Algebra and Trigonometry (6th Edition)
6th Edition
ISBN: 9780134463216
Author: Robert F. Blitzer
Publisher: PEARSON
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for example: f(x)=3*2^x write the thing and look at the picture asap ty
![### Problem Statement
Write an exponential function that contains the points \( (3,32) \) and \( (6,256) \).
### Solution
To find the exponential function that passes through the given points, we start with the general form of an exponential function:
\[ y = ab^x \]
Using the given points, we can set up a system of equations. For the point \( (3, 32) \):
\[ 32 = ab^3 \]
For the point \( (6, 256) \):
\[ 256 = ab^6 \]
By dividing the second equation by the first, we eliminate \(a\) and solve for \(b\):
\[ \frac{256}{32} = \frac{ab^6}{ab^3} \]
\[ 8 = b^3 \]
Taking the cube root of both sides:
\[ b = \sqrt[3]{8} \]
\[ b = 2 \]
Now that we have \( b \), we can substitute it back into the first equation to solve for \(a\):
\[ 32 = a(2)^3 \]
\[ 32 = a \cdot 8 \]
\[ a = \frac{32}{8} \]
\[ a = 4 \]
Thus, the exponential function that contains the points \( (3, 32) \) and \( (6, 256) \) is:
\[ y = 4 \cdot 2^x \]
### Verification
To ensure our solution is correct, let's verify the function with the given points:
- For \( (3, 32) \):
\[ y = 4 \cdot 2^3 = 4 \cdot 8 = 32 \]
- For \( (6, 256) \):
\[ y = 4 \cdot 2^6 = 4 \cdot 64 = 256 \]
Both points satisfy the equation, confirming that the derived exponential function \( y = 4 \cdot 2^x \) is accurate.](https://content.bartleby.com/qna-images/question/541e9811-c8f1-44f7-9808-3fa399aa819f/d1314776-1983-48d9-97af-da8557fd2d63/gsiyyyd_processed.png)
Transcribed Image Text:### Problem Statement
Write an exponential function that contains the points \( (3,32) \) and \( (6,256) \).
### Solution
To find the exponential function that passes through the given points, we start with the general form of an exponential function:
\[ y = ab^x \]
Using the given points, we can set up a system of equations. For the point \( (3, 32) \):
\[ 32 = ab^3 \]
For the point \( (6, 256) \):
\[ 256 = ab^6 \]
By dividing the second equation by the first, we eliminate \(a\) and solve for \(b\):
\[ \frac{256}{32} = \frac{ab^6}{ab^3} \]
\[ 8 = b^3 \]
Taking the cube root of both sides:
\[ b = \sqrt[3]{8} \]
\[ b = 2 \]
Now that we have \( b \), we can substitute it back into the first equation to solve for \(a\):
\[ 32 = a(2)^3 \]
\[ 32 = a \cdot 8 \]
\[ a = \frac{32}{8} \]
\[ a = 4 \]
Thus, the exponential function that contains the points \( (3, 32) \) and \( (6, 256) \) is:
\[ y = 4 \cdot 2^x \]
### Verification
To ensure our solution is correct, let's verify the function with the given points:
- For \( (3, 32) \):
\[ y = 4 \cdot 2^3 = 4 \cdot 8 = 32 \]
- For \( (6, 256) \):
\[ y = 4 \cdot 2^6 = 4 \cdot 64 = 256 \]
Both points satisfy the equation, confirming that the derived exponential function \( y = 4 \cdot 2^x \) is accurate.
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