Question 1

What is the amplitude and period of the function $$y = 3\sin(2x)$$?

Accepted Answer

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

Question 2

Which equation represents a sine function with amplitude 4, period $$\pi$$, and a phase shift of $$\frac{\pi}{2}$$ units to the right?

Accepted Answer

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

Question 3

A city experiences average monthly temperatures that oscillate between 10°C and 30°C throughout the year. If $$y$$ represents the temperature in month $$x ($$with $$x=1$$ for January), which sine function best models this behavior?

Accepted Answer

$$y = 10\sin(\frac{\pi}{6}x - \frac{\pi}{2}) + 20$$

Question 4

What is the range of the function $$y = -2\cos(x) + 1$$?

Accepted Answer

$$[-3, 1]$$

Question 5

If the graph of $$y = \sin(x) is $$shifted upward by 3 units, what is the equation of the new graph?

Accepted Answer

$$y = \sin(x) + 3$$

Question 6

Which of the following describes the effect of increasing the value of $$B in$$ $$y = \sin(Bx)$$?

Accepted Answer

The period decreases.

Question 7

A Ferris wheel rotates such that a rider's height above the ground (in meters) is modeled by $$y = 8\cos(\frac{\pi}{15}t) + 10$$, where $$t is $$time in seconds. What is the maximum height the rider reaches?

Accepted Answer

$$18$$ meters

Question 8

What is the phase shift of the function $$y = 2\sin(3x - \frac{\pi}{2})$$?

Accepted Answer

$$\frac{\pi}{6}$$ units to the right

Question 9

Which of the following is the period of $$y = \cos(\frac{x}{4})$$?

Accepted Answer

$$4\pi$$

Question 10

A sound wave can be modeled by $$y = 0.5\sin(440\pi t)$$, where $$t is in $$seconds. What is the amplitude and period of this wave?

Accepted Answer

Amplitude: $$0.5$$, Period: $$\frac{1}{220}$$

Question 11

Which of the following best describes the graph of $$y = -3\sin(x)$$ compared to $$y = \sin(x)$$?

Accepted Answer

It is reflected over the x-axis and stretched vertically by a factor of 3.

Question 12

If $$y = 2\cos(4x - \pi)$$, what is the amplitude, period, and phase shift?

Accepted Answer

Amplitude: 2, Period: $$\frac{\pi}{2}$$, Phase shift: $$\frac{\pi}{4}$$ right

Question 13

Which of the following is the equation for a cosine function with amplitude 5, period $$3\pi$$, and a vertical shift of 2 units down?

Accepted Answer

$$y = 5\cos(\frac{2\pi}{3}x) - 2$$

Question 14

A daylight model for a city is $$y = 2\sin(\frac{\pi}{6}x - \frac{\pi}{2}) + 12. $$What is the minimum number of daylight hours predicted by this model?

Accepted Answer

10 hours

Question 15

Which transformation will change the graph of $$y = \sin(x) to$$ $$y = \sin(x) - 4$$?

Accepted Answer

A vertical shift 4 units downward

Question 16

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

Accepted Answer

$$\frac{\pi}{2}$$ units to the left

Question 17

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

Accepted Answer

They have the same amplitude and period.

Question 18

A sinusoidal function has amplitude 6, period $$\pi$$, and is shifted $$\frac{\pi}{4}$$ units to the right. Which equation matches this description?

Accepted Answer

$$y = 6\sin(2x - \frac{\pi}{2})$$

Question 19

Which of the following best describes the effect of the parameter $$D in$$ $$y = A\sin(Bx - C) + D$$?

Accepted Answer

It shifts the graph vertically.

Graphs of Sine and Cosine Functions (Precalculus Essentials, Ch. 4.5)

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