Calculus for Business Name:____________________________________UNIT 0- Review Material: Functions and Graphs

Date:_____________________________________

Objective: To review prerequisite Precalculus content.

LINEAR FUNCTIONS:

Forms of Linear Equations:

slope-intercept form: standard form: point-slope form:

1. Write the linear equation in slope intercept, standard, and point-slope form given the line passes through (5, 2) and (7, 9)

2. Write the equation of the horizontal line that passes through (-9, 2)

InterceptsFinding x-intercepts: replace y with zero and solveFinding y-intercepts: replace x with zero and solve

3. Find the intercepts of the following linear equations(if any):a. y = 3x + 2 b. 3x + 4y = 11

4. Find the intercepts of the following non-linear equations(if any):a. y=(x – 2)(x – 6) b. xy=9

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Parallel & Perpendicular LinesParallel lines have __________ slopes.Perpendicular lines have slopes that are ______________________ _______________________.

5. Write the linear equation in standard form given that the line passes through (-2, 10) and is parallel to

the graph of

6. Write the equation of the line that passes through (6, -5) and is perpendicular to the graph of

Intersection of Linear FunctionsMethods

Elimination Substitution Graphical1. Cancel a variable and solve

for the other.2. Sub in result and solve for

the other variable*Great method when neither equation is in y=mx+b form

1. Plug one equation in the other and solve for the variable

2. Sub in result and solve for the other variable

*Great method when given only one equation in y=mx+b form

1. Graph both functions and see where they cross.

*Great method when given two lines in y=mx+b form

Examples:2x + 3y = 66x + 7y = 9

y = ½ x + 3x + y = 4

y = ½ x – 2y = -2x + 3

Intersection of Other FunctionsGenerally we stick with substitution and graphing here (may want to use separate paper for this one):

Example:

Calculus for Business Name:__________________________________2

Lesson- Absolute Value FunctionsDate:___________________________________

Objective: To learn how to graph absolute value functions

Do Now:Find the intersection between y = x + 1 & y = -3x + 1

______________________________________________________________________________________

Absolute Value Functions: Standard form: f(x) = a| x - h |+k; where ( h, k) is the vertex and “a” is the slope

if “a” is + then:if “a” is – then:

Graph the Following

1.

2.3.

3

x

y

x

y

x

y

4. What are some things you notice about the graph of the absolute value function as a, h, k vary?

Calculus for Business Name:__________________________________Lesson- Functions Rules and Notation

Date:___________________________________

Objective: To review basic function rules and function notation

IDENTIFYING FUNCTIONS

Definitions:

Relation:

Function:

Domain:

Range:

Examples1. State the domain and range of each relation, and state whether the relation is a function or not:a. {(-2, 0), (3, 2), (4, 5)} b. {(6, -2), (3, 4), (6, -6), (-3, 0)}

2. Which relation is a function? Why? What is the shortcut rule for determining a function?

(a) (b) (c) (d)

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3. Find the Domain for each:

a. b.

c. d.

Evaluating Functions: substitute numerical value or variable into function equation and simplify

4. find f(-1) if f(x) = x2 – 1

5. find g(m) if g(x) = 2x6 – 10x4 – x2 + 5

6. find k(w + 2) if k(x) = 3x + 4

7. find h(a – 2) if h(x) = 2x2 – x + 3

Operations with Functions: given functions f and gsum: difference:

product: quotient:

Given functions f and g: (a) perform each of the basic operations, (b) find the domain for each

8. 9.

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Composition of Functions: given functions f and g notation:

Find and for each f(x) and g(x):

10. 11.

12. 13.

14. 15.

One-to-one FunctionsA function is one-to-one when no two ordered pairs in the function have the same ordinate and different abscissas. The best way to check for one-to-oneness is to apply the vertical line test and the horizontal line test.

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If it passes both, then the function is one-to-one. (**Note: if a function is not one-to-one, it does not have an inverse**)

Examples: Determine whether the following functions are one-to-one.16. 17. 18.

Inverse Functions

Steps:1. Write the equation in terms of x and y.2. Switch the x with the y.3. Solve for y.

Examples:19. Find the inverse of 20. Find the inverse of

21. Graph each of the lines in examples one and two and their inverses on the same set of axes and identify any interesting characteristics.

22. Given the function

(a) Algebraically, find f -1(x).

(b) Algebraically, verify your answer to part (a).

