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Pre-Class Problems 14 for Monday, March 18

These are the type of problems that you will be working on in class. These problems are from Lesson 8.

Solution to Problems on the Pre-Exam.

You can go to the solution for each problem by clicking on the problem letter.

Objective of the following problems: To sketch the graph of two cycles of the tangent and cotangent functions and label the numbers on the x- and y-axes as needed.

1. Identify the amplitude, period, and phase shift. Sketch two cycles of the graph of the following functions. Label the numbers on the x- and y-axes as needed. Label where each cycle begins and ends. Then label the number between these numbers

a. [pic] b. [pic] c. [pic]

d. [pic] e. [pic]

f. [pic] g. [pic]

h. [pic] i. [pic]

Additional problems available in the textbook: Page 537 … 41 – 54. Example 3 on page 533.

Solutions:

COMMENT: The graph of both the tangent and cotangent functions can be obtained by using the vertical asymptotes and the x-intercepts of the functions. For each function, the distance between two consecutive vertical asymptotes is the period of the function. That is, if you know where one vertical asymptote is located, then you can move to the next vertical asymptote by going a distance of the period of the function. For each function, each x-intercept is the midpoint of two consecutive vertical asymptotes.

For an unshifted tangent function, the first two consecutive vertical asymptotes are symmetric with the y-axis. For a shifted tangent function, shift one of the vertical asymptotes for the unshifted function and use the period of the function to locate the next vertical asymptotes of the shifted function. For an unshifted cotangent function, the first vertical asymptote is the y-axis. For a shifted cotangent function, shift this vertical asymptote and use the period of the function to locate the next vertical asymptotes of the shifted function.

1a. [pic] Back to Problem 1.

Amplitude: None Period = [pic] Phase Shift: None

[pic] period = [pic]

y

x

[pic] [pic] [pic] [pic]

NOTE: Starting Point 1 + [pic]Period = [pic] = [pic]

NOTE: Starting Point 1 [pic] [pic]Period = [pic] = [pic]

NOTE: Starting Point 2 + Period = [pic] = [pic] = [pic] = [pic]

NOTE: The midpoint of [pic] and [pic] is 0.

NOTE: The midpoint of [pic] and [pic] is [pic] = [pic] = [pic] = [pic].

1b. [pic] Back to Problem 1.

Amplitude: None Period = [pic] = [pic] = [pic]

Phase Shift: None

y

x

[pic] [pic] [pic] [pic]

NOTE: Starting Point 1 + Period = [pic] = [pic]

NOTE: Starting Point 2 + Period = [pic] = [pic]

NOTE: The midpoint of 0 and [pic] is [pic].

NOTE: The midpoint of [pic] and [pic] is [pic].

1c. [pic] Back to Problem 1.

NOTE: Because of the multiplication by the number [pic], the tangent cycles will be inverted.

Amplitude: None Period = [pic] = [pic] = [pic]

Phase Shift: None

[pic] period = [pic]

y

x

[pic] [pic] [pic] [pic]

NOTE: Starting Point 1 + [pic]Period = [pic] = [pic]

NOTE: Starting Point 1 [pic] [pic]Period = [pic] = [pic]

NOTE: Starting Point 2 + Period = [pic] = [pic]

NOTE: The midpoint of [pic] and [pic] is 0.

NOTE: The midpoint of [pic] and [pic] is [pic].

1d. [pic] Back to Problem 1.

NOTE: The cotangent function is an odd function. That is, [pic]

[pic]. Thus,

[pic]

NOTE: Because of the multiplication by the number [pic], the cotangent cycles will be inverted.

Amplitude: None Period = [pic] = [pic] = [pic] = [pic]

Phase Shift: None

y

x

[pic] [pic] [pic] [pic]

NOTE: Starting Point 1 + Period = [pic] = [pic]

NOTE: Starting Point 2 + Period = [pic] = [pic] = 5

NOTE: The midpoint of 0 and [pic] is [pic].

NOTE: The midpoint of [pic] and 5 is [pic] = [pic] = [pic].

1e. [pic] Back to Problem 1.

Amplitude: None Period = [pic]

Phase Shift: [pic] units to the right

[pic] period = [pic]

For the unshifted tangent graph, we have a vertical asymptote of [pic] shown below.

y

[pic] x

We will shift this vertical asymptote to the right a distance of [pic]. Thus,

Starting Point + Shift = [pic] = [pic] = [pic] = [pic]

Thus, the first vertical asymptote in the function [pic] is

[pic].

y

x

[pic] [pic] [pic] [pic] [pic]

NOTE: Starting Point 1 + Period = [pic] = [pic] = [pic]

NOTE: Starting Point 2 + Period = [pic] = [pic] = [pic]

NOTE: The midpoint of [pic] and [pic] is [pic].

NOTE: The midpoint of [pic] and [pic] is [pic].

1f. [pic] Back to Problem 1.

[pic]

Amplitude: None Period = [pic]

Phase Shift: [pic] units to the left

For the unshifted cotangent graph, we have a vertical asymptote of the y-axis. The equation of the y-axis is [pic].

