AP Precalculus -3.5 Sinusoidal Functions- MCQ Exam Style Questions - Effective Fall 2023

AP Precalculus -3.5 Sinusoidal Functions- MCQ Exam Style Questions – Effective Fall 2023

AP Precalculus -3.5 Sinusoidal Functions- MCQ Exam Style Questions – AP Precalculus- per latest AP Precalculus Syllabus.

AP Precalculus – MCQ Exam Style Questions- All Topics

Question

The figure shows the graph of a sinusoidal function $ g $. What are the values of the period and amplitude of $ g $?

(A) The period is 4, and the amplitude is 3.
(B) The period is 8, and the amplitude is 3.
(C) The period is 4, and the amplitude is 6.
(D) The period is 8, and the amplitude is 6.

▶️ Answer/Explanation

Detailed solution

From the graph:
– Maximum $ g_{\text{max}} = 3 $, Minimum $ g_{\text{min}} = -3 $
– Amplitude = $ \frac{g_{\text{max}} – g_{\text{min}}}{2} = \frac{3 – (-3)}{2} = 3 $
– Consecutive maxima occur at $ x = 3 $ and $ x = 11 $, so period = $ 11 – 3 = 8 $
– Consecutive minima occur at $ x = -1 $ and $ x = 7 $, also giving period = $ 7 – (-1) = 8 $
✅ Answer: (B)

Question

The graph of the function $ h $ is given in the $ xy $-plane. If $ h(x) = a \tan(bx) + 10 $, where $ a $ and $ b $ are constants, which of the following is true?

(A) $ a > 0 $ and $ b > 1 $
(B) $ a > 0 $ and $ 0 < b < 1 $
(C) $ a < 0 $ and $ b > 1 $
(D) $ a < 0 $ and $ 0 < b < 1 $

▶️ Answer/Explanation

Detailed solution

From the graph:
– The standard tangent curve increases through its midline; here it also increases (no vertical reflection), so $ a > 0 $.
– The period of $ h $ is shorter than $ \pi $ (the period of $ \tan x $). Since period = $ \frac{\pi}{b} $, we have $ \frac{\pi}{b} < \pi $ → $ b > 1 $.
Thus $ a > 0 $ and $ b > 1 $.
✅ Answer: (A)

Question

The function $ g $ is given by $ g(x) = 3 \csc(\pi(x + 2)) – 1 $. Which of the following describes the range of $ g $?

(A) The range of $ g $ is $[-3, 3]$.
(B) The range of $ g $ is $(-\infty, -2] \cup [4, \infty)$.
(C) The range of $ g $ is $(-\infty, -3] \cup [3, \infty)$.
(D) The range of $ g $ is $(-\infty, -4] \cup [2, \infty)$.

▶️ Answer/Explanation

Detailed solution

$ g(x) = 3\csc(\pi(x+2)) – 1 $.
Cosecant function: $ \csc u $ has range $ (-\infty, -1] \cup [1, \infty) $.
Multiplying by 3 stretches vertically: range becomes $ (-\infty, -3] \cup [3, \infty) $.
Subtracting 1 shifts down by 1: range becomes $ (-\infty, -4] \cup [2, \infty) $.
✅ Answer: (D)

Question

The function $ p $ is given by $ p(\theta) = 3\tan\left(\frac{\pi}{2}(\theta + 1)\right) – 4 $. What is the period of $ p $?

(A) $ \frac{2}{\pi} $
(B) $ \frac{\pi}{2} $
(C) 2
(D) 4

▶️ Answer/Explanation

Detailed solution

For $ y = \tan(k\theta) $, the period is $ \frac{\pi}{|k|} $.
Here $ p(\theta) = 3\tan\left(\frac{\pi}{2}(\theta + 1)\right) – 4 $.
The coefficient of $ \theta $ inside the tangent is $ \frac{\pi}{2} $, so $ k = \frac{\pi}{2} $.
Period $ = \frac{\pi}{k} = \frac{\pi}{\pi/2} = 2 $.
✅ Answer: (C)

Question

The function $ f $ is given by $ f(x) = a\tan(bx) $, where $ a $ and $ b $ are constants. Which of the following is true about the period of $ f $?

(A) Both $ a $ and $ b $ have an impact on the period.
(B) Only $ a $ has an impact on the period.
(C) Only $ b $ has an impact on the period.
(D) Neither $ a $ nor $ b $ has an impact on the period.

