mathinstruction red hook ny · 2017-11-18 · unit 1 lesson 1 – variables, terms, and expressions...

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.



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 .



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



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



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



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


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



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.



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



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


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



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



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



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 ?


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



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?



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



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.


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?



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



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))?



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?


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



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



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



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.


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



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



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



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?


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



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



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



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?


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



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



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



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


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)



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



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



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


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.



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.



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.



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



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



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



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)


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



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.



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


form, 1 1y y m x x .

(b) Write the equation of this line in slope-

intercept form, . y mx b


form, 1 1y y m x x .

(b) Write the equation of this line in slope-

intercept form, . y mx b



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


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



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?


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




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



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


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.



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



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



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


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



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?



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



HOMEWORK (cont.)


2 5 35

3 4 31

3 2 2 23

x y z

x y z

x y z


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



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) _______________



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)



HOMEWORK (cont.)

2. 3. 4.

5. 6. 7.

8. 9. 10.


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



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



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.



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.


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

=



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



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



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) = _____________


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



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



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



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


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)



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



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



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


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



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



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



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


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.



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



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.



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)


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



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



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.



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


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.



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%



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



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).


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.



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.



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.



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



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



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



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


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



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



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



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


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



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



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



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.


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.



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.



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



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


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



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) 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.



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.



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.


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



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.



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?



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


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).



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.



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.



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?



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.


c. Write an equivalent function, C(t), it terms of the monthly rate of interest for the account.


(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.)



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.


c. Write an equivalent function, C(t), it terms of the daily rate of decrease for the town.




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.



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?



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



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



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



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, ...


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 =



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.



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.



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


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).



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).



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



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 .


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.



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 .



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.



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.


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:



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)



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



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.


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



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:



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?



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



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



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



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


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



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



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.



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


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



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



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



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


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



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



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



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


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.



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



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



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


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



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



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



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


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



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?



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



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.


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



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



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



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.


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



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



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



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


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



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.



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?



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


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



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



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



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:

http://www.google.com/url?sa=i&rct=j&q=&esrc=s&source=images&cd=&cad=rja&uact=8&ved=0ahUKEwj46M6Fxb7KAhVMVT4KHejkD10QjRwIBw&url=http://worksheets.tutorvista.com/12-basic-functions-worksheet.html&psig=AFQjCNGillROSrAkAJ-iMDfWSSss_QR-_w&ust=1453590501911869

https://www.mathsisfun.com/algebra/equation-symmetry.html

http://www.google.com/url?sa=i&rct=j&q=&esrc=s&source=images&cd=&cad=rja&uact=8&ved=0ahUKEwjk38bfx77KAhULFz4KHQ_WBkAQjRwIBw&url=http://www.johndcook.com/blog/2011/03/31/saved-by-symmetry/&psig=AFQjCNFQbNBURpHVIRD0DBXPxffC5tBTVQ&ust=1453591071659516





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)

http://www.google.com/url?sa=i&rct=j&q=&esrc=s&source=images&cd=&cad=rja&uact=8&ved=0ahUKEwigqvTNyb7KAhXJwj4KHUyRAGYQjRwIBw&url=http://www.softmath.com/tutorials-3/reducing-fractions/math-1100-exam-review-2.html&psig=AFQjCNGZeVMaeL16rH1J8iyMzEkb3F2prg&ust=1453591771981685

http://www.google.com/url?sa=i&rct=j&q=&esrc=s&source=images&cd=&cad=rja&uact=8&ved=0ahUKEwiwwZaOyb7KAhXIPT4KHQJLAU0QjRwIBw&url=http://ap-calculus.david-s.org/definite-integrals-involving-odd-functions/&psig=AFQjCNGkntwRu8McN-PU9AVIxB9kt2sP6w&ust=1453591647596452

http://www.google.com/url?sa=i&rct=j&q=&esrc=s&source=images&cd=&cad=rja&uact=8&ved=0ahUKEwiwwZaOyb7KAhXIPT4KHQJLAU0QjRwIBw&url=http://ap-calculus.david-s.org/definite-integrals-involving-odd-functions/&psig=AFQjCNGkntwRu8McN-PU9AVIxB9kt2sP6w&ust=1453591647596452

http://www.google.com/url?sa=i&rct=j&q=&esrc=s&source=images&cd=&cad=rja&uact=8&ved=0ahUKEwji_p6pyL7KAhUCTj4KHWjzCksQjRwIBw&url=http://math.tutorvista.com/algebra/quintic-function.html&psig=AFQjCNFQbNBURpHVIRD0DBXPxffC5tBTVQ&ust=1453591071659516



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.



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



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?


