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 $y < -1$ within $0 \leq x \leq 2$?

Accepted Answer

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

Graphs of Secant and Cosecant Functions

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