Q: Given the function $$y = ext{sec}(x)$$, at which $$x-$$values within $$0 \\leq x \\leq 2$$ does the graph have vertical asymptotes?

$$x = \\frac{}{2}, \\frac{3}{2}$$

Q: If $$y = ext{csc}(x)$$, which of the following describes the domain of the function?

All real numbers except where $$x = n$$, $$n is an $$integer.

Q: A medical device uses the function $$y = ext{csc}(x) to $$model the periodic signal of a sensor. If the sensor is calibrated so that $$x is $$measured in radians, at which values will the device experience undefined output?

At $$x = n$$, where $$n is $$any integer

Q: Which of the following is the correct reciprocal identity for the secant function?

$$ ext{sec}(x) = \\frac{1}{ ext{cos}(x)}$$

Q: A sound engineer is analyzing a signal modeled by $$y = ext{sec}(x - /2). $$What is the phase shift of the signal?

The graph is shifted right by $$/2$$

Q: Which of the following is true about the graph of $$y = ext{sec}(x)$$?

It is undefined at $$x = \\frac{}{2} + n$$, where $$n is an $$integer

Question 1

Which of the following statements best explains why the graphs of $$y = 	ext{sec}(x)$$ and $$y = 	ext{csc}(x)$$ have vertical asymptotes?

Accepted Answer

Because the secant and cosecant functions are undefined wherever the cosine and sine functions are zero, resulting in vertical asymptotes.

Question 2

Given the function $$y = 	ext{sec}(x)$$, at which $$x-$$values within $$0 \leq x \leq 2$$ does the graph have vertical asymptotes?

Accepted Answer

$$x = \frac{}{2}, \frac{3}{2}$$

Question 3

If $$y = 	ext{csc}(x)$$, which of the following describes the domain of the function?

Accepted Answer

All real numbers except where $$x = n$$, $$n is an $$integer.

Question 4

Which transformation is applied to the graph of $$y = 	ext{sec}(x) to $$obtain the graph of $$y = 	ext{sec}(x - b)$$?

Accepted Answer

A horizontal shift to the right by $$b$$ units

Question 5

Suppose the graph of $$y = 	ext{csc}(bx)$$ has vertical asymptotes at $$x = 0$$, $$x = \frac{}{b}$$, and $$x = \frac{2}{b}. $$What is the value of $$b$$?

Accepted Answer

$$b = 1$$

Question 6

Given the function $$y = 	ext{sec}(x)$$, what is the range of the function?

Accepted Answer

$$(-, -1] \cup [1, )$$

Question 7

A medical device uses the function $$y = 	ext{csc}(x) to $$model the periodic signal of a sensor. If the sensor is calibrated so that $$x is $$measured in radians, at which values will the device experience undefined output?

Accepted Answer

At $$x = n$$, where $$n is $$any integer

Question 8

Which of the following is the correct reciprocal identity for the secant function?

Accepted Answer

$$	ext{sec}(x) = \frac{1}{	ext{cos}(x)}$$

Question 9

If $$y = 	ext{sec}(x)$$, what is the period of the function?

Accepted Answer

$$2$$

Question 10

A sound engineer is analyzing a signal modeled by $$y = 	ext{sec}(x - /2). $$What is the phase shift of the signal?

Accepted Answer

The graph is shifted right by $$/2$$

Question 11

Which of the following best describes the relationship between the graphs of $$y = 	ext{sin}(x)$$ and $$y = 	ext{csc}(x)$$?

Accepted Answer

The graph of $$y = 	ext{csc}(x)$$ has vertical asymptotes wherever $$y = 	ext{sin}(x) is $$zero, and its maximum and minimum points are reciprocal to those of $$y = 	ext{sin}(x).$$

Question 12

If $$y = 	ext{sec}(x)$$, what is the value of $$y$$ when $$x = 0$$?

Accepted Answer

$$y = 1$$

Question 13

A physicist models a wave using $$y = 	ext{csc}(x). If $$the amplitude of the wave is doubled, which equation best represents the new model?

Accepted Answer

$$y = 2 	ext{csc}(x)$$

Question 14

Which of the following is true about the graph of $$y = 	ext{sec}(x)$$?

Accepted Answer

It is undefined at $$x = \frac{}{2} + n$$, where $$n is an $$integer

Question 15

If $$y = 	ext{sec}(x)$$ and $$y = 	ext{csc}(x)$$ are graphed on the same axes, which points will both graphs be undefined?

Accepted Answer

There are no points where both are undefined simultaneously

Question 16

A student sketches the graph of $$y = 	ext{sec}(x - b)$$ and observes that the vertical asymptotes are at $$x = b + \frac{}{2}$$ and $$x = b + \frac{3}{2}. $$What is the value of $$b if $$one asymptote is at $$x = 2$$?

Accepted Answer

$$b = 2 - \frac{}{2}$$

Question 17

Which of the following best describes the behavior of $$y = 	ext{csc}(x)$$ near $$x = 0$$?

Accepted Answer

It approaches infinity as $$x$$ approaches $$0$$

Question 18

If $$y = 	ext{sec}(x)$$, which of the following intervals contains all values of $$x$$ for which $$y \u003c -1$$ within $$0 \leq x \leq 2$$?

Accepted Answer

$$x \in (, \frac{3}{2})$$

Graphs of Secant and Cosecant Functions

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