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AP Precalculus -2.4 Exponential Function Manipulation- MCQ Exam Style Questions - Effective Fall 2023

AP Precalculus -2.4 Exponential Function Manipulation- MCQ Exam Style Questions – Effective Fall 2023

AP Precalculus -2.4 Exponential Function Manipulation- MCQ Exam Style Questions – AP Precalculus- per latest AP Precalculus Syllabus.

AP Precalculus – MCQ Exam Style Questions- All Topics

Question 

Iodine-131 has a half-life of 8 days. In a particular sample, the amount of iodine-131 remaining after \( d \) days can be modeled by the function \( h \) given by \( h(d) = A_0(0.5)^{(d/8)} \), where \( A_0 \) is the amount of iodine-131 in the sample at time \( d = 0 \). Which of the following functions \( k \) models the amount of iodine-131 remaining after \( t \) hours, where \( A_0 \) is the amount of iodine-131 in the sample at time \( t = 0 \)? (There are 24 hours in a day, so \( t = 24d \).)
(A) \( k(t) = A_0(0.5)^{(t/24)} \)
(B) \( k(t) = A_0\left(0.5^{(1/24)}\right)^{(8t)} \)
(C) \( k(t) = A_0\left(0.5^{(24)}\right)^{(t/8)} \)
(D) \( k(t) = A_0\left(0.5^{(1/192)}\right)^t \)
▶️ Answer/Explanation
Detailed solution

Given \( h(d) = A_0(0.5)^{d/8} \) with \( t = 24d \), so \( d = \frac{t}{24} \).
Substitute into \( h \):
\( k(t) = A_0(0.5)^{\frac{t/24}{8}} = A_0(0.5)^{t/(192)} \)
Rewrite exponent: \( 0.5^{t/192} = \left(0.5^{1/192}\right)^t \).
Thus \( k(t) = A_0\left(0.5^{1/192}\right)^t \).
✅ Answer: (D)

Question 

The value, in millions of dollars, of transactions processed by an online payment platform is modeled by the function \( M \). The value is expected to increase by 6.1% each quarter of a year. At time \( t = 0 \) years, 54 million dollars of transactions were processed. If \( t \) is measured in years, which of the following is an expression for \( M(t) \)?
(Note: A quarter is one fourth of a year.)
(A) \( 54(0.061)^{(t/4)} \)
(B) \( 54(0.061)^{(4t)} \)
(C) \( 54(1.061)^{(t/4)} \)
(D) \( 54(1.061)^{(4t)} \)
▶️ Answer/Explanation
Detailed solution

Growth per quarter: multiply by \( 1 + 0.061 = 1.061 \).
Each year has 4 quarters, so in \( t \) years there are \( 4t \) quarters.
Starting value \( A_0 = 54 \).
Thus \( M(t) = 54 \cdot (1.061)^{4t} \).
✅ Answer: (D)

Question 

The function \( f \) is given by \( f(x) = 2^x \), and the function \( g \) is given by \( g(x) = \frac{f(x)}{8} \). For which of the following transformations is the graph of \( g \) the image of the graph of \( f \)?
(A) A horizontal translation to the left 3 units
(B) A horizontal translation to the right 3 units
(C) A vertical translation up \( \frac{1}{8} \) unit
(D) A vertical translation down \( \frac{1}{8} \) unit
▶️ Answer/Explanation
Detailed solution

\( g(x) = \frac{2^x}{8} = \frac{2^x}{2^3} = 2^{x-3} \).
This is \( f(x-3) \), which corresponds to shifting the graph of \( f \) horizontally to the right by 3 units.
✅ Answer: (B)

Question 

The function \( m \) is given by \( m(x) = 36^{(x/2)} \). Which of the following expressions could also define \( m(x) \)?
(A) \( 6^x \)
(B) \( 6 \cdot 6^x \)
(C) \( 18^x \)
(D) \( 18 \cdot 36^x \)
▶️ Answer/Explanation
Detailed solution

\( 36^{(x/2)} = (36^{1/2})^x \).
Since \( 36^{1/2} = 6 \), we have \( 6^x \).
✅ Answer: (A)

Question 

The function \( k \) is given by \( k(x) = 9^x \). Which of the following expressions also defines \( k(x) \)?
(A) \( 2^{(3x)} \)
(B) \( 3^{(2x)} \)
(C) \( 3^{(3x)} \)
(D) \( 3^{(x/2)} \)
▶️ Answer/Explanation
Detailed solution

\( 9^x = (3^2)^x = 3^{2x} \).
✅ Answer: (B)

Question 

The function \( f \) is given by \( f(x) = 3^x \). The function \( g \) is given by \( g(x) = (f(x))^b \), where \( b < 0 \). Which of the following describes the relationship between the graphs of \( f \) and \( g \)?
(A) The graph of \( g \) is a combination of a horizontal dilation of the graph of \( f \) and a reflection over the \( x \)-axis.
(B) The graph of \( g \) is a combination of a horizontal dilation of the graph of \( f \) and a reflection over the \( y \)-axis.
(C) The graph of \( g \) is a combination of a vertical dilation of the graph of \( f \) and a reflection over the \( x \)-axis.
(D) The graph of \( g \) is a combination of a vertical dilation of the graph of \( f \) and a reflection over the \( y \)-axis.
▶️ Answer/Explanation
Detailed solution

\( g(x) = (3^x)^b = 3^{bx} \), with \( b < 0 \).
Let \( b = -c \) where \( c > 0 \), then \( g(x) = 3^{-cx} = (3^{-c})^x \).
The transformation \( x \to -cx \) represents a horizontal stretch/compression (dilation) and a reflection over the \( y \)-axis (due to negative sign).
✅ Answer: (B)

Question 

The functions \( f \) and \( g \) are given by \( f(x) = 2^x \) and \( g(x) = 2^x \cdot 2^a \), where \( a > 0 \). Which of the following describes the relationship between the graph of \( f \) and the graph of \( g \)?
(A) The graph of \( g \) is a vertical translation of the graph of \( f \) by \( a \) units.
(B) The graph of \( g \) is a horizontal translation of the graph of \( f \) by \( a \) units.
(C) The graph of \( g \) is a vertical translation of the graph of \( f \) by \( -a \) units.
(D) The graph of \( g \) is a horizontal translation of the graph of \( f \) by \( -a \) units.
▶️ Answer/Explanation
Detailed solution

\( g(x) = 2^x \cdot 2^a = 2^{x+a} = f(x+a) \).
This corresponds to shifting the graph of \( f \) left by \( a \) units (horizontal translation by \( -a \) units).
✅ Answer: (D)

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