Q: What is the domain of the function $$y = \csc x$$?

$$\{ x \mid x \neq n\pi, \text{ where } n \text{ is an integer} \}$$

Question 1

Which of the following statements correctly describes the symmetry of the secant and cosecant functions?

Accepted Answer

The secant function is even and symmetric about the y-axis, while the cosecant function is odd and symmetric about the origin.

Question 2

What is the domain of the function $$y = \csc x$$?

Accepted Answer

$$\{ x \mid x 
eq n\pi, 	ext{ where } n 	ext{ is an integer} \}$$

Question 3

Which equation represents a secant function with period $$4\pi$$ and amplitude 2?

Accepted Answer

$$y = 2 \sec \frac{1}{2}x$$

Question 4

For the function $$y = a \csc (x - d)$$, what is the effect of the parameter $$d$$?

Accepted Answer

It shifts the graph horizontally by $$d$$ units.

Question 5

Which of the following is the correct range for the function $$y = \sec x$$?

Accepted Answer

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

Question 6

If $$y = 	an 2x$$, what is the period of the function?

Accepted Answer

$$\frac{\pi}{2}$$

Question 7

Which of the following functions is odd?

Accepted Answer

$$y = \csc x$$

Question 8

What is the equation for a cotangent function with period $$\frac{\pi}{2}$$ and amplitude $$\frac{1}{2}$$?

Accepted Answer

$$y = \frac{1}{2} \cot 2x$$

Question 9

Which of the following statements about the secant function is true?

Accepted Answer

The secant function has vertical asymptotes at $$x = (2n+1)\frac{\pi}{2}.$$

Question 10

If $$y = -2 	an x$$, which transformation is applied to the basic tangent function?

Accepted Answer

Vertical stretch by 2 and reflection across the x-axis.

Question 11

For the function $$y = \csc x$$, what happens to $$|y| as$$ $$|\sin x|$$ approaches 0?

Accepted Answer

$$|y|$$ gets larger and approaches infinity.

Question 12

Which of the following is the correct equation for a secant function translated 1 unit upward?

Accepted Answer

$$y = \sec x + 1$$

Question 13

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

Accepted Answer

$$\frac{\pi}{b}$$

Question 14

Which of the following describes the graph of $$y = \csc x$$?

Accepted Answer

It has vertical asymptotes at $$x = n\pi$$ and no x-intercepts.

Question 15

If $$y = a \sec bx$$, which function should be used as a guide when sketching its graph?

Accepted Answer

$$y = a \cos bx$$

Question 16

A real-world scenario: A Ferris wheel rotates such that the height of a passenger above the ground can be modeled by $$y = 10 \sec (\frac{\pi}{4} t)$$, where $$t is $$time in minutes. What is the period of the passenger's height function?

Accepted Answer

$$8$$

Question 17

Which of the following is the correct equation for a cotangent function reflected across the x-axis, translated down 2 units, and shifted $$\frac{\pi}{4}$$ units to the right?

Accepted Answer

$$y = -2 - \cot (x - \frac{\pi}{4})$$

Question 18

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

Accepted Answer

It has amplitude.

Question 19

If $$y = 3 \csc (x - \frac{\pi}{2})$$, what is the amplitude and phase shift of the corresponding guide function?

Accepted Answer

Amplitude $$3$$, phase shift $$\frac{\pi}{2}$$ units right

9.4 Graphs of the Other Circular Functions: Secant, Cosecant, Tangent, and Cotangent

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