
Exercise #1: Using the quadratic formula shown above, solve the equation x2 -10x + 23= 0 . You should get
the same solution as you did in the last lesson.
How can you tell from your solutions that the quadratic equation, x2 -10x + 23= 0 is not factorable?
Exercise #2: Which of the following represents the solutions to the equation 2 10 20 0x x ?
(1) 5 10x (3) 5 10x
(2) 5 5x (4) 5 5x
Although the quadratic formula is most helpful when a quadratic expression is prime (not factorable), it can
be used as a replacement for the Zero Product Law in cases where the quadratic can be factored.
THE QUADRATIC FORMULA
The solutions to the quadratic equation , assuming , are given by
A l g e b r a 2 U n i t 6 - Quadratic Functions and Their Algebra
Unit 6 Lesson 7 – The Quadratic Formula Page 191
Exercise #3: Solve the quadratic equation shown below in two different ways – (a) by factoring and (b) by
using the quadratic formula.
(a) 22 11 6 0x x by factoring (b)
22 11 6 0x x by the quadratic formula
(c)Where will the function, f (x) = 2x2 +11x - 6 intersect the x-axis?
The quadratic formula is very useful in algebra - it should be committed to memory with practice.
Exercise #4: Solve each of the following quadratic equations by using the quadratic formula. Some answers
will be purely rational numbers and some will involve irrational numbers. Place all answers in simplest form.
(a) 23 5 2 0x x (b)
2 8 13 0x x
(c) 22 2 5 0x x (d)
25 8 4 0x x
Exercise #5: A shot-put throw can be modeled using the equation y = -.0241x2 + x +5.5 where x is the
distance traveled (in feet) and y is the height (also in feet). How long was the throw?
A l g e b r a 2 U n i t 6 - Quadratic Functions and Their Algebra
Unit 6 Lesson 7 – The Quadratic Formula Page 192
UNIT 6 LESSON 7 - THE QUADRATIC FORMULA
HOMEWORK
1. Solve each of the following quadratic equations using the quadratic formula. Express all answers in
simplest form.
(a) 2 7 18 0x x (b)
2 2 1 0x x
(c) 2 8 13 0x x (d)
23 2 3 0x x
(e) 26 7 2 0x x (f)
25 3 4 0x x
A l g e b r a 2 U n i t 6 - Quadratic Functions and Their Algebra
Unit 6 Lesson 7 – The Quadratic Formula Page 193
HOMEWORK (cont.)
2. Which of the following represents all solutions of 2 4 1 0x x ?
(1) 2 5 (3) 2 10
(2) 2 5 (4) 2 12
3. Which of the following is the solution set of the equation 24 12 19 0x x ?
(1) 5
32 (3)
37
2
(2) 2
23
(4) 7
63
4. Rounded to the nearest hundredth the larger root of 2 22 108 0x x is
(1) 18.21 (3) 6.74
(2) 13.25 (4) 14.61
5. Algebraically find the x-intercepts of the quadratic function whose equation is 2 4 6y x x .
Express your answers in simplest radical form.
6. A missile is fired such that its height above the ground is given by 29.8 38.2 6.5h t t , where t
represents the number of seconds since the rocket was fired. Using the quadratic formula, determine, to
the nearest tenth of a second, when the rocket will hit the ground.
A l g e b r a 2 U n i t 6 - Quadratic Functions and Their Algebra
Unit 6 Lesson 8 –More Work with Quadratic Equations Page 194
UNIT 6 LESSON 8 - MORE WORK WITH QUADRATIC EQUATIONS
Exercise #1: You have seen that some quadratics are factorable and some are not. You also know that
certain methods can always be used to solve a quadratic.
a) What are these methods?
b) Let’s say you used the quadratic formula to solve a quadratic. How can you tell from your answers
when you could have also factored the quadratic?
c) Decide if each set of numbers is rational, irrational, or nonreal.
i. 3 7 3 7
,2 2
ii. 3 25 3 25
,2 2
iii.
3 0 3 0,
2 2
iv.
3 9 3 9,
2 2
d) When do you usually see numbers in the form above?
e) What part of each number dictates what type of number it is?
In the quadratic formula, 2 4b ac is the expression in the radical. It is known as the discriminant because
helps you discriminate (differentiate) between quadratics that can be factored and those that cannot be
factored. (It also gives other information that will be covered later.
Exercise #2: Use the discriminant, , to quickly determine if each of the following quadratics can be
factored. Indicate if the quadratic has nonreal solutions.
(a) 2x² − 3x + 4 = 0 (b) 3x² − 7x – 6 = 0 (c) x² − 2 = 5x (d) x² − 6x = −9
Exercise #3: Which of the following sets represents the x-intercepts of 23 19 6y x x ?
(1) 1 7,
2 3
(3) 2 5, 2 5
(2) 1 17 1 17,
6 2 6 2
(4) 1, 6
3
2 4b ac
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Unit 6 Lesson 8 –More Work with Quadratic Equations Page 195
20
50 y
x
5
15 y
x
Exercise #4: Consider the quadratic function 2 4 36f x x x .
(a) Algebraically determine this function’s x-intercepts
using the quadratic formula. Express your answers in
simplest radical form.
(b) Express the x-intercepts of the quadratic to the nearest
hundredth.
(c) Using your calculator, sketch a graph of the quadratic on the axes given. Use the intersect command on
your calculator to verify your answers from part (b). (Remember to put y2=0) Label the zeros on the graph.
Exercise #5: The Crazy Carmel Corn company modeled the percent of popcorn kernels that would pop, P, as
a function of the oil temperature, T, in degrees Fahrenheit using the equation
212.8 394
250P T T
The company would like to find the lowest temperature that ensures that at least 50% of the kernels will pop.
Write an equation to model this situation. Solve this equation with the help of the quadratic formula. Round
the temperature to the nearest tenth of a degree.
Exercise #6: Find the intersection points of the linear-quadratic system shown below algebraically. Then,
use you calculator to help produce a sketch of the system. Label the intersection points you found on your
graph.
24 6 2 and 6 3y x x y x
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Unit 6 Lesson 8 –More Work with Quadratic Equations Page 196
UNIT 6 LESSON 8 - MORE WORK WITH QUADRATIC EQUATIONS
HOMEWORK
1. Use the discriminant, , to quickly determine if each of the following quadratics can be factored.
If the equation can be factored, solve by factoring. If the equation cannot be factored, choose a different
method to solve it.
(a) 5x² − 6x + 2 = 0 (b) 2x² − 11x + 14 = 0
2. Which of the following represents the solutions to 2 4 12 6 2x x x ?
(1) 4 7x (3) 5 22x
(2) 5 11x (4) 4 13x
3. The smaller root, to the nearest hundredth, of 22 3 1 0x x is
(1) 0.28 (3) 1.78
(2) 0.50 (4) 3.47
4. The x-intercepts of 22 7 30y x x are
(1) 7 191
2x
(3)
56 and
2x
(2) 3 and 5x (4) 3 131x
2 4b ac
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Unit 6 Lesson 8 –More Work with Quadratic Equations Page 197
HOMEWORK (cont.)
5. Solve the following equation for all values of x. Express your answers in simplest radical form.
24 4 5 8 6x x x
6. Algebraically solve the system of equations shown below.
26 19 15 and 12 15y x x y x
7. The Celsius temperature, C, of a chemical reaction increases and then decreases over time according to the
formula 218 93
2C t t t , where t represents the time in minutes. Use the Quadratic Formula to
determine the least amount of time, to the nearest tenth of a minute, it takes for the reaction to reach 110
degrees Celsius.
