Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.7: Applications
Problem 16EQ
Related questions
Question
![### Understanding Exponential Functions
An exponential function is a mathematical expression in the form \( f(x) = ab^x \), where:
- \( a \) is a constant that represents the initial amount (when \( x = 0 \)).
- \( b \) is the base of the exponential, representing the growth factor if \( b > 1 \) or the decay factor if \( 0 < b < 1 \).
- \( x \) is the variable.
#### Example Problem:
Determine the values of \( a \) and \( b \) for the exponential function \( f(x) = ab^x \) that passes through the points \( (0, 4) \) and \( (3, 108) \).
To solve this, follow these steps:
1. **Substitute the points into the function:**
For the point \( (0, 4) \):
\[ 4 = ab^0 \]
\[ 4 = a \cdot 1 \]
\[ a = 4 \]
For the point \( (3, 108) \):
\[ 108 = 4b^3 \]
2. **Solve for \( b \):**
\[ 108 = 4b^3 \]
\[ b^3 = \frac{108}{4} \]
\[ b^3 = 27 \]
\[ b = \sqrt[3]{27} \]
\[ b = 3 \]
Therefore, the values are:
\[ a = 4 \]
\[ b = 3 \]
### Input Section
Please enter the values of \( a \) and \( b \) in the boxes provided below:
```
a = [ ]
b = [ ]
```
**Question Help:**
For more assistance, click on the video link below:
[](video_link)
### Explanation:
- The **initial point (0, 4)** helps determine the value of \( a \).
- The **second point (3, 108)** helps to solve for \( b \) when substituted back into the function after solving for \( a \).
- The function \( f(x) \) is completely defined once the values of \( a \) and \( b \) are known.
This foundational understanding of exponential functions is critical for algebra and pre](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff7167f26-c4aa-4cbc-ab2c-28c429d612af%2F8d7dfc95-230c-46ac-9eb1-41f23447990a%2Fx5tsypp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Understanding Exponential Functions
An exponential function is a mathematical expression in the form \( f(x) = ab^x \), where:
- \( a \) is a constant that represents the initial amount (when \( x = 0 \)).
- \( b \) is the base of the exponential, representing the growth factor if \( b > 1 \) or the decay factor if \( 0 < b < 1 \).
- \( x \) is the variable.
#### Example Problem:
Determine the values of \( a \) and \( b \) for the exponential function \( f(x) = ab^x \) that passes through the points \( (0, 4) \) and \( (3, 108) \).
To solve this, follow these steps:
1. **Substitute the points into the function:**
For the point \( (0, 4) \):
\[ 4 = ab^0 \]
\[ 4 = a \cdot 1 \]
\[ a = 4 \]
For the point \( (3, 108) \):
\[ 108 = 4b^3 \]
2. **Solve for \( b \):**
\[ 108 = 4b^3 \]
\[ b^3 = \frac{108}{4} \]
\[ b^3 = 27 \]
\[ b = \sqrt[3]{27} \]
\[ b = 3 \]
Therefore, the values are:
\[ a = 4 \]
\[ b = 3 \]
### Input Section
Please enter the values of \( a \) and \( b \) in the boxes provided below:
```
a = [ ]
b = [ ]
```
**Question Help:**
For more assistance, click on the video link below:
[](video_link)
### Explanation:
- The **initial point (0, 4)** helps determine the value of \( a \).
- The **second point (3, 108)** helps to solve for \( b \) when substituted back into the function after solving for \( a \).
- The function \( f(x) \) is completely defined once the values of \( a \) and \( b \) are known.
This foundational understanding of exponential functions is critical for algebra and pre
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