In this section:

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2.

3.

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8.

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12.

13.

Function Summaries

Linear Functions

Power Functions

Exponential Functions

Transformations of Exponential Functions

Comparing Exponential and Power Functions

Absolute Value Function

Quadratic Functions

Polynomial and Rational Functions

Logarithmic Functions

Sine, Cosine, and Tangent Functions

Inverse Trig Functions

Sinusoidal Models

Function Summaries

In the following pages you will find a summary of almost all of the functions you have studied in your precalculus sequence. First you will see a table for the family of functions with all of the important characteristics of that family summarized. This is followed by an application problem involving modeling using that family. Make sure that you are completely familiar with these functions as you move into calculus. Linear Functions

FUNCTION TYPE: LINEAR f (x) mx b; m 0

Sample Graphs:

NOTES: rise m

(slope)

run

Domain:

x-intercept(s): x b/m

( , )

Range:

NONE

Horizontal Asymptote:

( , ) y-intercept: NONE

Symmetry

(Even/Odd?):

Vertical Asymptote(s):

(0, b)

Odd if b = 0 m 0

As x m 0

As x

,y

m 0

As x

,y

m 0

As x

,y

,y

2

Linear Functions

A motor oil producing company wants to study the connection between the number of minutes of advertising per day on a certain website for their product and the number of oil cases sold per month. Their market research produces the following information in tabular form.

TV ads

1

(min/day)

Units sold (in

1

millions/month)

2

3

3.5

5.5

6.2

2.5

3.7

4.2

7

8.7

(A) The market research people say that they think this relationship is approximately linear. What do you think? Base your conclusions upon some quantitative reasoning. We left this open ended so that you can make your own arguments.

(B) Find a linear function that you think approximately models the data. You might choose two of the 6 points and use them to determine a straight line, or you might take an “average slope” of some kind.

(C) What is the significance of the slope of this line in the context of the problem?

(D) If you know how, use the linear regression feature on your calculator to find a line of “best fit” to this data.

(E) Use the line of best fit to predict the quantity sold if advertising time is increased to 7 minutes per day.

Power Functions

FUNCTION TYPE: Even positive integer powers of x f (x)

2n

x , where n 1, 2, 3, ...

Sample Graphs:

NOTES:

x-intercept(s):

All functions pass through (0, 0)

(0, 0), ( 1, 1) and (1, 1) .

Graphs become more

“square looking” as n increases. Domain:

Vertical Asymptote(s):

( , )

Range:

NONE

Horizontal Asymptote:

[0, ) y-intercept: NONE

Symmetry

(Even/Odd?):

(0, 0)

Even

As x

FUNCTION TYPE: Odd positive integer powers of x (≥ 3) f (x)

x 2n 1 , where n 1, 2, 3, ...

Sample Graphs:

,y

NOTES:

As x

,y

x-intercept(s):

All functions pass through (0, 0)

(0, 0), ( 1, 1) and (1, 1) .

Graphs become more

“square looking” as n increases. Domain:

Vertical Asymptote(s):

( , )

Range:

NONE

Horizontal Asymptote:

( , ) y-intercept: NONE

Symmetry

(Even/Odd?):

(0, 0)

Odd

As x

,y

As x

,y

4

FUNCTION TYPE: Even root functions f (x)

2n

1

2n

x

x , where n 1, 2, 3, ...

Sample Graphs:

NOTES:

x-intercept(s):

All functions pass through (0, 0)

(1, 1) . Graphs become more “square looking” as n increases.

Domain:

Vertical Asymptote(s):

[0, )

Range:

NONE

Horizontal Asymptote:

[0, ) y-intercept: NONE

Symmetry

(Even/Odd?):

(0, 0)

NONE

As x

,y

N/A

As x

,y

Increasing concave down function

FUNCTION TYPE: Odd root functions f (x)

2n 1

x

x

1

2n 1

Sample Graphs:

, where n 1, 2, 3, ...