A l g e b r a 2 U n i t 6 - Quadratic Functions and Their Algebra
Unit 6 Lesson 9 – Imaginary Numbers Page 198
UNIT 6 LESSON 9 IMAGINARY NUMBERS
Recall that in the Real Number System, it is not possible to take the square root of a negative quantity because
whenever a real number is squared it is non-negative. This fact has a ramification for finding the x-intercepts
of a parabola, as Exercise #1 will illustrate.
Exercise #1: On the axes below, a sketch of 2y x is shown. Now, consider the parabola whose equation is
given in function notation as 2 1f x x .
Since we cannot solve this equation using Real Numbers, we introduce a new number, called i, the basis of
imaginary numbers. Its definition allows us to now have a result when finding the square root of a negative
real number. Its definition is given below.
Exercise #2: Simplify each of the following square roots in terms of i.
(a) 9 (b) 100 (c) 32 (d) 18
THE DEFINITION OF THE IMAGINARY NUMBER i
(c) What can be said about the x-intercepts of the
function ?
(d) Algebraically, show that these intercepts do not
exist, in the Real Number System, by solving the
incomplete quadratic .
(a) How is the graph of shifted to produce
the graph of ? (b) Create a quick sketch of on the axes
below. y
x
A l g e b r a 2 U n i t 6 - Quadratic Functions and Their Algebra
Unit 6 Lesson 9 – Imaginary Numbers Page 199
Exercise #3: Solve each of the following incomplete quadratics. Place your answers in simplest radical form.
(a) 25 8 12x (b) 2120 2
2x (c) 22 10 36x
Exercise #4: Which of the following is equivalent to 5 6i i ?
(1) 30i (3) 30
(2) 11i (4) 11
Powers of i display a remarkable pattern that allow us to simplify large powers of i into one of 4 cases. This
pattern is discovered in Exercise #4.
Exercise #5: Simplify each of the following powers of i.
1i i 2i 3i 4i
5i 6i 7i 8i
We see, then, from this pattern that every power of i is either 1,1, , or i i . And the pattern will repeat.
Exercise #6: Simplify each of the following large powers of i.
1. Divide the large power by 4 noting the remainder
2. Write as iremainder and simplify
(a) 38i (b) 21i (c) 83i (d) 40i
Exercise #7: Which of the following is equivalent to 16 23 265 3i i i ?
(1) 8 2i (3) 5 4i
(2) 4 3i (4) 2 7i
A l g e b r a 2 U n i t 6 - Quadratic Functions and Their Algebra
Unit 6 Lesson 9 – Imaginary Numbers Page 200
UNIT 6 LESSON 9 IMAGINARY NUMBERS
HOMEWORK
1. The imaginary number i is defined as
(1) 1 (3) 4
(2) 1 (4) 2
1
2. Which of the following is equivalent to 128 ?
(1) 8 2 (3) 8 2
(2) 8i (4) 8 2i
3. The sum 9 16 is equal to
(1) 5 (3) 7i
(2) 5i (4) 7
4. Which of the following powers of i is not equal to one?
(1) 16i (3) 32i
(2) 26i (4) 48i
5. Which of the following represents all solutions to the equation 21
10 73
x ?
(1) 3x i (3) x i
(2) 5x i (4) 2x i
6. Solve each of the following incomplete quadratics. Express your answers in simplest radical form.
(a) 22 100 62x (b) 22
20 23
x
A l g e b r a 2 U n i t 6 - Quadratic Functions and Their Algebra
Unit 6 Lesson 9 – Imaginary Numbers Page 201
HOMEWORK (cont.)
7. Which of the following represents the solution set of 2112 37
2x ?
(1) 7i (3) 5 2i
(2) 7 2i (4) 3 2i
8. Simplify each of the following powers of i into either 1,1, , or i i .
(a) 2i (b) 3i (c) 4i (d) 11i
(e) 41i (f) 30i (g) 25i (h) 36i
(i) 51i (j) 45i (k) 80i (l) 70i
9. Which of the following is equivalent to 7 8 9 10i i i i ?
(1) 1 (3) 1 i
(2) 2 i (4) 0
10. When simplified the sum 18 25 28 435 7 2 6i i i i is equal to
(1) 2 4i (3) 5 7i
(2) 3 i (4) 8 i
11. The product 6 2 4 3i i can be written as
(1) 24 6i (3) 2 5i
(2) 18 10i (4) 30 10i
A l g e b r a 2 U n i t 6 - Quadratic Functions and Their Algebra
Unit 6 Lesson 10 – Complex Numbers Page 202
UNIT 6 LESSON 10 COMPLEX NUMBERS
Complex numbers can always be thought of as a combination of a real number with an imaginary number and
will have the form:
where and a bi a b are real numbers
We say that a is the real part of the number and bi is the imaginary part of the number. These two parts, the
real and imaginary, cannot be combined. Like real numbers, complex numbers may be added and subtracted.
The key to these operations is that real components can combine with real components and imaginary with
imaginary.
Exercise #1: Find each of the following sums and differences.
(a) 2 7 6 2i i (b) 8 4 12i i (c) 5 3 2 7i i (d) 3 5 8 2i i
Exercise #2: Which of the following represents the sum of 6 2 and 8 5i i ?
(1) 5i (3) 2 3i
(2) 2 3i (4) 5i
Adding and subtracting complex numbers is straightforward because the process is similar to combining
algebraic expressions that have like terms.
Exercise #3: Find the following products. Write each of your answers as a complex number in the form a bi
.
(a) 3 5 7 2i i (b) 2 6 3 2i i (c) 4 5 3i i
A l g e b r a 2 U n i t 6 - Quadratic Functions and Their Algebra
Unit 6 Lesson 10 – Complex Numbers Page 203
Exercise #4: Consider the more general product a bi c di where constants a, b, c and d are real numbers.
(c) Under what conditions will the product of two complex numbers always be a purely imaginary number?
Check by generating a pair of complex numbers that have this type of product.
Exercise #5: Determine the result of the calculation below in simplest a bi form.
5 2 3 4 2 3i i i i
Exercise #6: Which of the following products would be a purely real number?
(1) 4 2 3i i (3) 5 2 5 2i i
(2) 3 2 4i i (4) 6 3 6 3i i
(a) Show that the real component of this product will
always be ac bd .
(b) Show that the product of 2 3i
and 4 6i results in a purely real number.
A l g e b r a 2 U n i t 6 - Quadratic Functions and Their Algebra
Unit 6 Lesson 10 – Complex Numbers Page 204
UNIT 6 LESSON 10 COMPLEX NUMBERS
HOMEWORK
1. Find each of the following sum or difference.
(a) 6 3 2 9i i (b) 7 3 5i i (c) 10 3 6 8i i
(d) 2 7 15 6i i (e) 15 2 5 5i i (f) 1 5 6i i
2. Which of the following is equivalent to 3 5 2 2 3 6i i ?
(1) 9 18i (3) 9 6i
(2) 21 8i (4) 21 2i
3. Find each of the following products in simplest a bi form.
(a) 5 2 1 7i i (b) 3 9 2 4i i (c) 4 2 6i i
4. Complex conjugates are two complex numbers that have the form a bi and a bi . Find the following
products of complex conjugates:
(a) 5 7 5 7i i (b) 10 10i i (c) 3 8 3 8i i
(d) What's true about the product of two complex conjugates?
A l g e b r a 2 U n i t 6 - Quadratic Functions and Their Algebra
Unit 6 Lesson 10 – Complex Numbers Page 205
HOMEWORK (cont.)
