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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)

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