Q: If $y = 1 - \tan(x - \dfrac{\pi}{2})$, what is the $y$-intercept of the graph?

$y = 1 - \tan(-\dfrac{\pi}{2})$ (undefined)

Q: Which of the following statements is true about the graph of $y = \tan(x)$ and $y = \cot(x)$?

$y = \tan(x)$ has vertical asymptotes at $x = \dfrac{\pi}{2} + n\pi$, while $y = \cot(x)$ has vertical asymptotes at $x = n\pi$.

Q: A student graphs $y = \tan(x)$ and notices the function is undefined at $x = \dfrac{\pi}{2}$. What is the reason for this?

Because $\cos(x) = 0$ at $x = \dfrac{\pi}{2}$, making $\tan(x)$ undefined.

Question 1

Which of the following is the period of the function $y = 	an(bx)$?

Accepted Answer

$\dfrac{\pi}{b}$

Question 2

For the function $y = 	an(x)$, at which $x$-values do vertical asymptotes occur?

Accepted Answer

$x = \dfrac{\pi}{2} + n\pi$, where $n$ is any integer

Question 3

Given the function $y = -2\cot\left(\dfrac{x}{2}\right)$, what is its period?

Accepted Answer

$4\pi$

Question 4

Which transformation is applied to the graph of $y = 	an(x)$ to obtain $y = 	an(x - \dfrac{\pi}{2})$?

Accepted Answer

A horizontal shift to the right by $\dfrac{\pi}{2}$ units

Question 5

If $y = 	an(3x)$, what is the distance between consecutive vertical asymptotes?

Accepted Answer

$\dfrac{\pi}{3}$

Question 6

Which of the following is the reciprocal identity for cotangent?

Accepted Answer

$\cot(x) = \dfrac{1}{	an(x)}$

Question 7

A medical device uses the function $y = 	an(x)$ to model the angle of a rotating arm. If the device is programmed to operate only between $x = -\dfrac{\pi}{2}$ and $x = \dfrac{\pi}{2}$, what is the range of $y$ in this interval?

Accepted Answer

$(-\infty, \infty)$

Question 8

For the function $y = \cot(x)$, where do its vertical asymptotes occur?

Accepted Answer

$x = n\pi$, where $n$ is any integer

Question 9

A sound engineer is analyzing a signal modeled by $y = 	an(x - \dfrac{\pi}{4})$. What is the phase shift of the signal?

Accepted Answer

$\dfrac{\pi}{4}$ units to the right

Question 10

Which of the following best describes the graph of $y = 	an(x)$?

Accepted Answer

It is an odd function with vertical asymptotes at $x = \dfrac{\pi}{2} + n\pi$ and repeats every $\pi$ units.

Question 11

If $y = 1 - 	an(x - \dfrac{\pi}{2})$, what is the $y$-intercept of the graph?

Accepted Answer

$y = 1 - 	an(-\dfrac{\pi}{2})$ (undefined)

Question 12

A physicist models a pendulum's displacement using $y = \cot(2x)$. What is the period of this function?

Question 13

Which of the following statements is true about the graph of $y = 	an(x)$ and $y = \cot(x)$?

Accepted Answer

$y = 	an(x)$ has vertical asymptotes at $x = \dfrac{\pi}{2} + n\pi$, while $y = \cot(x)$ has vertical asymptotes at $x = n\pi$.

Question 14

If $y = 	an(2x)$, what is the period and where are the vertical asymptotes located?

Question 15

A student graphs $y = 	an(x)$ and notices the function is undefined at $x = \dfrac{\pi}{2}$. What is the reason for this?

Accepted Answer

Because $\cos(x) = 0$ at $x = \dfrac{\pi}{2}$, making $	an(x)$ undefined.

Question 16

Which of the following describes the effect of the coefficient $a$ in $y = a	an(x)$?

Accepted Answer

It stretches or compresses the graph vertically by a factor of $|a|$.

Question 17

If $y = 	an(x)$ models the angle of a ramp, and the ramp is rotated so the function becomes $y = 	an(x + \dfrac{\pi}{3})$, what transformation has occurred?

Accepted Answer

A phase shift of $\dfrac{\pi}{3}$ units to the left

Question 18

Which of the following is true about the symmetry of $y = 	an(x)$?

Accepted Answer

$y = 	an(x)$ is an odd function, so $	an(-x) = -	an(x)$

Graphs of Tangent and Cotangent Functions

Study Guide - Practice Questions

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