5. Show that the product of a bi and a bi is the purely real number 2 2a b .
6. The product of 8 2i and its conjugate is equal to
(1) 64 4i (3) 68
(2) 60 (4) 60 4i
7. The complex computation 6 2 6 2 3 4 3 4i i i i can be simplified to
(1) 15 (3) 10
(2) 39 (4) 35
8. Perform the following complex calculation. Express your answer in simplest a bi form.
8 5 3 2 4 4i i i i
9. Perform the following complex calculation. Express your answer in simplest a bi form.
7 3 5 4 2 6 7i i i
10. Simplify the following complex expression. Write your answer in simplest a bi form.
2 2
5 2 2i i
A l g e b r a 2 U n i t 6 - Quadratic Functions and Their Algebra
Unit 6 Lesson 11 –Solving Quad. Equations w/complex solutions Page 206
UNIT 6 LESSON 11
SOLVING QUADRATIC EQUATIONS WITH COMPLEX SOLUTIONS
As we saw in the last unit, the roots or zeroes of any quadratic equation can be found using the quadratic formula:
2 4
2
b b acx
a
Since this formula contains a square root, it is fair to investigate solutions to quadratic equations now when the
quantity 2 4b ac , known as the discriminant, is negative. Up to this point, we would have concluded that if the
discriminant was negative, the quadratic had no (real) solutions. But, now it can have complex solutions.
Exercise #1: Use the quadratic formula to find all solutions to the following equation. Express your answers in
simplest a bi form. 2 4 29 0x x
As long as our solutions can include complex numbers, then any quadratic equation can be solved for two roots.
Exercise #2: Solve each of the following quadratic equations. Express your answers in simplest a bi form.
(a) 2 5 30 7 10x x x (b)
2 16 15 10 4x x x
A l g e b r a 2 U n i t 6 - Quadratic Functions and Their Algebra
Unit 6 Lesson 11 –Solving Quad. Equations w/complex solutions Page 207
There is an interesting connection between the x-intercepts (zeroes) of a parabola and complex roots with non-
zero imaginary parts. The next exercise illustrates this important concept.
Exercise #3: Consider the parabola whose equation is 2 6 13y x x .
Exercise #4: Use the discriminant of each of the following quadratics to determine whether it has x-intercepts.
(a) 2 3 10y x x (b)
2 6 10y x x (c) 22 3 5y x x
Exercise #5: Which of the following quadratic functions, when graphed, would not cross the x-axis?
(1) 22 5 3y x x (3)
24 4 5y x x
(2) 2 6y x x (4)
23 13 4y x x
(a) Algebraically find the x-intercepts of this
parabola. Express your answers in simplest a bi
form.
(b) Using your calculator, sketch a graph of the
parabola on the axes below. Use the window
indicated.
(c) From your answers to (a) and (b), what can be
said about parabolas whose zeroes are complex
roots with non-zero imaginary parts?
y
x
20
10
A l g e b r a 2 U n i t 6 - Quadratic Functions and Their Algebra
Unit 6 Lesson 11 –Solving Quad. Equations w/complex solutions Page 208
UNIT 6 LESSON 11 SOLVING QUADRATIC EQUATIONS WITH COMPLEX SOLUTIONS
HOMEWORK
1. Solve each of the following quadratic equations. Express your solutions in simplest a bi form. Check.
(a) 2 4 20 12 5x x x (b)
2 1x x
(c) 22 25 27 15 10x x x (d)
28 36 24 12 5x x x
(e) 2 6 15 8 2x x x (f)
24 38 50 10 35x x x
A l g e b r a 2 U n i t 6 - Quadratic Functions and Their Algebra
Unit 6 Lesson 11 –Solving Quad. Equations w/complex solutions Page 209
HOMEWORK (cont.)
2. Which of the following represents the solution set to the equation 2 2 2 0x x ?
(1) 1 or 2x (3) 2x i
(2) 1 2x i (4) 1x i
3. The solutions to the equation 2 6 11 0x x are
(1) 3 2x i (3) 6 11x i
(2) 3 2 2x i (4) 6 2 11x i
4. Using the discriminant, 2 4b ac , determine whether each of the following quadratics has real or imaginary
zeroes.
(a) 22 7 6y x x (b)
23 2 1y x x (c) 2 8 14y x x
(d) 22 12 26y x x (e)
22 6 5y x x (f) 24 4 1y x x
5. Which of the following quadratics, if graphed, would lie entirely above the x-axis? Try to use the discriminant
to solve this problem and then graph to check.
(1) 22 21y x x (3)
2 4 7y x x
(2) 2 6y x x (4)
2 10 16y x x
6. For what values of c will the quadratic 2 6y x x c have no real zeroes? Set up and solve an inequality
for this problem.
A l g e b r a 2 U n i t 7 - Graphic Characteristics of Functions
Unit 7 Lesson 1 – Even and Odd Functions Page 210
U n i t 7 - Graphic Characteristics of Functions
LESSON 1 - EVEN AND ODD FUNCTIONS Even Functions
A function is “EVEN” when its graph has symmetry about the y-axis (like a reflection).
They were named "even" functions because the functions x2, x4, x6, x8, etc. are symmetric across the
y-axis, but there are other functions that behave like that too, such as:
For every point (x,y) on the graph, (-x, y) is also on the graph.
An even exponent does not necessarily make an even function, for example y = (x+4)2 is not an even
function. Sketch it and show why.
Odd Functions
A function is "odd" when it looks the same upside down, in other words, it is symmetric with respect to the
origin, or has 180º rotational symmetry.
For every point (x,y) on the graph, (-x, -y) is also on the graph.
They were named "odd" because the functions x, x3, x5, x7, etc. have origin symmetry, but there are other
functions that have 180º rotational symmetry:
A l g e b r a 2 U n i t 7 - Graphic Characteristics of Functions
Unit 7 Lesson 1 – Even and Odd Functions Page 211
An odd exponent does not necessarily make an odd function, for example x3+1 is not an odd function.
Sketch it and show why.
Neither Odd nor Even Don't be misled by the names "odd" and "even" ... they are just names ... and a function does not
have to be even or odd.
In fact most functions are neither odd nor even. For example,
Example 1: Decide if the following graphs are even, odd, or neither. Explain.
(F) (G) (H)
A l g e b r a 2 U n i t 7 - Graphic Characteristics of Functions
Unit 7 Lesson 1 – Even and Odd Functions Page 212
2. Graph f(x) = -x2 + 7 and tell if the function is odd, even, or neither.
3. Graph g(x) = 0.5x3 – 5x and tell if the function is odd, even, or neither.
4. Graph h(x) = 2x - 6 tell if the function is odd, even, or neither.
A l g e b r a 2 U n i t 7 - Graphic Characteristics of Functions
Unit 7 Lesson 1 – Even and Odd Functions Page 213
UNIT 7 LESSON 1 - EVEN AND ODD FUNCTIONS
HOMEWORK
1. Half of the graph of f x is shown below. Sketch the other half based on the function type.
(a) Even (b) Odd
2. Sketch the graph of the function and determine if the function is even, odd, or neither.
a. xxf2
3)( b. 4)( 2 xxf
c. xxxf 4)( 2 d. 23)( 23 xxxf
e. 1)( 2 xxg f. xxg 1)(
y
x
y
x
A l g e b r a 2 U n i t 7 - Graphic Characteristics of Functions
Unit 7 Lesson 1 – Even and Odd Functions Page 214
HOMEWORK (cont.)
g. 2
3
)( xxg h. 3 1)( xxf
j. 3)( xxf k. 2)( xxf
3. If g x is an odd, one-to-one function and if 7 2g , then which of the following points must lie on the
graph of the inverse of g x , 1g x . Explain how you made your choice.