We will shift this vertical asymptote to the left a distance of [pic]. Thus,

Starting Point [pic] Shift = [pic] = [pic]

Thus, the first vertical asymptote in the function [pic] is

[pic].

y

x

[pic] [pic] [pic] [pic] [pic]

NOTE: Starting Point 1 [pic] Period = [pic] = [pic] =

[pic]

NOTE: Starting Point 2 [pic] Period = [pic] = [pic] =

[pic] = [pic]

NOTE: The midpoint of [pic] and [pic] is [pic] =

[pic] = [pic].

NOTE: The midpoint of [pic] and [pic] is [pic] =

[pic] = [pic].

1g. [pic] Back to Problem 1.

NOTE: The tangent function is an odd function. That is, [pic]

[pic]. Thus,

[pic]

[pic]

NOTE: Because of the multiplication by the number [pic], the tangent cycles will be inverted.

Amplitude: None Period = [pic]

Phase Shift: [pic] units to the left

[pic] period = [pic]

For the unshifted tangent graph, we have a vertical asymptote of [pic] shown below.

y

[pic] x

We will shift this vertical asymptote to the left a distance of [pic]. Thus,

Starting Point [pic] Shift = [pic] = [pic] = [pic]

Thus, the first vertical asymptote in the function [pic] is [pic].

y

x

[pic] [pic] [pic] [pic] [pic]

NOTE: Starting Point 1 [pic] Period = [pic] = [pic] =

[pic]

NOTE: Starting Point 2 [pic] Period = [pic] = [pic] =

[pic]

NOTE: The midpoint of [pic] and [pic] is [pic] =

[pic] = [pic] = [pic].

NOTE: The midpoint of [pic] and [pic] is [pic] =

[pic] = [pic] = [pic].

1h. [pic] Back to Problem 1.

[pic]

NOTE: The [pic] was obtained by [pic] = [pic] = [pic] =

[pic].

Amplitude: None Period = [pic] = [pic] = [pic]

Phase Shift: [pic] units to the right

For the unshifted cotangent graph, we have a vertical asymptote of the y-axis. The equation of the y-axis is [pic].

We will shift this vertical asymptote to the right a distance of [pic]. Thus,

Starting Point + Shift = [pic] = [pic]

Thus, the first vertical asymptote in the function [pic]

is [pic].

y

[pic] [pic] [pic] [pic] [pic] x

NOTE: Starting Point 1 + Period = [pic] = [pic] =

[pic]

NOTE: Starting Point 2 + Period = [pic] = [pic] =

[pic]

NOTE: The midpoint of [pic] and [pic] is [pic] =

[pic] = [pic].

NOTE: The midpoint of [pic] and [pic] is [pic] =

[pic] = [pic].

1i. [pic] Back to Problem 1.

[pic]

NOTE: The [pic] was obtained by [pic] = [pic] = [pic] =

[pic].

Amplitude: None Period = [pic] = [pic] = [pic]

Phase Shift: [pic] units to the right

[pic] period = [pic]

For the unshifted tangent graph, we have a vertical asymptote of [pic] shown below.

y

[pic] x

We will shift this vertical asymptote to the right a distance of [pic]. Thus,

Starting Point + Shift = [pic] = [pic] = [pic]

Thus, the first vertical asymptote in the function [pic] is

[pic].

y

[pic] [pic] [pic] [pic] [pic] x

NOTE: Starting Point 1 + Period = [pic] = [pic] = [pic]

NOTE: Starting Point 2 + Period = [pic] = [pic] = [pic]

NOTE: The midpoint of [pic] and [pic] is [pic] = [pic]

= [pic] = [pic].

NOTE: The midpoint of [pic] and [pic] is [pic] =

[pic] = [pic] = [pic] = [pic].

Solution to Problems on the Pre-Exam: Back to Page 1.

17. Sketch the graph of two cycles of the following function on the same side of the y-axis. Label the numbers on the x- and y-axes as needed. Label where the cycles begin and end. Then label the numbers between these numbers. (10 pts.)

[pic]

[pic]

NOTE: The [pic] was obtained by [pic] = [pic] = [pic] = [pic].

Amplitude: None Period = [pic] = [pic] = [pic]

Phase Shift: [pic] units to the left

For the unshifted cotangent graph, we have a vertical asymptote of the y-axis. The equation of the y-axis is [pic].

We will shift this vertical asymptote to the left a distance of [pic]. Thus,

Starting Point [pic] Shift = [pic] = [pic]

Thus, the first vertical asymptote in the function [pic] is

[pic].

y

x

[pic] [pic] [pic] [pic] [pic]

NOTE: Starting Point 1 [pic] Period = [pic] = [pic] =

[pic]

NOTE: Starting Point 2 [pic] Period = [pic] = [pic] =

[pic]

NOTE: The midpoint of [pic] and [pic] is [pic] =

[pic] = [pic] = [pic].

NOTE: The midpoint of [pic] and [pic] is [pic] =

[pic] = [pic] = [pic].

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