Question 1

Which of the following best describes the domain and range of the function $$y = \sin x$$?

Accepted Answer

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

Question 2

What is the amplitude and period of the function $$y = -4 \cos(2x)$$?

Accepted Answer

Amplitude: $$4$$, Period: $$\pi$$

Question 3

A Ferris wheel rotates such that the height $$h (in $$meters) of a passenger above the ground at time $$t (in $$minutes) is modeled by $$h = 10 \sin\left(\frac{\pi}{5} t\right) + 12. $$What is the maximum height reached by the passenger?

Accepted Answer

$$22$$

Question 4

Which transformation occurs when the function $$y = \sin(x) is $$changed to $$y = 3\sin(2x - \pi)$$?

Accepted Answer

Vertical stretch by 3, period halved, phase shift right by $$\frac{\pi}{2}$$

Question 5

Given $$y = a \sin[b(x - c)] + d$$, which parameter controls the vertical shift of the graph?

Accepted Answer

$$d$$

Question 6

If $$y = \sin(x) is an $$odd function, which of the following is true for all $$x$$?

Accepted Answer

$$\sin(-x) = -\sin(x)$$

Question 7

A ball attached to a spring is pulled down 8 cm and released. Its motion is modeled by $$y = -8 \cos(\omega t). If $$the period is 6 seconds, what is the value of $$\omega$$?

Accepted Answer

$$\omega = \frac{2\pi}{6}$$

Question 8

Which of the following equations represents a cosine function with amplitude 5, period $$\pi$$, and phase shift left by $$\frac{\pi}{4}$$?

Accepted Answer

$$y = 5 \cos\left(2x + \frac{\pi}{4}\right)$$

Question 9

The function $$y = 2 \sin\left(\frac{\pi}{6}x - \frac{\pi}{3}\right)$$ models the average temperature (in $$^\circC) in a $$city as a function of month $$x$$ ($$x=1$$ for January, $$x=12$$ for December). What is the period of this function in months?

Accepted Answer

$$12$$

Question 10

Which of the following statements about the graphs of $$y = \sin x$$ and $$y = \cos x is $$true?

Accepted Answer

Both have a period of $$2\pi$$ and range $$[-1, 1]$$

Question 11

A city’s daylight hours $$y as a $$function of month $$x is $$modeled by $$y = 4 \sin\left(\frac{\pi}{6}(x - 3)\right) + 12. $$What is the minimum number of daylight hours in a year?

Accepted Answer

$$8$$

Question 12

Which of the following best describes the effect of a negative amplitude in a sine or cosine function, such as $$y = -2 \sin x$$?

Accepted Answer

The graph is reflected about the x-axis

Question 13

If $$y = 3 \cos(4x - \pi)$$, what is the phase shift of the function?

Accepted Answer

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

Question 14

Which of the following is the correct formula for the period of $$y = a \sin(bx)$$?

Accepted Answer

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

Question 15

A patient’s heart rate oscillates according to $$H(t) = 10 \sin(\pi t) + 70$$, where $$t is in $$seconds. What is the patient’s maximum heart rate?

Accepted Answer

$$80$$

Question 16

Which of the following is true about the function $$y = \cos x$$?

Accepted Answer

It is an even function

Question 17

A mass on a spring moves according to $$y = 5 \sin(3t). $$What is the frequency of the motion?

Accepted Answer

$$\frac{3}{2\pi}$$

Question 18

If $$y = 2 \cos(\pi x)$$, what is the period of the function?

Accepted Answer

$$2$$

Question 19

Which of the following best describes a periodic function?

Accepted Answer

A function that repeats its values at regular intervals

Question 20

Given $$y = 3 \sin(2x + \frac{\pi}{2})$$, what is the phase shift and in which direction?

Accepted Answer

$$\frac{\pi}{4}$$ units to the left

Graphs and Properties of Sine and Cosine Functions

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