(1) 7, 2 (3) 2, 7
(2) 2, 7 (4) 7, 2
4. Which of the following functions is even?
(1) 2 4y x x (3) 29y x
(2) 6y x (4) 4xy
5. The function 24 2x
f xx
is either even or odd. Determine which.
6. Even functions have symmetry across the y-axis. Odd function have symmetry across the origin. Can a
function have symmetry across the x-axis? Why or why not?
A l g e b r a 2 U n i t 7 - Graphic Characteristics of Functions
Unit 7 Lesson 2 – Transformation of Functions Page 215
UNIT 7 LESSON 2 - TRANSFORMATION OF FUNCTIONS
Transformations of Graphs of Absolute Value and Quadratic Functions (use colored pencils)
1. a. Sketch the graph of xy
b. Sketch the graph of 4 xy
c. Sketch the graph of 5 xy
d. Describe the graph of axy
in terms of the graph of xy
2. a. Sketch the graph of xy
b. Sketch the graph of 3 xy
c. Sketch the graph of 6 xy
d. Describe the graph of axy
in terms of the graph of xy
3. a. Sketch the graph of xy
b. Sketch the graph of xy
c. Describe the graph of xy
in terms of the graph of xy
A l g e b r a 2 U n i t 7 - Graphic Characteristics of Functions
Unit 7 Lesson 2 – Transformation of Functions Page 216
4. a. Sketch the graph of xy
b. Sketch the graph of xy 2
c. Sketch the graph of xy2
1
d. Describe the graph of xay
in terms of the graph of xy
5. a. Sketch the graph of 2xy
b. Sketch the graph of 22xy
c. Sketch the graph of
2
2
1
xy
d. Describe the graph of 2)(axy
in terms of the graph of 2xy
Recall transformations, such as line reflections, translations, and dilations. These transformations can be applied to basic functions. Parent Functions and Transformations There are certain basic functions whose graphs should be easily recognizable. Transformations will be applied to the graphs of these. These basic functions are also called parent functions. Below is a set of parent functions that will be used with transformations.
f(x) = x² f(x) = x³ f(x) = x f(x) = log2(x)
A l g e b r a 2 U n i t 7 - Graphic Characteristics of Functions
Unit 7 Lesson 2 – Transformation of Functions Page 217
Applying transformations to f(x) = log2(x)
Two different translations, or shifts, can be applied to a function – a vertical shift or a horizontal shift.
A vertical shift is when a constant is added/subtracted to f(x) = log2(x) as follows:
f(x) + 2 = log2(x) + 2 f(x) – 1 = log2(x) – 1
shifted up 2 shifted down 1
A horizontal shift happens when a constant is added/subtracted to the x inside parentheses: Note the difference from above!
f(x+3) = log2(x+3) f(x–2) = log2(x–2) shifted 3 to the left shifted 2 to the right Two different reflections can be applied: in the x-axis (negate the y-values) or in the y-axis (negate the x-values).
–f(x) = – log2(x) f(–x) = log2(–x)
A l g e b r a 2 U n i t 7 - Graphic Characteristics of Functions
Unit 7 Lesson 2 – Transformation of Functions Page 218
A vertical dilation occurs when the y-values of each point are multiplied by a constant.
3f(x) = 3x² is a vertical STRETCH because |a| >1 (“taller”) (note: If |a| <1 it would be “shorter”) Also known as a vertical compression
A horizontal dilation occurs when the x-values of each point are divided by a constant.
xf
3
1 =
2
3
1
x is a horizontal STRETCH because |a| <1 (“fatter”)
xf 3 = 23x is a horizontal compression, SHRINK because |a| > 1 (“skinnier”)
A l g e b r a 2 U n i t 7 - Graphic Characteristics of Functions
Unit 7 Lesson 2 – Transformation of Functions Page 219
Performing transformations with functions and relations Know: f(x + c) f(x) + c f(-x) -f(x) cf(x) f(cx)
Given the function f(x) state the transformation. a. f(x – 3) b. f(x) + 2 c. 2f(x) d. f(-x) 1. Given the graph f(x) shown below, sketch the graphs of each and describe the transformation:
a. f(x) + 5 b. f(x – 4) f(x) f(x) c. ‒f(x) d. f(-x) e. 3f(x) g. f(½x) Function? _________ One-to one? _________ Original f(x) Domain and Range____________ 2. Given the graph f(x) shown below, sketch the graphs of each and describe the transformation: a. f(x) + 6 b. f(x + 5) c. f(-x) f(x) f(x) d. –f(x) e. ½ f(x) g. f(x – 1) + 5 Function? _________ One-to one? _________ Original f(x) Domain and Range____________
A l g e b r a 2 U n i t 7 - Graphic Characteristics of Functions
Unit 7 Lesson 2 – Transformation of Functions Page 220
3. Given the graph f(x) shown below, sketch the graphs of each and describe the transformation: a. f(x) – 4 b. f(x + 6) c. f(-x) f(x) f(x)
d. –f(x) e. 2f(x) g. f(x – 7) + 8 Function? _________ One-to one? _________ Original f(x) Domain and Range____________
4. Given the graph f(x) = x shown below, sketch the graphs of each and describe the
transformation: f(x) f(x) a. f(x) – 8 b. f(x + 6) c. f(-x) d. –f(x) e. ½f(x) g. f(x – 1) + 6 Function? _________ One-to one? _________ Original f(x) Domain and Range____________
A l g e b r a 2 U n i t 7 - Graphic Characteristics of Functions
Unit 7 Lesson 2 – Transformation of Functions Page 221
UNIT 7 LESSON 2 - TRANSFORMING FUNCTIONS
HOMEWORK
1. Given the function f x shown graphed on the grid, create a
graph for each of the following functions and label on the grid.
(a) 2g x f x
(b) 3h x f x
(c) 1 4k x f x
2. If the quadratic function f x has a turning point at 3, 7 then where does the quadratic function g defined
by 4 5g x f x have a turning point?
(1) 7,12 (3) 4, 5
(2) 1,12 (4) 4, 5
3. If the domain of f x is 3 9x and the range of f x is 2 15y , then which of the following
statements is correct about the domain and range of 2 8g x f x ?
(1) Its domain is 1 11x and its range is 10 23y .
(2) Its domain is 5 7x and its range is 6 7y .
(3) Its domain is 1 11x and its range is 6 7y .
(4) Its domain is and its range is 10 23y .
4. Which of the following equations would represent the graph of the parabola 23 4 1y x x after a
reflection in the x-axis?
(1) 23 4 1y x x (3) 23 4 1y x x
(2) 23 4 1y x x (4) 23 4 1y x x
y
x
A l g e b r a 2 U n i t 7 - Graphic Characteristics of Functions
Unit 7 Lesson 2 – Transformation of Functions Page 222
HOMEWORK (cont.)
5. The graph of 210y x represents the graph of 2y x after
(1) a vertical shift upwards of 10 units followed by a reflection in the x-axis.
(2) a reflection in the x-axis followed by a vertical shift of 10 units upward.
(3) a leftward shift of 10 units followed by a reflection in the y-axis.
(4) a reflection across the x-axis followed by a rightward shift of 10 units.
6. If 22 5 3f x x x and g x is the reflection of f x across the y-axis, then an equation of g is which
of the following?
(1) 22 5 3g x x x (3) 22 5 3g x x x
(2) 22 5 3g x x x (4) 22 5 3g x x x
7. If the function 4y f x were graphed, it would represent which of the following transformations to the
graph of y f x ?