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x

y

Calculus for Business Name:____________________________________Lesson- Evaluate functions with variables;

difference quotient Date:_____________________________________

Objectives: evaluate functions as expressions that involve one or more variables explore functions by evaluating and simplifying a difference quotient

Evaluating & Simplifying a Difference Quotient:

1. For f(x) = x2 + 3x + 7, evaluate and simplify:

a. b.

For each of the following, evaluate and simplify:a.

b.

c.

2. f(x) = 3x2 – 2 3. f(x) = 4x – 78

4. For each given function, find and simplify:

a. f(x) = -x2 – 2x – 4

b. f(x) = -2x + 5

c. f(x) = x2 + 4x + 5

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Calculus for Business Name:____________________________________Lesson- Symmetry, Odd/Even/Neither Functions

Date:_____________________________________

Objectives: ~To review the algebraic method for testing for symmetry with respect to axes and origin~To review the algebraic method for determining whether functions are even/odd/neither~To learn shortcut approaches for the above

Odd Functions symmetric with respect to the origin TEST: f(-x) = -f(x)

Even Functions symmetric with respect to the y-axis TEST: f(x) = f(-x)

Symmetry Tests

symmetric with respect to the: the given equation is equivalent when:

y-axis x is replaced with -xx-axis y is replaced with –yorigin x and y are replaced with –x and -y

Examples:

1. f(x) = -x2 – 2x – 4 2. f(x) = -2x + 5

3. g(x) = 2x6 – 10x4 – x2 + 5 4.

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Calculus for Business Name:______________________________Review of Factoring

Date:_______________________________

Objective: Review methods of factoring including: Perfect square trinomials Difference of two squares Sum of two squares * Difference of two cubes Sum of two cubes Factoring by grouping Undefined terms Reducing to lowest terms Multiplying Rational Expressions Dividing Rational Expressions Adding and Subtracting Rational Expressions Solving Rational Equations Simplifying Complex Fractions

Perfect Square TrinomialTo factor a perfect square trinomial follow the steps below:

1) Create two empty binomials as indicated on the right ( )( )2) Take the square root of the first term of the given trinomial3) Take the square root of the last term of the given trinomial4) Take the result of step 2 and put in the 1st position in each binomial5) Take the result of step 3 and put in the 2nd position in each binomial6) The signs in the binomials should be the same as the middle term of the binomial

Example 1:

Difference of Two SquaresTo factor a difference of two squares follow the steps below:

1) Create two empty binomials as indicated on the right ( )( )2) Take the square root of the first term of the given difference of two cubes3) Take the square root of the last term of the given difference of two cubes4) Take the result of step 2 and put in the 1st position in each binomial5) Take the result of step 3 and put in the 2nd position in each binomial6) Alternate the signs

Example 2:

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Sum of Two SquaresThe sum of squares is not factorable

Sum of Two CubesTo factor a sum of two cubes follow the steps below:

1) Create an empty binomial and an empty trinomial as indicated on the right ( )( )

2) Take the cube root of the first term of the given sum of two cubes**note- ignore all signs until the last step**

3) Take the cube root of the last term of the given sum of two cubes4) Take the result of step 2 and put in the 1st position in the binomial5) Take the result of step 3 and put in the 2nd position in the binomial6) Take the result of step 4, square it and put it in the first position of the trinomial7) Take the result of step 4 and multiply by the result of step 5 and put it in the middle position of the

trinomial8) Take the result of step 5, square it and put it in the last position of the trinomial9) Arrange the signs as follows ( + )( − + )

Example 3:

Difference of Two CubesThe only difference between a difference of two cubes and a sum of two cubes is the sign arrangement.

Arrange the signs as follows ( − )( + + )

Example 4:

Factoring by GroupingSteps:

1) Find a convenient point in the polynomial to partition2) Factor within each group3) Factor across the groups

Example 5: Example 6:

Example 7:

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Example 8. Find the value of x that makes the expression undefined:

Reducing- to reduce (or simplify) a rational expression means to write the answer in lowest terms. This only applies when working with a single fraction, a product of fractions or a quotient of fractions (you cannot reduce across a sum or difference of two or more fractions.

Example 9. Reduce to lowest terms:

If there is an addition or subtraction sign in the numerator we must factor the numerator first and then cancel with like factors in the denominator!!!

