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AP Precalculus -3.8 The Tangent Function- MCQ Exam Style Questions - Effective Fall 2023

AP Precalculus -3.8 The Tangent Function- MCQ Exam Style Questions – Effective Fall 2023

AP Precalculus -3.8 The Tangent Function- MCQ Exam Style Questions – AP Precalculus- per latest AP Precalculus Syllabus.

AP Precalculus – MCQ Exam Style Questions- All Topics

Question 

Which of the following describes the graph of \( f(x) = \cot x \)?
(A) The graph has vertical asymptotes at \( x = \frac{\pi}{2} + n\pi \), where \( k \) is any integer, and the function is increasing on all intervals in its domain.
(B) The graph has vertical asymptotes at \( x = \frac{\pi}{2} + n\pi \), where \( k \) is any integer, and the function is decreasing on all intervals in its domain.
(C) The graph has vertical asymptotes at \( x = \pi + n\pi \), where \( k \) is any integer, and the function is increasing on all intervals in its domain.
(D) The graph has vertical asymptotes at \( x = \pi + n\pi \), where \( k \) is any integer, and the function is decreasing on all intervals in its domain.
▶️ Answer/Explanation
Detailed solution

\( f(x) = \cot x = \frac{\cos x}{\sin x} \).
Vertical asymptotes occur where \( \sin x = 0 \) → \( x = 0 + n\pi = n\pi \) (or \( x = \pi + n\pi \) is same as \( x = n\pi \) if \( n \) starts from 0). In list form: \( x = k\pi \) for integer \( k \). Option (D) says \( x = \pi + n\pi \), which is equivalent if \( n \) ranges over all integers.
Cotangent is decreasing on each interval \( (k\pi, (k+1)\pi) \).
Thus (D) matches: asymptotes at \( x = \pi + n\pi \) (i.e., \( k\pi \)) and decreasing.
✅ Answer: (D)

Question 

Which of the following is the graph of \( f(x) = 3\sin(2x) \) in the \( xy \)-plane?
Key features of \( f(x) = 3\sin(2x) \):
Amplitude = 3, Period = \( \frac{2\pi}{2} = \pi \).
Graph passes through origin, oscillates between \( y = 3 \) and \( y = -3 \), completes one full cycle from \( x = 0 \) to \( x = \pi \).

▶️ Answer/Explanation
Detailed solution

\( f(x) = 3\sin(2x) \) has:
Amplitude \( = 3 \), period \( = \frac{2\pi}{2} = \pi \).
The graph starts at \( (0,0) \), increases to a maximum at \( x = \frac{\pi}{4} \), returns to 0 at \( x = \frac{\pi}{2} \), reaches a minimum at \( x = \frac{3\pi}{4} \), and returns to 0 at \( x = \pi \).
Among the options, the correct graph shows this behavior with amplitude 3 and period \( \pi \).
✅ Answer: (D)

Question 

The function \( g \) is given by \( g(\theta) = \tan(2\pi\theta) \). Which of the following statements about the graph of \( g \) in the \( xy \)-plane is true?
(A) The vertical asymptotes occur at \( \theta = 0 + \frac{1}{2}k \), where \( k \) is an integer.
(B) The vertical asymptotes occur at \( \theta = \frac{1}{4} + \frac{1}{2}k \), where \( k \) is an integer.
(C) The vertical asymptotes occur at \( \theta = \frac{1}{4} + k \), where \( k \) is an integer.
(D) The vertical asymptotes occur at \( \theta = \frac{1}{2} + 2k \), where \( k \) is an integer.
▶️ Answer/Explanation
Detailed solution

For \( g(\theta) = \tan(2\pi\theta) \), the period is \( \frac{\pi}{2\pi} = \frac{1}{2} \).
Vertical asymptotes occur where \( \cos(2\pi\theta) = 0 \), i.e., where \( 2\pi\theta = \frac{\pi}{2} + \pi k \).
Solving: \( \theta = \frac{1}{4} + \frac{1}{2}k \), where \( k \) is an integer.
✅ Answer: (B)

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