Question 1

Given the function $$y = 3 \sin(2x) - 1$$, what is the amplitude, period, and midline of the graph?

Accepted Answer

Amplitude: $$3$$, Period: $$\pi$$, Midline: $$y = -1$$

Question 2

Which transformation occurs when the equation $$y = \cos(x) - 2 is $$graphed?

Accepted Answer

A vertical shift down by 2 units

Question 3

If the period of $$y = \sin(Bx) is$$ $$\pi$$, what is the value of $$B$$?

Accepted Answer

$$B = 2$$

Question 4

A population of rabbits in a park is modeled by $$R(t) = 500 \sin\left(\frac{\pi}{12} t\right) + 800. $$What is the period of the population cycle?

Accepted Answer

$$24$$ months

Question 5

Which of the following equations represents a cosine function with amplitude $$4$$, period $$\pi$$, and midline $$y = 2$$?

Accepted Answer

$$y = 4 \cos(2x) + 2$$

Question 6

Solve for $$x in$$ $$\sin(x) = \frac{\sqrt{2}}{2}$$ for $$x in$$ $$[0, 2\pi].$$

Accepted Answer

$$x = \frac{\pi}{4}, \frac{3\pi}{4}$$

Question 7

What is the reference angle for $$	heta = \frac{5\pi}{6}$$?

Accepted Answer

$$\frac{\pi}{6}$$

Question 8

Which statement best explains why trigonometric functions are appropriate for modeling the population cycles of foxes and rabbits?

Accepted Answer

Because their populations show periodic, repeating behavior over time

Question 9

Given $$y = 2 \sin\left(4\left(x - \frac{\pi}{2}\right)\right) + 1$$, what is the horizontal shift?

Accepted Answer

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

Question 10

Solve the equation $$2\sin(2x) - 1 = 0$$ for $$x in$$ $$[0, 2\pi].$$

Accepted Answer

$$x = \frac{\pi}{12}, \frac{5\pi}{12}, \frac{13\pi}{12}, \frac{17\pi}{12}$$

Question 11

If $$\cos(	heta) is $$negative and $$\sin(	heta) is $$positive, in which quadrant does $$	heta$$ lie?

Accepted Answer

Quadrant II

Question 12

What is the period of the function $$y = 5 \sin\left(\frac{x}{3}\right)$$?

Accepted Answer

$$6\pi$$

Question 13

Which of the following is the correct solution set for $$\cos^2$(x) = \cos(x)$$ for $$x in$$ $$[0, 2\pi]$$?

Accepted Answer

$$x = 0, 2\pi, \pi$$

Question 14

For the function $$y = A \sin(B(x - C)) + D$$, which parameter controls the vertical shift?

Accepted Answer

$$D$$

Question 15

What is the sign of $$\sin(	heta)$$ when $$	heta is in $$Quadrant III?

Accepted Answer

Negative

Question 16

Which of the following is a correct method to solve $$2\cos(x) + 1 = 0$$ for $$x in$$ $$[0, 2\pi]$$?

Accepted Answer

Set $$\cos(x) = -\frac{1}{2}$$ and find $$x in $$Quadrants II and III

Question 17

If $$y = 7 \sin(3x) + 2$$, what is the maximum value of $$y$$?

Accepted Answer

$$9$$

Question 18

Which equation has a period of $$\frac{2\pi}{5}$$?

Accepted Answer

$$y = \sin(5x)$$

Question 19

What is the solution to $$4\sin(2x) - 2 = 0$$ for $$x in$$ $$[0, 2\pi]$$?

Accepted Answer

$$x = \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}$$

Study Notes: Period, Amplitude, Midline, and Solving Trigonometric Equations

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