$$x \in \mathbb{R},\ x \neq \dfrac{\pi}{2} + k\pi$$ for any integer $$k$$

Question 1

Which of the following best describes the domain of the tangent function $$y = 	an x$$?

Accepted Answer

$$x \in \mathbb{R},\ x 
eq \dfrac{\pi}{2} + k\pi$$ for any integer $$k$$

Question 2

What is the period of the cotangent function $$y = \cot x$$?

Accepted Answer

$$\pi$$

Question 3

Which transformation occurs when graphing $$y = 2	an(3x)$$ compared to $$y = 	an x$$?

Accepted Answer

Vertical stretch by a factor of 2 and horizontal compression by a factor of 3

Question 4

Which of the following is the range of the cosecant function $$y = \csc x$$?

Accepted Answer

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

Question 5

Where do the vertical asymptotes of the secant function $$y = \sec x$$ occur?

Accepted Answer

$$x = \dfrac{\pi}{2} + k\pi$$ for any integer $$k$$

Question 6

Suppose a sound wave can be modeled by $$y = 3	an(2x) + 1. $$What is the period of this function?

Accepted Answer

$$\dfrac{\pi}{2}$$

Question 7

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

Accepted Answer

It has vertical asymptotes at $$x = k\pi$$ for any integer $$k$$

Question 8

If $$y = -2\cot(\dfrac{1}{2}x) + 3$$, what is the vertical shift and the period of the function?

Accepted Answer

Vertical shift up 3 units, period $$2\pi$$

Question 9

Which of the following best describes the range of the secant function $$y = \sec x$$?

Accepted Answer

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

Question 10

A Ferris wheel's height above the ground can be modeled by $$y = 10\csc(x) + 20. $$What is the minimum possible height?

Accepted Answer

$$30$$

Question 11

Which of the following is true about the tangent function $$y = 	an x$$?

Accepted Answer

It is periodic with period $$\pi$$

Question 12

Given $$y = 2\sec(4x) - 1$$, what is the period of the function?

Accepted Answer

$$\dfrac{\pi}{2}$$

Question 13

Which of the following is NOT a property of the tangent function $$y = 	an x$$?

Accepted Answer

It has vertical asymptotes at integer multiples of $$\pi$$

Question 14

If $$y = 3\csc(2x)$$, what are the vertical asymptotes of the function?

Accepted Answer

$$x = \dfrac{k\pi}{2}$$ for any integer $$k$$

Question 15

A medical device uses the function $$y = 5	an(x - \dfrac{\pi}{4}) to $$model a signal. What is the horizontal shift?

Accepted Answer

Right $$\dfrac{\pi}{4}$$ units

Question 16

Which of the following is the correct equation for a cotangent function with period $$\dfrac{\pi}{3}$$?

Accepted Answer

$$y = \cot(\dfrac{x}{3})$$

Question 17

Which of the following best describes the domain of the cosecant function $$y = \csc x$$?

Accepted Answer

$$x \in \mathbb{R},\ x 
eq k\pi$$ for any integer $$k$$

Question 18

If $$y = -\sec(x) + 2$$, what is the range of the function?

Accepted Answer

$$(-\infty, 1] \cup [3, \infty)$$

Graphs of Tangent, Cotangent, Cosecant, and Secant Functions in Precalculus 5.6

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