Question 1

Which of the following best describes a periodic function?

Accepted Answer

A function that repeats its values at regular intervals, so $$f(x) = f(x + p)$$ for all $$x$$ and some positive $$p.$$

Question 2

What is the period of the function $$y = 	an x$$?

Accepted Answer

$$\pi$$

Question 3

Given the function $$y = 3\cos(2x)$$, what is its amplitude and period?

Accepted Answer

Amplitude: $$3$$, Period: $$\pi$$

Question 4

Which transformation occurs when the graph of $$y = \sin(x) is $$changed to $$y = \sin(x - \frac{\pi}{4})$$?

Accepted Answer

A horizontal shift to the right by $$\frac{\pi}{4}$$ units.

Question 5

If $$y = -2\sin(3x)$$, which statement is true about its graph?

Accepted Answer

It is reflected across the $$x-$$axis, has amplitude $$2$$, and period $$\frac{2\pi}{3}.$$

Question 6

Which of the following is the correct equation for a sine function with amplitude $$5$$, period $$\pi$$, and phase shift $$\frac{\pi}{2} to $$the right?

Accepted Answer

$$y = 5\sin\left(2\left(x - \frac{\pi}{2}\right)\right)$$

Question 7

The graph of $$y = \cos x is $$related to the graph of $$y = \sin x by $$which transformation?

Accepted Answer

A horizontal shift to the left by $$\frac{\pi}{2}$$ units.

Question 8

For the function $$y = a\sin(bx + c) + d$$, which parameter controls the vertical translation?

Accepted Answer

$$d$$

Question 9

A temperature model is given by $$f(x) = 34\sin\left(\frac{\pi}{6}(x - 4.3)\right) + 51. $$What is the average annual temperature predicted by this model?

Accepted Answer

$$51^\circF$$

Question 10

Which of the following statements about the graph of $$y = 	an x is $$correct?

Accepted Answer

It has vertical asymptotes at $$x = \frac{\pi}{2} + n\pi$$, where $$n is an $$integer.

Question 11

What is the effect of the parameter $$b in $$the function $$y = \sin(bx)$$?

Accepted Answer

It changes the period of the function to $$\frac{2\pi}{b}.$$

Question 12

Which of the following is true about the symmetry of the sine and cosine functions?

Accepted Answer

$$\sin(-x) = -\sin(x)$$, so sine is odd; $$\cos(-x) = \cos(x)$$, so cosine is even.

Question 13

If the function $$y = 2\cos(3x - \pi) is $$graphed, what is the phase shift?

Accepted Answer

$$\frac{\pi}{3}$$ units to the right

Question 14

Which of the following best describes the graph of $$y = -3\cos(\pi x)$$?

Accepted Answer

It has amplitude $$3$$, period $$2$$, and is reflected across the $$x-$$axis.

Question 15

Which statement is true about the domain and range of $$y = \sin x$$?

Accepted Answer

Domain: $$(-\infty, \infty)$$; Range: $$[-1, 1]$$

Question 16

A function $$y = c + a\cos(bx) is $$graphed. What does the parameter $$c$$ represent?

Accepted Answer

The vertical shift of the graph.

Question 17

Which of the following is the correct period for $$y = \sin(4x)$$?

Accepted Answer

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

Question 18

The graph of $$y = \cot x$$ has vertical asymptotes at which values?

Accepted Answer

$$x = n\pi$$, \; n \in \mathbb{Z}$

Question 19

Which of the following best describes the effect of increasing the amplitude in a sine function model for temperature?

Accepted Answer

The difference between the maximum and minimum temperatures increases.

Graphs of Sine, Cosine, Tangent, and Cotangent Functions: Periodic Behavior and Transformations

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