(1) A rightward shift of 4 units followed by a reflection in the x-axis.
(2) A rightward shift of 4 units followed by a reflection in the y-axis.
(3) A downward shift of 4 units followed by a reflection in the x-axis.
(4) A leftward shift of 4 units followed by a reflection in the y-axis.
8. After a reflection in the x-axis, the parabola 2 4y x would have the equation
(1) 2 4y x (3) 24y x
(2) 2 4y x (4) 2 8y x
9. If the point 6,10 lies on the graph of y f x then which of the following points must lie on the graph
of 1
2y f x ?
(1) 3, 5 (3) 6, 5
(2) 3,10 (4) 12, 20
10. If the function h x is defined as vertical stretch by a factor of 2 followed by a reflection in the x-axis of the
function f x then h x
(1) 2 f x (3) 1
2f x
(2) 1
2f x (4) 2 f x
A l g e b r a 2 U n i t 7 - Graphic Characteristics of Functions
Unit 7 Lesson 3 – Definition of a Parabola , part 1 Page 223
UNIT 7 LESSON 3 – GEOMETRIC DEFINITION OF A PARABOLA, PART 1
Materials: 1 sheet patty paper, 1 pencil, 3 colored pencils, ruler
Use ruler to draw a straight line one ruler width from the bottom of the patty paper
Fold paper in half, folding line upon itself making a crease, mark a point above the line on this crease
Make several creases in which the line coincides with the point
Using a blue pencil outline the shape that the creases form
A parabola is a special curve shaped like an arch.
Any point on a parabola is at an equal distance from…..
*a fixed point, the _______________ and
* a fixed straight line, ___________________
The axis of symmetry is the line that divides a parabola into two parts that are mirror images. The vertex of a
parabola is the point at which the parabola intersects the axis of symmetry.
A l g e b r a 2 U n i t 7 - Graphic Characteristics of Functions
Unit 7 Lesson 3 – Definition of a Parabola , part 1 Page 224
DEFINITION OF A PARABOLA
A parabola is the set of all points (x,y) in a plane that are equidistant from a fixed line (directrix) and a
fixed point (focus) not on the line.
The midpoint between the focus and the directrix is called the vertex, and the line passing through the focus and
the vertex is called the axis of symmetry. The directed distance from the focus to the vertex is p. This new variable p is one you'll need to be able to work with when writing equations of parabolas; it represents the distance between the vertex and the focus, and also the distance between the vertex and the directrix so 2p is the distance between the focus and the directrix.
The standard form of the equation of a parabola with a vertical axis of symmetry when the vertex (h,k) and the
p value are known is khxp
y 2)(4
1
When the axis of symmetry is a horizontal axis, the variables switch and we get:
The important difference in the two equations is in which variable is squared: for regular (vertical) parabolas, the x part
is squared; for sideways (horizontal) parabolas, the y part is squared.
For each of the following equations of a parabola (a) state whether it opens up or down, (b) vertex, (c) p-value,
(d) focus, and (e) find the equation of the directrix.
1. 4)3(12
1 2 xy
(a) Opens up or down
(b) Vertex
(c) p-value
(d) Focus
(e) Equation of directrix
If lead coefficient is positive: open up , if negative: open down
If lead coefficient is positive: open right , if negative: open left
"regular", or vertical, parabola; the focus "inside" the parabola, the directrix below the graph, the axis of symmetry passing through the focus and the vertex
"sideways", or horizontal, parabola; the focus "inside" the parabola, the directrix to the left of the graph, the axis of symmetry passing through the focus and the vertex
A l g e b r a 2 U n i t 7 - Graphic Characteristics of Functions
Unit 7 Lesson 3 – Definition of a Parabola , part 1 Page 225
2. 2)1(4
16 xy
(a) Opens up or down
(b) Vertex
(c) p-value
(d) Focus
(e) Equation of directrix
3. 2)7()8(8 xy
(a) Opens up or down
(b) Vertex
(c) p-value
(d) Focus
(e) Equation of directrix
4. 2)3( 2 yx
(a) Opens up or down
(b) Vertex
(c) p-value
(d) Focus
(e) Equation of directrix
A l g e b r a 2 U n i t 7 - Graphic Characteristics of Functions
Unit 7 Lesson 3 – Definition of a Parabola , part 1 Page 226
UNIT 7 LESSON 3 – DEFINITION OF A PARABOLA PART 1
HOMEWORK For each of the following equations of a parabola (a) state whether it opens up or down, (b) vertex, (c) p-value,
(d) focus, and (e) find the equation of directrix.
1. 1)3(8
1 2 xy
(a) Opens up or down
(b) Vertex
(c) p-value
(d) Focus
(e) Equation of directrix
2. 2)4(20
14 xy
(a) Opens up or down
(b) Vertex
(c) p-value
(d) Focus
(e) Equation of directrix
3. 2)4()6(12 xy
(a) Opens up or down
(b) Vertex
(c) p-value
(d) Focus
(e) Equation of directrix
A l g e b r a 2 U n i t 7 - Graphic Characteristics of Functions
Unit 7 Lesson 3 – Definition of a Parabola , part 1 Page 227
HOMEWORK (cont.)
4. 4)1(2 2 yx
(a) Opens up or down
(b) Vertex
(c) p-value
(d) Focus
(e) Equation of directrix
5. Write the equation of a vertical parabola whose vertex is (-1,4) and p-value is 3.
6. Write the equation of a parabola whose vertex is (-1,4) and focus is (-1,6).
7. Write the equation of a parabola whose vertex is (3,1) and focus is (3,4).
8. Write the equation of a parabola whose vertex is (2,-3) and focus is (2,-5).
A l g e b r a 2 U n i t 7 - Graphic Characteristics of Functions
Unit 7 Lesson 4 – Definition of a Parabola , part 2 Page 228
UNIT 7 LESSON 4 – GEOMETRIC DEFINITION OF A PARABOLA, PART 2 Write the general equation of a parabola that has a vertical axis of symmetry.
Write the general equation of a parabola that has a horizontal axis of symmetry.
Exercises #1-4:
For each of the following equations of a parabola (a) state whether it opens right, left, up or down, (b) vertex,
(c) p-value, (d) focus, and (e) find the equation of the directrix.
1. 3)2(4
1 2 yx
(a) Opens right or left
(b) Vertex
(c) p-value
(d) Focus
(e) Equation of directrix
2. 2)2(16
15 yx
(a) Opens
(b) Vertex
(c) p-value
(d) Focus
(e) Equation of directrix
A l g e b r a 2 U n i t 7 - Graphic Characteristics of Functions
Unit 7 Lesson 4 – Definition of a Parabola , part 2 Page 229
3. 2)2()6(4 yx
(a) Opens
(b) Vertex
(c) p-value
(d) Focus
(e) Equation of directrix
4. 248)1( 2 yx
(a) Opens
(b) Vertex
(c) p-value
(d) Focus
(e) Equation of directrix
5. Write an equation for the set of points which are equidistant from the origin and the line x = -2.
6. Write an equation for the set of points which are equidistant from (4,-2) and the line y = 4.
7. Write an equation for the set of points which are equidistant from (0,2) and the x-axis.
A l g e b r a 2 U n i t 7 - Graphic Characteristics of Functions
Unit 7 Lesson 4 – Definition of a Parabola , part 2 Page 230
UNIT 7 LESSON 4 – DEFINITION OF A PARABOLA PART 2
HOMEWORK
For each of the following equations of a parabola (a) state whether it opens right, left, up or down, (b) vertex,
(c) p-value, (d) focus, and (e) find the equation of directrix.
