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AP Precalculus -2.2 Linear and Exponential Change- MCQ Exam Style Questions - Effective Fall 2023

AP Precalculus -2.2 Linear and Exponential Change- MCQ Exam Style Questions – Effective Fall 2023

AP Precalculus -2.2 Linear and Exponential Change- MCQ Exam Style Questions – AP Precalculus- per latest AP Precalculus Syllabus.

AP Precalculus – MCQ Exam Style Questions- All Topics

Question 

Which of the following tables provides evidence that \( f \) is an exponential function if \( y = f(x) \)?
(A) \[\begin{array}{c|c} x & y \\ \hline 1 & 3 \\ 2 & 6 \\ 3 & 9 \\ 4 & 12 \\ \end{array}\]
(B) \[\begin{array}{c|c} x & y \\ \hline 5 & 30 \\ 6 & 42 \\ 7 & 56 \\ 8 & 72 \\ \end{array}\]
(C) \[\begin{array}{c|c} x & \ln y \\ \hline 1 & 1 \\ 2 & 3 \\ 3 & 9 \\ 4 & 27 \\ \end{array}\]
(D) \[\begin{array}{c|c} x & \ln y \\ \hline 5 & 50 \\ 6 & 60 \\ 7 & 70 \\ 8 & 80 \\ \end{array}\]
▶️ Answer/Explanation
Detailed solution

A function is exponential if the natural logarithm of its output values \( \ln y \) is linear in \( x \).
In option (C), \( \ln y \) values are 1, 3, 9, 27 — these are not linear in \( x \).
In option (D), \( \ln y \) values are 50, 60, 70, 80 — these form an arithmetic sequence, so \( \ln y \) is linear in \( x \).
Thus \( y = e^{mx+b} \) is exponential.
✅ Answer: (D)

Question 

The function \( g \) is a function of the form \( g(x) = a \cdot b^x \), where \( a \neq 0 \) and \( b > 0 \). The function \( f \) is given by \( f(x) = g(x) + 4 \). Which of the following statements is true?
(A) The output values of both \( f \) and \( g \) are proportional over equal-length input-value intervals.
(B) The output values of \( f \) only, not \( g \), are proportional over equal-length input-value intervals.
(C) The output values of \( g \) only, not \( f \), are proportional over equal-length input-value intervals.
(D) The output values of neither \( f \) nor \( g \) are proportional over equal-length input-value intervals.
▶️ Answer/Explanation
Detailed solution

Exponential functions \( g(x) = a \cdot b^x \) have outputs that are proportional over equal-length input intervals (ratio constant).
Adding a constant \( +4 \) to \( g \) gives \( f(x) \), which is no longer a pure exponential; its successive ratios are not constant.
Thus only \( g \) has proportional output changes over equal-length input intervals.
✅ Answer: (C)

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