▶️ Answer/Explanation

Detailed solution

For $ y = \tan(bx) $, the period is $ \frac{\pi}{|b|} $.
The amplitude-like factor $ a $ affects vertical stretching but not the period.
Thus only $ b $ affects the period.
✅ Answer: (C)

Question

The function $ f $ is defined by $ f(x) = a \sin(b(x + c)) + d $, for constants $ a, b, c, $ and $ d $. In the $ xy $-plane, the points (2, 2) and (4, 4) represent a minimum value and a maximum value, respectively, on the graph of $ f $. What are the values of $ a $ and $ d $?

(A) $ a = 1 $ and $ d = 3 $
(B) $ a = 1 $ and $ d = 2 $
(C) $ a = 2 $ and $ d = 3 $
(D) $ a = 2 $ and $ d = 2 $

▶️ Answer/Explanation

Detailed solution

Midline $ d $ is halfway between max and min:
\[ d = \frac{4 + 2}{2} = 3 \]
Amplitude $ a $ is half the difference:
\[ a = \frac{4 – 2}{2} = 1 \]
Thus $ a = 1 $, $ d = 3 $.
✅ Answer: (A)

Question

The function $ f $ is given by $ f(x) = 2\sin(4x) + \cos(2x) $. Using the period of $ f $, which of the following is the number of complete cycles of the graph of $ f $ in the $ xy $-plane on the interval $ 0 \leq x \leq 1000 $?

(A) 159
(B) 318
(C) 602
(D) 636

▶️ Answer/Explanation

Detailed solution

Find the period of $ f(x) = 2\sin(4x) + \cos(2x) $.
• $ 2\sin(4x) $ has period $ \frac{2\pi}{4} = \frac{\pi}{2} $.
• $ \cos(2x) $ has period $ \frac{2\pi}{2} = \pi $.
The combined function $ f $ is periodic with period equal to the least common multiple of $ \frac{\pi}{2} $ and $ \pi $, which is $ \pi $.
Thus the period of $ f $ is $ \pi $.
Number of complete cycles in $ 0 \leq x \leq 1000 $:
\[ \frac{1000}{\pi} \approx 318.31 \]
So there are 318 complete cycles.
✅ Answer: (B)

Question

Is the function $f(x) = \sin 3\theta – \cos 3\theta$ a sinusoidal function on all its domain?

a. No, since it has both a sine and a cosine, while a sinusoidal function can only have one sine or one cosine function.

b. No, since it has a sine being subtracted by a cosine function, while a sinusoidal function has it being added together.

c. Yes, since the sine and cosine functions have the same input angle, $3\theta$.

d. Yes, since it incorporates trigonometric functions and not inverse trigonometric functions.

▶️ Answer/Explanation

Detailed solution

The correct answer is c.
Any linear combination of a sine and cosine with the same frequency results in a single shifted sinusoid.
Using the identity $A\sin(Bx) + B\cos(Bx) = R\sin(Bx + \phi)$, we can rewrite the function.
In this case, both terms share the same angular frequency (or input angle) of $3\theta$.
The function $f(x) = \sin 3\theta – \cos 3\theta$ can be simplified to $\sqrt{2}\sin(3\theta – 45^\circ)$.
Because it can be expressed as a single sine wave, it is classified as a sinusoidal function.
Options a and b are incorrect because they misunderstand the additive properties of waves.

Question

Which of the following characteristics of $f(x) = -3\sin(3x – 3\pi)$ is true?

i. $f(x)$ is odd
ii. $f(x)$ has an amplitude of $6$
iii. $f(x)$ has a period of $\frac{2\pi}{3}$

a. i and iii only
b. i only
c. iii only
d. i, ii, and iii only

▶️ Answer/Explanation

Detailed solution

The function is $f(x) = -3\sin(3(x – \pi))$, which simplifies using $\sin(\theta – 3\pi) = -\sin(\theta)$ to $f(x) = 3\sin(3x)$.
Checking (i): $f(-x) = 3\sin(-3x) = -3\sin(3x) = -f(x)$, so the function is odd.
Checking (ii): The amplitude is $|a| = |-3| = 3$, so the statement “amplitude of $6$” is false.
Checking (iii): The period is calculated as $T = \frac{2\pi}{|b|} = \frac{2\pi}{3}$, so this statement is true.
Since statements i and iii are correct, the correct option is a.

AP Precalculus -3.5 Sinusoidal Functions- MCQ Exam Style Questions - Effective Fall 2023