1. 2)3(12
1 2 yx
(a) Opens
(b) Vertex
(c) p-value
(d) Focus
(e) Equation of directrix
2. 2)3(20
16 yx
(a) Opens
(b) Vertex
(c) p-value
(d) Focus
(e) Equation of directrix
3. 1)1( 2 yx
(a) Opens
(b) Vertex
(c) p-value
(d) Focus
(e) Equation of directrix
A l g e b r a 2 U n i t 7 - Graphic Characteristics of Functions
Unit 7 Lesson 4 – Definition of a Parabola , part 2 Page 231
HOMEWORK (cont.)
4. )5(8)3( 2 xy
(a) Opens
(b) Vertex
(c) p-value
(d) Focus
(e) Equation of directrix
5. Write the equation of the parabola with focus (1,6) and directrix x = 10.
6. Write the equation of the parabola with focus (-2,0) and directrix the y-axis.
7. Write the equation of a parabola whose focus is (3,1) and directrix is y = 5.
8. Write the equation of a parabola whose focus is (2,3) and directrix is y = -3
A l g e b r a 2 U n i t 7 - Graphic Characteristics of Functions
Unit 7 Lesson 5 – Center-Radius Equation of Circles Page 232
LESSON 5 - CENTER-RADIUS EQUATION OF CIRCLES
Exercise #1: Which of the following equations would have a center of 3, 6 and a radius of 3?
(1) 2 2
3 6 9x y (3) 2 2
3 6 3x y
(2) 2 2
3 6 9x y (4) 2 2
3 6 3x y
Exercise #2: For each of the following equations of circles, determine both the circle’s center and its radius. If
its radius is not an integer, express it in decimal form rounded to the nearest tenth.
(a) 2 2
2 7 100x y (b) 2 2
5 8 4x y (c) 2 2 121x y
(d) 2 2
1 2 1x y (e) 22 3 49x y (f)
2 26 5 18x y
(g) 2 2 64x y (h) 2 2
4 2 20x y (i) 2 2 57x y
Exercise #3: Write equations for circles A and B shown below. Show how you arrive at your answers.
THE CENTER-RADIUS EQUATION OF A CIRCLE
A circle whose center is at and whose radius is r is:
y
x
B
A
A l g e b r a 2 U n i t 7 - Graphic Characteristics of Functions
Unit 7 Lesson 5 – Center-Radius Equation of Circles Page 233
Exercise #4 - Write each equation of a circle in center-radius form and identify the center and radius. a. x² + y² – 2x + 6y + 9 = 0 b. x² + y² – 10x – 6y + 25 = 0 c. 4x² + 4y² + 12x – 24y + 41 = 0 Systems of Equations Involving a Circle and a Line – Must solve for x and y - (x,y) Identify the ordered pairs which mark any intersection of a circle and a line. Here are three types of situations.
Exercise #5: Solve the following system of equations algebraically AND graphically:
42
4
22
yx
xy
y
x
A l g e b r a 2 U n i t 7 - Graphic Characteristics of Functions
Unit 7 Lesson 5 – Center-Radius Equation of Circles Page 234
LESSON 5 – CENTER-RADIUS EQUATION OF CIRCLES
HOMEWORK
1. Each of the following is an equation of a circle. State the circle’s center and radius. In the cases where the
radius is not an integer, give its value rounded to the nearest tenth.
(a) 2 2 144x y (b) 36)7()3( 22 yx (c) 2 2
5 1 64x y
(d) 2 2
2 9 100x y (e) 2 2 1x y (f) 22 5 25x y
(g) 2 2 50x y (h) 2 23 200x y (i)
2 26 6 20x y
2. Which of the following is true about a circle whose equation is 2 2
5 3 36x y ?
(1) It has a center of 5, 3 and an area of 12 .
(2) It has a center of 5, 3 and a diameter of 6.
(3) It has a center of 5, 3 and an area of 36 .
(4) It has a center of 5, 3 and a circumference of 12 .
3. Which of the following represents the equation of the circle shown graphed below?
(1) 2 2
2 3 16x y
(2) 2 2
2 3 4x y
(3) 2 2
2 3 4x y
(4) 2 2
2 3 16x y
4. By completing the square on each of the quadratic expressions, determine the center and radius of a circle
whose equation is shown below.
2 26 10 66x x y y
y
x
A l g e b r a 2 U n i t 7 - Graphic Characteristics of Functions
Unit 7 Lesson 5 – Center-Radius Equation of Circles Page 235
HOMEWORK (cont.)
5. Circles are described below by the coordinates of their centers, C, and one point on their circumference, A.
Determine an equation for each circle in center-radius form.
(a) 5, 2 and 11,10C A (b) 2, 5 and 3, 17C A (c) 5, 1 and 2, 5C A
6. Solve the following system of equations graphically.
2 2
2
25
5
x y
y x
7. Find the intersection of the circle 2 2 29 and 3x y y x algebraically.
7. Jonas is designing a circular garden whose equation is 2 2 49x y . He wishes to place a walkway within the
garden at all points within the circle that satisfy the
inequality 2 2y . Graph the circle on the grid to the
right and shade in all points that represent the walkway.
y
x
y
x
A l g e b r a 2 U n i t 8 - Extension Lessons for Honors course
Unit 8 Lesson 1 – Log Rules Page 236
U n i t 8 - Extension Lessons for Honors course LESSON 1 – LOG RULES
Recall: Rules of Exponents Let a and b be positive integers and x and y real numbers:
Multiplication Rule ba xx ___________ ex. a4 ∙ a2 = _____________
Division Rule b
a
x
x ___________ ex. _____________
Power Rule bax )( ___________ ex. (a4)2 = _____________
Recall: Log form of an exponential equation Exponential Form Logarithmic Form 2³ = 8 log28 = 3 25 = 32 5 = Write each equation in logarithmic form.
62554 36216
Log Rules
1. Product Rule logb(AB) = _____________________ or ln(AB) = _________________
2. Quotient Rule logb
B
A= _____________________ or ln
B
A= __________________
3. Power Rule logb xA _____________________ or ln xA __________________ *Note* There is no rule for logb(A+B) or logb(A–B)
These rules can be combined in a single expression.
Also, always re-write radicals in exponential form ex. x = 2
1
x
3 x 5 2x
4
15
a
a
A l g e b r a 2 U n i t 8 - Extension Lessons for Honors course
Unit 8 Lesson 1 – Log Rules Page 237
Examples: A. Expand each expression using the above properties: 1. log4(4xy) _____________________________ 2. ln(x³) _____________________________
3. log2
a
16 ____________________________________________ 4. ln(a² x ) __________________________
5. log6
z
xy2
____________________________ 6. ln
4
2
z
yx ________________________________________
7. log5
2
25
xy ___________________________________________ 8. ln
e
1 ___________________________
B. Condense each expression using the above properties: 1. log4(4x) + log4(x) _________________________________ 2. 2ln(x) – 3ln(y) ________________________________
3. 3
1log(x) + 3log(y) – 4log(z) ________________________________
A l g e b r a 2 U n i t 8 - Extension Lessons for Honors course
Unit 8 Lesson 1 – Log Rules Page 238
LESSON 1 –LOG RULES
HOMEWORK
4. ln(a) + 2
1ln(b) – ( ln(c) + ln(d) ) ________________________________
5. log2(512) – log2(64) __________________________________ 6. ln(e) + ln(e) + ln(e) _____________________________ 7. log2(x² – 16) – log2(x – 4) _____________________________ Set A, do #s 24-35
A l g e b r a 2 U n i t 8 - Extension Lessons for Honors course
Unit 8 Lesson 1 – Log Rules Page 239
HOMEWORK (cont.)
