Essay about Trigonometric Functions and Vertical Asymptote

A Summary of the Important Functions For calculus
In this section:
1.
2.
3.
4.
5.
6.
7.
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10.
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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, ...

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