Example 10. Reduce to lowest terms:

Sometimes you may also have to factor the denominator in order to reduce…

Example 11. Reduce to lowest terms:

Multiplying Rational Expressions:Steps:

Factor each numerator and denominator completely Cancel any like factor in any numerator with any like factor in any denominator Multiply the remaining expressions in each numerator Multiply the remaining expressions in each denominator Reduce if possible

Example 12.

Dividing Rational Expressions:Steps:

Multiply the first fraction by the reciprocal of the second (KCF) Factor each numerator and denominator completely Cancel any like factor in any numerator with any like factor in any denominator Multiply the remaining expressions in each numerator Multiply the remaining expressions in each denominator

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Reduce if possible

Example 13.

Adding and Subtracting Rational ExpressionsSteps:

Find the least common denominator among all fractions (if none already exists) Multiply each denominator by an appropriate factor to make it equivalent to the LCD Multiply each numerator by the same factor that you multiplied its denominator by Combine all numerators (make sure the signs are placed appropriately) and simplify and put over LCD Reduce if possible

Combine each set of rational expressions and simplify.

Example 14: Example 15:

Solving Rational EquationsSteps:

Find the LCD Multiply each fraction by this LCD Cancel all denominators Solve for the variable

Solve each rational equation

Example 16. Example 17.

Simplifying complex fractionsSteps:

Find the grand LCD Multiply the numerator and denominator by the GLCD (will cancel out denominators within

the subfractions) Follow the method for reducing fractions

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Example 18.

Calculus for Business Name:__________________________________Lesson- Polynomial Functions including quadratics

Date:___________________________________

Do Now:

1. State the degree and leading coefficient of each polynomial:

a. 3b2 – 7b5 – 2b3 + 5 b. 6a4 + a3 – 2a

2. For each polynomial equation, sketch the graph and state the number of x-intercepts. Based on your answers, do you notice a pattern?

a. y = 2x + 3 b. y = x2 + x – 2 c. y = x3 – 3x2 – x + 2

3. Write the polynomial equation with the given roots:Recall: Imaginary Roots Theorem: Imaginary roots occur in complex conjugate pairs.

8 and -9 3, 4i, and -4

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i, -i, 5i, and -5i 1, 0, and 2i

4. Write P(x) as a product of first-degree factors using the given zero (use synthetic division):a. P(x) = 2x3 – x2 – 26x + 40; 2 is a zero

b. P(x) = 3x3 – 8x2 + 7x – 2; 1 is a double zero

c. P(x) = x3 – 9x2 + x + 111; -3 is a zero

5. Solve the following quadratic equations by completing the square:2x2 – 5x – 12 = 0 4x2 – 16x + 20 = 0

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6. Complete the square to change the following quadratic functions to the form: and state the vertex for each parabola:

Rational Roots Theorem : Possible roots   , where p represents factors of the constant term and q

represents factors of the leading coefficient.

Process:a. Use Rational Roots Theorem to find potential rational roots.

b. Use synthetic division, or long division, to find an actual root.

c. Repeat step 2 until the polynomial is of degree 2.

d. Factor the remaining quadratic polynomial (it is possible to get imaginary answers).

e. List all factors in simplest factored form

Find all of the factors of each:

7. x3 – 6x2 + 25x – 150 8. x3 – 2x2 – 23x + 60

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9. x3 – 7x + 6 10. x3 – 11x – 10

Calculus for Business Name:__________________________________Lesson- Graphing Rational Functions

Date:___________________________________

Objective: To review graphing a rational function

Objectives: Graphing Rational Functions without the graphing calculator by:a) determining zeroes of the rational functionb) determining vertical and horizontal asymptotesc) determining slant asymptotesd) describing the behavior of the function around vertical asymptotes

Do Now: Given the function f(x) = x3 – 6x2 + 10x – 8

(1) What is the degree of this polynomial?

(2) What is the leading coefficient?

(3) Determine if 4 is a zero of this function f(x):

Graph each (on separate graph paper) of the following functions (and check with the graphing calculator) by:

a) determining zeroes of the rational functionb) determining vertical and horizontal asymptotesc) determining slant (oblique) asymptotesd) describing the behavior of the function around vertical asymptotes

(increasing/decreasing)

1.

4.

18

x

y

x

y

5.

6.

19

x

y

x

y

Calculus for Business Name:__________________________________Lesson- Graphing Piecewise Functions

Date:___________________________________

Objective: To review graphing a piecewise function

Do Now:

State the domain in interval notation and determine any asymptotes for the

______________________________________________________________________________________

Piecewise functionsThis is a graph that is exactly what it sounds like. It is a graph that is basically in pieces.