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


b. Sketch the graph of 3 xy

c. Sketch the graph of 6 xy

d. Describe the graph of axy



b. Sketch the graph of xy

c. Describe the graph of xy





b. Sketch the graph of xy 2

c. Sketch the graph of xy2

1

d. Describe the graph of xay


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)

https://www.google.com/url?sa=i&rct=j&q=&esrc=s&source=images&cd=&cad=rja&uact=8&ved=0ahUKEwiXopf07M_KAhVDFj4KHdIdAvcQjRwIBw&url=https://en.wikipedia.org/wiki/Logarithm&psig=AFQjCNHdsViKmSAqxOxH3nzVmTtbQ1KZiQ&ust=1454184152048993



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)

https://www.google.com/url?sa=i&rct=j&q=&esrc=s&source=images&cd=&cad=rja&uact=8&ved=0ahUKEwiXopf07M_KAhVDFj4KHdIdAvcQjRwIBw&url=https://en.wikipedia.org/wiki/Logarithm&psig=AFQjCNHdsViKmSAqxOxH3nzVmTtbQ1KZiQ&ust=1454184152048993



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”)

http://www.google.com/url?sa=i&rct=j&q=&esrc=s&source=images&cd=&cad=rja&uact=8&ved=0ahUKEwj44N38v77KAhXDMz4KHZYiDU0QjRwIBw&url=http://www.bbc.co.uk/schools/gcsebitesize/maths/algebra/transformationhirev2.shtml&bvm=bv.112454388,d.cWw&psig=AFQjCNF6eSfqF_qW2ygABSBKMuYx9Vzamg&ust=1453589152583805

http://www.google.com/url?sa=i&rct=j&q=&esrc=s&source=images&cd=&cad=rja&uact=8&ved=0ahUKEwi8xpu9wL7KAhUBHj4KHS7lAEYQjRwIBw&url=http://www.bbc.co.uk/schools/gcsebitesize/maths/algebra/transformationhirev2.shtml&bvm=bv.112454388,d.cWw&psig=AFQjCNF6eSfqF_qW2ygABSBKMuYx9Vzamg&ust=1453589152583805



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____________



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____________



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 .



(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



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


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.



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



2. 2)1(4

16 xy


(b) Vertex

(c) p-value

(d) Focus


3. 2)7()8(8 xy


(b) Vertex

(c) p-value

(d) Focus


4. 2)3( 2 yx


(b) Vertex

(c) p-value

(d) Focus




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


(b) Vertex

(c) p-value

(d) Focus


2. 2)4(20

14 xy


(b) Vertex

(c) p-value

(d) Focus


3. 2)4()6(12 xy


(b) Vertex

(c) p-value

(d) Focus




HOMEWORK (cont.)

4. 4)1(2 2 yx


(b) Vertex

(c) p-value

(d) Focus


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).



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


2. 2)2(16

15 yx

(a) Opens

(b) Vertex

(c) p-value

(d) Focus




3. 2)2()6(4 yx

(a) Opens

(b) Vertex

(c) p-value

(d) Focus


4. 248)1( 2 yx

(a) Opens

(b) Vertex

(c) p-value

(d) Focus


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.



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


2. 2)3(20

16 yx

(a) Opens

(b) Vertex

(c) p-value

(d) Focus


3. 1)1( 2 yx

(a) Opens

(b) Vertex

(c) p-value

(d) Focus




HOMEWORK (cont.)

4. )5(8)3( 2 xy

(a) Opens

(b) Vertex

(c) p-value

(d) Focus


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


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



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



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



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



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) ________________________________



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



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)


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



2. Expand (a – 3)4 3. Expand (2x + 3y)5 4. Expand (3x – )6



LESSON 2 –PASCAL’S TRIANGLE AND BINOMIAL EXPANSION

HOMEWORK

1. Expand (2 – x)5 2. Expand (x + 5y)4 3. Expand (a – 3)5



HOMEWORK (cont.)

4. Expand (a – 1)6 5. Expand (2x + y)5 6. Expand (a – b)6


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



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



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



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)

http://www.google.com/url?sa=i&rct=j&q=&esrc=s&source=images&cd=&cad=rja&uact=8&ved=0ahUKEwiV-vyFuu3KAhWJVT4KHfk4CmkQjRwIBw&url=http://blogs.mathworks.com/loren/2012/03/16/new-regression-capabilities-in-release-2012a/&psig=AFQjCNGnZytKiAzKQr59CpudfSjaN0R8hQ&ust=1455202527533094

https://www.google.com/url?sa=i&rct=j&q=&esrc=s&source=images&cd=&cad=rja&uact=8&ved=0ahUKEwiQt-6gu-3KAhWBFT4KHfbNDWgQjRwIBw&url=https://www.spcforexcel.com/knowledge/root-cause-analysis/scatter-diagrams&bvm=bv.113943665,d.cWw&psig=AFQjCNHHfwO548IbpPPyXc08wh_1Bd9U4Q&ust=1455202747720558



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

http://www.google.com/url?sa=i&rct=j&q=&esrc=s&source=images&cd=&cad=rja&uact=8&ved=0ahUKEwi2-4rExO3KAhUIMj4KHXHyDmUQjRwIBw&url=http://www.regentsprep.org/regents/math/algebra/ad4/PracPlot.htm&psig=AFQjCNHHfwO548IbpPPyXc08wh_1Bd9U4Q&ust=1455202747720558



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.



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


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.



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.

http://www.ck12.org/analysis/Function-Families



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



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.



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



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

mathinstruction red hook ny · 2017-11-18 · unit 1 lesson 1 – variables, terms, and expressions...

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