Q: Which of the following is the correct domain for the parent tangent function $$y = \tan x$$?

$$\left\{ x \mid x \neq \frac{\pi}{2} + k\pi,\ k \in \mathbb{Z} \right\}$$

Question 1

Which of the following correctly describes the amplitude and period of the function $$y = -3 \sin(2x) + 1$$?

Accepted Answer

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

Question 2

A Ferris wheel's height above the ground is modeled by $$y = 10 \sin\left(\frac{\pi}{6} t\right) + 20. $$What is the maximum height the Ferris wheel reaches?

Accepted Answer

$$30$$ meters

Question 3

What is the phase shift of the function $$y = 2 \cos\left(3x - \frac{\pi}{2}\right) - 4$$?

Accepted Answer

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

Question 4

Which of the following is the correct domain for the parent tangent function $$y = 	an x$$?

Accepted Answer

$$\left\{ x \mid x 
eq \frac{\pi}{2} + k\pi,\ k \in \mathbb{Z} ight\}$$

Question 5

Given the graph of $$y = A \sin(Bx) + D$$ with maximum value $$7$$ and minimum value $$3$$, what is the amplitude and midline?

Accepted Answer

Amplitude: $$2$$, Midline: $$5$$

Question 6

Which of the following statements about the parent cosine function $$y = \cos x is $$TRUE?

Accepted Answer

It starts at a maximum.

Question 7

A spring oscillates according to $$y = 5 \cos(4t). $$What is the period of the motion?

Accepted Answer

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

Question 8

Which of the following is the correct range for the parent secant function $$y = \sec x$$?

Accepted Answer

$$y \leq -1 or$$ $$y \geq 1$$

Question 9

If $$y = 2 \sin\left(x - \frac{\pi}{4}\right) + 3$$, what is the vertical shift and the phase shift?

Accepted Answer

Vertical shift: $$3$$, Phase shift: $$\frac{\pi}{4}$$ right

Question 10

Which of the following is the correct range for the inverse sine function $$y = \sin$$^{-1}$$ x$$?

Accepted Answer

$$[-\frac{\pi}{2}, \frac{\pi}{2}]$$

Question 11

A tide rises and falls according to $$y = 2 \sin\left(\frac{\pi}{12} t\right) + 5. $$How many hours does it take for one full cycle?

Accepted Answer

$$24$$ hours

Question 12

Which of the following is TRUE about the cotangent function $$y = \cot x$$?

Accepted Answer

It has vertical asymptotes at $$x = k\pi$$

Question 13

If $$\sin$$^{-1}$$\left(\frac{\sqrt{3}}{2}\right) is $$evaluated, what is the exact value?

Accepted Answer

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

Question 14

A right triangle has legs of length $$5$$ and $$12. $$What is the length of the hypotenuse?

Accepted Answer

$$13$$

Question 15

Which transformation is applied to $$y = \sin x to $$obtain $$y = -2 \sin(x - \pi) + 3$$?

Accepted Answer

Vertical stretch by $$2$$, reflection over x-axis, phase shift $$\pi$$ right, vertical shift $$3 up$$

Question 16

If $$y = \cos x is $$transformed to $$y = \cos(2x)$$, what happens to the period?

Accepted Answer

It halves.

Question 17

A pilot flies at a bearing of $$N30^\circ E. $$What is the angle measured from the north direction?

Accepted Answer

$$30^\circ$$ east of north

Question 18

Which of the following is TRUE about the inverse tangent function $$y = 	an$$^{-1}$$ x$$?

Accepted Answer

Its domain is $$(-\infty, \infty)$$ and range is $$\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$$

Question 19

If $$y = 4 \sin(3x)$$, how many times does the function cross its midline in one period?

Accepted Answer

Four times

Study Guide: Trigonometric Functions and Their Applications

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