Set B Expand and simplify when possible. 1. ln 4e 2. ln 4e2
3. ln (4e)2 4. ln e3
5. ln
e
1 6. ln ex
Condense.
7. 2
1log x + 5log y – 3log z 8. ln x - 2ln y
9. ln e + ln e 10. 2log x – ½ (log y + log z)
A l g e b r a 2 U n i t 8 - Extension Lessons for Honors course
Unit 8 Lesson 2 – Pascal’s Triangle and Binomial Expansion Page 240
LESSON 2 – PASCAL’S TRIANGLE AND BINOMIAL EXPANSION From earlier work in algebra, you are familiar with the term binomial, an expression consisting of two terms, such as x + y, or x² – 5. These are binomials because they all show two monomial terms being combined by addition or subtraction. Binomial expansion is taking that two-term expression and raising it to successive powers, as shown below
(a + b) 0 = 1 (a + b) 1 = 1a + 1b (a + b) 2 = 1a² + 2ab + 1b² (a + b) 3 = 1a3 + 3a²b1 + 3a1b² + 1b3 (a + b) 4 = 1a4 + 4a3b1 + 6a²b² + 4a1b3 + 1b4 (a + b) 5 = 1a5 + 5a4b1 + 10a3b² + 10a²b3 + 5a1b4 + 1b5
Notice that a binomial raised to the nth power has n + 1 terms. Look at the coefficients of each term in the expansion above and see if you notice a pattern.
This arrangement of numbers is known as Pascal’s triangle after the French mathematician and physicist Blaise Pascal. The following formula can be used to expand a binomial.
(x + y) n = nCoxny0 + nC1xn-1y1 +nC2xn-2y2 + …+ nCn-1x1yn-1 + nCnx0yn 1. Expand (m + 2)3
A l g e b r a 2 U n i t 8 - Extension Lessons for Honors course
Unit 8 Lesson 2 – Pascal’s Triangle and Binomial Expansion Page 241
2. Expand (a – 3)4 3. Expand (2x + 3y)5 4. Expand (3x – )6
A l g e b r a 2 U n i t 8 - Extension Lessons for Honors course
Unit 8 Lesson 2 – Pascal’s Triangle and Binomial Expansion Page 242
LESSON 2 –PASCAL’S TRIANGLE AND BINOMIAL EXPANSION
HOMEWORK
1. Expand (2 – x)5 2. Expand (x + 5y)4 3. Expand (a – 3)5
A l g e b r a 2 U n i t 8 - Extension Lessons for Honors course
Unit 8 Lesson 2 – Pascal’s Triangle and Binomial Expansion Page 243
HOMEWORK (cont.)
4. Expand (a – 1)6 5. Expand (2x + y)5 6. Expand (a – b)6
A l g e b r a 2 U n i t 8 - Extension Lessons for Honors course
Unit 8 Lesson 3 – Binomial Expansion , cont. Page 244
LESSON 3 – BINOMIAL EXPANSION, CONT.
Sometimes you are asked for only one term of a binomial expansion. You can expand the entire expression and select the particular term: 1. Find the third term of the expansion (3x + 2)5 2. Find the middle term of the expansion (2m – 5)6
3. Find the last term of the expansion (3x + 2y)5
A l g e b r a 2 U n i t 8 - Extension Lessons for Honors course
Unit 8 Lesson 3 – Binomial Expansion , cont. Page 245
4. Find the middle term of the expansion (2m – 1)8
5. Find the fourth term of the expansion (x + 2i)5 6. Find the middle term of the expansion (m – 3i)6
A l g e b r a 2 U n i t 8 - Extension Lessons for Honors course
Unit 8 Lesson 3 – Binomial Expansion , cont. Page 246
LESSON 3 – BINOMIAL EXPANSION, CONT.
HOMEWORK
1. Find the last term of the expansion (3x - 4)7 2. Find the third term of the expansion (2i – 6)5
3. Find the middle term of the expansion (3x - 2y)6
A l g e b r a 2 U n i t 8 - Extension Lessons for Honors course
Unit 8 Lesson 3 – Binomial Expansion , cont. Page 247
HOMEWORK (cont.)
4. Find the third term of the expansion (x + 5y)5 5. Find the last term of the expansion (3m – i)4
6. Find the fifth term of the expansion (2x + 2y)6
A l g e b r a 2 U n i t 9 - Regression
Unit 9 Lesson 1 - Linear Regression and Predictions Page 248
U n i t 9 - Regression
LESSON 1 - LINEAR REGRESSION AND PREDICTIONS
A graph used to determine whether there is a relationship between paired data is called a ____________________.
Recall how to view scatter plots and writing an equation of a line or curve of best fit using your
graphing calculator: Calculator Steps: 1. Key the 2 sets of data into L1 and L2 Stat Edit 2. Turn stat plot 1 on. 2nd y= 3. Zoom 9 – look at the scatter plot and note its shape. 4. Turn diagnostic on if not on already. (CATALOG: 2nd 0 and scroll down) 5. Stat Calc (4:LinReg(ax+b)) for a Linear Regression (if points are in a straight line) Under Store RegEQ, press VARS > Y-VARS > 1:Function > Y1
Also recall: The number that is used to measure the strength and direction of a linear relationship is called the correlation coefficient, (denoted r). -1 ≤ r ≤ +1 If we have the ‘Diagnostic On’ the calculator will compute the correlation coefficient (must go to CATALOG to turn Diagnostic On). A linear correlation coefficient close to zero signifies no significant linear correlation while a correlation coefficient close to +1 or -1 indicates that the points in the scatter plot are close to the calculated line of best fit, or are strongly correlated linearly. Linear Regression example #1
Emma recently purchased a new car. She decided to keep track of how many gallons of gas she used on
five of her business trips. The results are shown in the table below.
(a) Write a linear regression function g(m) for these data where miles
driven, m, is the input, or independent variable. (Round all values to
the nearest hundredth). (b) State the value of the correlation
coefficient, r, to the nearest thousandth. (c) How many gallons would
be used if 225 miles are driven (m=225)? (d) How far could Emma
drive on her full tank of gas, if her tank holds 16 gallons?
a)
b)
c) d)
A l g e b r a 2 U n i t 9 - Regression
Unit 9 Lesson 1 - Linear Regression and Predictions Page 249
In the previous example you found a function value (using an input of m=225 miles) between given input values (150 ≤ m ≤ 1000). This process is called interpolation. Often we want to use data collected about past events to predict the future. The process of using given
data values to approximate values outside the given range of values is called extrapolation. The same
procedure is used, just find g(m) for any m value within or outside the given domain. Linear Regression example #2 2. The table below shows the attendance at a museum in select years from 2007 to 2013.
(a) State the linear regression function a(t) represented by the data table when t = 7 is used to represent
the year 2007 and a(t) is used to represent the attendance. Round all values to the nearest hundredth. (b)
State the correlation coefficient to the nearest hundredth and (c) determine whether the data suggest a
strong or weak association. (d) What does the model predict the attendance to be this year, in 2016?