Graph the following:

The procedure is to graph each part of the function separately.

Graph the Following

20

x

y

x

y

(1)

(2)

(3) (4)

21

x

y

x

y

x

y

x

y

22

(5)

(6)

(7) (8)

23

x

y

x

y

x

y

x

y

Calculus for Business Name:__________________________________Lesson- Regression and Modeling

Date:___________________________________

Correlation and RegressionCorrelation- When one variable is related to another in some way

Scatterplot- A plot on an x-y plane, where (x, y) are paired data plotted as a single point

Types of plots:

Linear Cases

Perfect + Strong + Moderate +

Perfect - Strong - Moderate -_______________________________________________________________________________________Non-Linear Cases- Only sketch the Perfect Correlations

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Exponential Quadratic Cubic

Quartic Natural Logarithmic None

Linear Correlation Coefficient (r) measures the strength of the linear relationship between the given variables (AKA Pearson’s Product Moment Correlation Coefficient)

; where n is the number of ordered pairs

(in percent form) is the percent of variation in the y variable that can be explained by variation in the x variable.

r Type of correlation1 Perfect +

.75 r < 1 Strong +.50 r < .75 Moderate +-.50 < r < .50 None-.75 < r -.50 Moderate --1 < r -.75 Strong -

-1 Perfect -

Ex 1: Construct a scatterplot and compute the linear correlation coefficient between x and y.x 1 2 3 5 9 11y 3 5 7 10 16 20

Linear RegressionAKA- Least Squares Line, Line of Best Fit, Linear Regression Equation

This is the equation that best fits the data set given. Calculator uses: y = ax + b where a = slope and b = y-intercept

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Ex 2: Using the data from Ex 1, write the equation of the line of best fit. Plot this line on your scatterplot as verification.

Using the Graphing Calculator: Note: You should only need to do steps 2-8, 10 once.1. Enter x values in and y values in 2. Press 3. Make sure all plots are off, if not, Press 4 (Plotsoff) Enter4. Press 5. Select 16. Turn this plot on by highlighting On and hitting Enter7. Press the Down Arrow once and Press Enter8. Verify that and are listed.9. Press Zoom 9 (On your graph you should see a scatterplot)10. Press DiagnosticsOn Press Enter twice11. Press Stat, Right Arrow, 4 (LinReg(ax + b))12. , , Enter (Correlation and Regression data is now presented)13. Zoom 9 (Line should appear on the graph verifying that equation is correct)

Making Predictions

To determine “y” given an “x” you can either: Sub the given “x” into the best fit model and solve for “y” or Enter the best fit equation into your calculator and hit: 2nd Trace “Value”. Type in the given “x” and hit

enter.

To determine “x” given a “y” you can either: Sub the given “y” into the best fit model and solve for “x” or Enter the best fit equation into your calculator. Enter the given “y” as an equation. Then find the

intersection (2nd Trace “Intersection” and follow the instructions on the screen).

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Other RegressionsThere are other regressions that can be determined using the graphing calculator.Quadratic: Cubic: Quartic :

Natural Logarithmic: Exponential: Power Regression:

Sine Regression:

Examples:

1. A study was conducted to determine if there is a relationship between the number of cookies eaten on a daily basis and an individual male’s weight. The data are given below.

# of cookies daily 12 13 22 31 41 53 65 73 83 99Weight in lb 120 140 170 195 205 225 250 295 310 325

a. Construct a scatterplot (complete, with appropriate labels)b. Visually determine the best fit model.c. Determine, using your calculator, the best regression model. Why is the one you chose the best?

Justify your answer by showing all work.d. Determine the equation of best fit.e. What weight would you expect a person who eats 100 cookoes per day to be?f. How many cookies per day would you expect a person who is 150 lb to eat?

2. The table below gives the average monthly temperature for Omaha, Nebraska, over a 30 year period. Jan Feb Ma Apr Ma Jun Jul Aug Sep Oct Nov Dec21.1 26.9 38.6 51.9 62.4 72.1 76.9 74.1 65.1 53.4 39 25.1

a. Put the data into your calculator and examine the resulting scatter plot.

b. Use the sinreg function to regress the data and generate an equation. Plot the equation with the scatterplot.

c. Write the equation here:

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d. Amplitude: _______ Period: _______ Frequency: _______

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