The table of ordered pairs (x, y) gives an exponential function. Write an equation for the function. X -1 0 y 1 4 2
The table of ordered pairs (x, y) gives an exponential function. Write an equation for the function. X -1 0 y 1 4 2
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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![### Writing an Exponential Function Rule from a Table of Ordered Pairs
The table of ordered pairs \((x, y)\) below gives an exponential function. Your task is to write an equation for this function.
| \(x\) | \(y\) |
|:--:|:--:|
| \(-1\) | \(\frac{1}{4}\) |
| \(0\) | \(2\) |
| \(1\) | \(16\) |
| \(2\) | \(128\) |
To begin formulating the exponential function, note that the function appears to follow the general form:
\[ y = ab^x \]
Where:
- \( a \) is the initial value when \( x = 0 \).
- \( b \) is the base or growth factor that can be determined by observing the consecutive terms of the function's output.
Using the data provided:
- When \( x = 0 \), \( y = 2 \), thus \( a = 2 \).
To find \( b \), observe the change in \( y \) as \( x \) increases by 1:
- From \( x = 0 \) to \( x = 1\), \( y \) changes from \( 2 \) to \( 16 \).
- Therefore, \( b = \frac{16}{2} = 8 \).
Now, plug these values into the general form:
\[ y = 2 \cdot 8^x \]
This equation represents the exponential function for the given data.
#### Diagram Description
The image does not contain any diagrams besides the table and the text.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F93976936-9bd0-47c5-8a3c-e89a2ab20129%2Fc8d1b34a-88b3-4cdf-b3fb-9d8cee26495c%2Fa1gc3ht_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Writing an Exponential Function Rule from a Table of Ordered Pairs
The table of ordered pairs \((x, y)\) below gives an exponential function. Your task is to write an equation for this function.
| \(x\) | \(y\) |
|:--:|:--:|
| \(-1\) | \(\frac{1}{4}\) |
| \(0\) | \(2\) |
| \(1\) | \(16\) |
| \(2\) | \(128\) |
To begin formulating the exponential function, note that the function appears to follow the general form:
\[ y = ab^x \]
Where:
- \( a \) is the initial value when \( x = 0 \).
- \( b \) is the base or growth factor that can be determined by observing the consecutive terms of the function's output.
Using the data provided:
- When \( x = 0 \), \( y = 2 \), thus \( a = 2 \).
To find \( b \), observe the change in \( y \) as \( x \) increases by 1:
- From \( x = 0 \) to \( x = 1\), \( y \) changes from \( 2 \) to \( 16 \).
- Therefore, \( b = \frac{16}{2} = 8 \).
Now, plug these values into the general form:
\[ y = 2 \cdot 8^x \]
This equation represents the exponential function for the given data.
#### Diagram Description
The image does not contain any diagrams besides the table and the text.
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