(e) What year would the attendance be 10.1 million? Linear Regression example #3
Write a linear regression function f(t) for the following scatter plot: (Round to nearest hundredth)
f(t)
t
A l g e b r a 2 U n i t 9 - Regression
Unit 9 Lesson 1 - Linear Regression and Predictions Page 250
LESSON 1 − LINEAR REGRESSION AND PREDICTIONS
HOMEWORK
1. In a math class of ten students, the teacher wanted to determine how a homework grade influenced a
student’s performance on the subsequent test. The homework grade and subsequent test grade for each
student are given in the accompanying table.
a) Give the equation of the linear regression line for this set of data. (Round to nearest hundredth)
b) State the correlation coefficient to the nearest hundredth
c) A new student comes to the class and earns a homework grade of
78. Based on the equation in part a, what grade would the teacher
predict the student would receive on the subsequent test, to the
nearest integer?
d) What would the homework grade need to be in order to get a test
grade of 80?
2. The data table below shows water temperatures at various depths in an ocean.
a) Write the linear regression equation for this set of data,
rounding all values to the nearest thousandth.
b) State the correlation coefficient to the nearest
thousandth
c) Using this equation, what would the depth of the water
be for a water temperature reading of 10ºC.
d) Using this equation, predict the temperature (ºC), to the nearest integer, at a water depth of 255
meters.
A l g e b r a 2 U n i t 9 - Regression
Unit 9 Lesson 1 - Linear Regression and Predictions Page 251
HOMEWORK (cont.)
3. The number of newly reported crime cases in a county in New York State is shown in the accompanying
table.
(a) Write the linear regression function, c(t), that represents this set of data.
(Let t = 0 represent 2009.)
(b) State the correlation coefficient to the nearest hundredth
(c) Using this equation, find the projected number of new cases for 2019, rounded to the nearest whole number.
Year New Cases
2009 440
2010 457
2011 369
2012 351
A l g e b r a 2 U n i t 9 - Regression
Unit 9 Lesson 2 - Non-Linear Regression and Predictions Page 252
LESSON 2 - NON-LINEAR REGRESSION AND PREDICTIONS
Using your knowledge and a graphing calculator, draw a sample graph of each of the following functions. Describe each graph. Quadratic function Cubic function
Exponential function Logarithmic function Draw and describe the graph of a model that best fits each set of data.
A l g e b r a 2 U n i t 9 - Regression
Unit 9 Lesson 2 - Non-Linear Regression and Predictions Page 253
Not all bivariate data can be represented by a linear function. Some data can be better approximated by a curve. There are a variety of non-linear functions that can be applied to non-linear data. Once you understand how to do linear regression with your calculator, you already know the technical
mechanics to perform other regressions in the [STAT] [CALC] menu. The most common regressions
correspond to the function families.
• QuadReg - Quadratic model fits your points on the parabola ax2 + bx + c
• CubicReg - Cubic model fits your points on the curve ax3 + bx2 + cx + d
• LnReg - Natural Log model fits points on the curve a + bln(x)
• ExpReg - Exponential model fits points on the curve a*bx
Use your best judgment when choosing a function family to model a given set of data and deciding how good a fit the model is.
A l g e b r a 2 U n i t 9 - Regression
Unit 9 Lesson 2 - Non-Linear Regression and Predictions Page 254
1. Key the 2 sets of data into L1 and L2 Stat Edit 2. Turn stat plot 1 on. 2nd y= 3. Zoom 9 – look at the scatter plot and note its shape. 4. Stat Calc 5. QuadReg or 6:CubicReg or 9:LnReg or 0:ExpReg (you decide based on scatter plot) Under Store RegEQ, press VARS > Y-VARS > 1:Function > Y1
Non-Linear Regression Example 1 Given the following data about SAT scores and number of hours slept the night before, use an
appropriate function family to produce a reasonable model. Defend your choice of regression.
1. What is the perfect amount of sleep to get before the SATs?
2. Calculate the score you are predicted to get if you get 5 hours of
sleep.
3. What is the domain of the model?
4. The average SAT score is about 1500. According to the model, what amount of sleep predicts this
score? Enter Y2 =1500. Then 2nd TRACE 5:intersect.
Does this number represent the average number of hours that people sleep before the SATs?
Example 2
The table below shows the concentration of a drug in a patient's bloodstream t hours after it was
administered.
Time (hours) .5 1 1.5 2 2.5 3
Concentration (mg/cc) .16 .19 .2 .19 .18 .17
Find a cubic model to fit this data.
When was the concentration .1825 mg/cc? Enter Y2 =.1825 Then 2nd TRACE 5:intersect.
# Hours Slept SAT Score
8.5 1840
10.9 1510
9.1 1900
7.5 2070
7.2 1550
6.0 1720
2.3 840
5.5 1230
A l g e b r a 2 U n i t 9 - Regression
Unit 9 Lesson 2 - Non-Linear Regression and Predictions Page 255
Example 3 About a year ago, Joey watched an online video of a band and noticed that it had been viewed only 843
times. One month later, Joey noticed that the band’s video had 1708 views. Joey made the table below
to keep track of the cumulative number of views the video was getting online.
a) Write a regression equation that best models these data.
Round all values to the nearest hundredth. Justify your
choice of regression equation.
b) As shown in the table, Joey forgot to record the number of views after the second month. Use the
equation from part a to estimate the number of full views of the online video that Joey forgot to record.
Example 4 The table below shows the yield (in mg) of a chemical reaction in the first 6 minutes.
Time (minutes) 1 2 3 4 5 6
Yield (mg) 1.2 6.9 9.3 12.7 14.1 15.7
a) Use the scatter plot to find the best model to fit this data, y(m). (Round to nearest hundredth)
b) Using that model, determine in how many minutes will the yield be 20 mg.
5. Write an exponential equation for the graph shown:
Explain how you determined the equation.
A l g e b r a 2 U n i t 9 - Regression
Unit 9 Lesson 2 - Non-Linear Regression and Predictions Page 256
LESSON 2 - NON-LINEAR REGRESSION AND PREDICTIONS
HOMEWORK
1.Determine the type of regression model that best fits the data. a) b) c) d) e) 2. An application developer released a new app to be downloaded. The table below gives the number of
downloads for the first four weeks after the launch of the app.
a) Write an exponential function that models these data, d(w)
b) Use this model to predict how many downloads the developer would expect in the 26th week if this trend
continues. Round your answer to the nearest download.
c) Would it be reasonable to use this model to predict the number of downloads past one year? Explain your
reasoning 3. For the table below, a) determine a regression model that is the most appropriate for the data.
b) write the regression equation, rounding the coefficients to three decimal places.
x 4 7 3 8 6 5 6 3 9 4.5
y 10 7 15 9 5 6 6 14 14 8
A l g e b r a 2 U n i t 9 - Regression
Unit 9 Lesson 2 - Non-Linear Regression and Predictions Page 257
HOMEWORK (cont.)
4. The accompanying table shows wind speed and the corresponding wind chill factor when the air temperature is 10ºF.
a) Write the logarithmic regression equation for this set of data,
rounding coefficients to the nearest ten thousandth.
b) Using this equation, find the wind chill factor, to the nearest
degree, when the wind speed is 50 miles per hour.
c) Based on your equation, if the wind chill factor is 0, what is the
wind speed, to the nearest mile per hour?
5. The data collected by a biologist showing the growth of a colony of bacteria at the end of each hour are
displayed in the table below.
a) Write an exponential regression equation to model these data. Round all
values to the nearest thousandth.
b) Assuming this trend continues, use this equation to estimate, to the nearest
ten, the number of bacteria in the colony at the end of 7 hours.
6. Alice is in Wonderland and drinks a potion that approximately halves her height for each sip she takes, as
shown in the table below.
a. Write an exponential regression function that best models these data, h(s).
Round all values to the nearest hundredth
b. How many sips did she take if she is 2 inches tall?
c. How tall will she be if she has 6 sips?
# of sips Height (inches)
0 60
1 29
2 16
3 8
4 4.1