Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![### Writing an Exponential Function
**Problem Statement:**
Write an exponential function \( y = ab^x \) whose graph passes through the points \( (3, 108) \) and \( (5, 972) \).
### Steps to Solve:
1. **Identify the points provided:**
- Point 1: \( (3, 108) \)
- Point 2: \( (5, 972) \)
2. **Set up the system of equations:**
- Using the point \( (3, 108) \):
\[
108 = ab^3
\]
- Using the point \( (5, 972) \):
\[
972 = ab^5
\]
3. **Solve the system of equations to find \( a \) and \( b \):**
Divide the second equation by the first equation to eliminate \( a \):
\[
\frac{972}{108} = \frac{ab^5}{ab^3}
\]
\[
9 = b^2
\]
\[
b = 3 \quad \text{(since \( b \) is a growth factor, it must be positive)}
\]
Substitute \( b = 3 \) back into one of the original equations to find \( a \):
\[
108 = a(3)^3
\]
\[
108 = a \cdot 27
\]
\[
a = 4
\]
4. **Write the final exponential function:**
\[
y = 4 \cdot 3^x
\]
### Final Answer:
\[
y = 4 \cdot 3^x
\]
This function represents the exponential curve that passes through the given points \( (3, 108) \) and \( (5, 972) \).
### Diagram Explanation:
- There are no graphs or diagrams included in this particular problem. The solution involves algebraic manipulation and solving a system of equations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1f0eb3d4-5c31-491e-b1a2-5741c316a209%2Fec6bcf9d-a411-4e37-bf03-c8312ed7e3a4%2Fbw8lnqv_processed.png&w=3840&q=75)
Transcribed Image Text:### Writing an Exponential Function
**Problem Statement:**
Write an exponential function \( y = ab^x \) whose graph passes through the points \( (3, 108) \) and \( (5, 972) \).
### Steps to Solve:
1. **Identify the points provided:**
- Point 1: \( (3, 108) \)
- Point 2: \( (5, 972) \)
2. **Set up the system of equations:**
- Using the point \( (3, 108) \):
\[
108 = ab^3
\]
- Using the point \( (5, 972) \):
\[
972 = ab^5
\]
3. **Solve the system of equations to find \( a \) and \( b \):**
Divide the second equation by the first equation to eliminate \( a \):
\[
\frac{972}{108} = \frac{ab^5}{ab^3}
\]
\[
9 = b^2
\]
\[
b = 3 \quad \text{(since \( b \) is a growth factor, it must be positive)}
\]
Substitute \( b = 3 \) back into one of the original equations to find \( a \):
\[
108 = a(3)^3
\]
\[
108 = a \cdot 27
\]
\[
a = 4
\]
4. **Write the final exponential function:**
\[
y = 4 \cdot 3^x
\]
### Final Answer:
\[
y = 4 \cdot 3^x
\]
This function represents the exponential curve that passes through the given points \( (3, 108) \) and \( (5, 972) \).
### Diagram Explanation:
- There are no graphs or diagrams included in this particular problem. The solution involves algebraic manipulation and solving a system of equations.
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