Question 1

Given the function $$y = 2 \sin(3x - \frac{\pi}{2}) + 1$$, what is the amplitude, period, and phase shift of the graph?

Accepted Answer

Amplitude: $$2$$, Period: $$\frac{2\pi}{3}$$, Phase shift: $$\frac{\pi}{6}$$ right, Vertical shift: $$1 up$$

Question 2

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

Accepted Answer

A horizontal shift to the left $$\pi$$ units

Question 3

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

Accepted Answer

$$\pi$$

Question 4

If $$y = -3 \cos(x - \frac{\pi}{4}) + 2$$, what is the maximum value of the function?

Accepted Answer

$$5$$

Question 5

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

Accepted Answer

$$220 Hz$$

Question 6

Which of the following best describes the effect of the constant $$A in$$ $$y = A \sin x$$?

Accepted Answer

It determines the amplitude and vertical stretch/shrink

Question 7

Given $$y = \sin(2x) + \sin(x)$$, what is the period of the sum?

Accepted Answer

$$4\pi$$

Question 8

A damped oscillation is modeled by $$y = e$$^{-0.5x}$$ \cos x. $$What happens to the amplitude as $$x$$ increases?

Accepted Answer

It decreases exponentially

Question 9

What is the phase shift of $$y = \sin(4x + \pi)$$?

Accepted Answer

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

Question 10

Which equation represents a cosine function with amplitude $$3$$, period $$4\pi$$, and phase shift $$\frac{\pi}{2} to $$the right?

Accepted Answer

$$y = 3 \cos\left(\frac{1}{2}(x - \frac{\pi}{2})\right)$$

Question 11

If $$y = \sin x is $$reflected across the x-axis and shifted down $$2$$ units, what is the equation of the new function?

Accepted Answer

$$y = -\sin x - 2$$

Question 12

Which of the following functions has a period of $$\pi$$ and an amplitude of $$3$$?

Accepted Answer

$$y = 3 \sin(2x)$$

Question 13

A Ferris wheel rotates such that the height $$h (in $$meters) of a seat above the ground at time $$t (in $$seconds) is modeled by $$h = 10 \cos\left(\frac{\pi}{15} t\right) + 12. $$What is the maximum height reached by the seat?

Accepted Answer

$$22$$ meters

Question 14

Which transformation will result in the graph of $$y = \sin x$$ being reflected across the y-axis?

Accepted Answer

Replace $$x$$ with $$-x to $$get $$y = \sin(-x)$$

Question 15

If $$y = 2 \sin(x) + 3 \cos(x)$$, what is the period of the function?

Accepted Answer

$$2\pi$$

Question 16

Which of the following best describes the effect of the constant $$D in$$ $$y = \sin x + D$$?

Accepted Answer

It translates the graph vertically by $$D$$ units

Question 17

A spring oscillates according to $$y = 5 \cos(\pi t). $$How long does it take to complete one full cycle?

Accepted Answer

$$2$$ seconds

Question 18

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

Accepted Answer

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

Question 19

If $$y = 2 \sin(x) \cos(x)$$, what is an equivalent expression for $$y$$?

Accepted Answer

$$\sin(2x)$$

Graphs of Transformed Sine and Cosine Functions: Precalculus Study Guide

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