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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Find the exponential function f(x)=ax whose graph is given. Function pass point (0,1) and (-2, 1/16).
![#1. Find the exponential function \( f(x) = a^x \) whose graph is given. The function passes through the points \( (0, 1) \) and \( \left(-2, \frac{1}{16}\right) \).
### Graph Description
The graph shows an exponential curve. It crosses the y-axis at the point \( (0, 1) \), indicating that when \( x = 0 \), \( f(x) = 1 \). This point suggests that the base \( a \) raised to the power of 0 is equal to 1, which is true for any non-zero value of \( a \).
Another indicated point on the graph is \( \left(-2, \frac{1}{16}\right) \). This point helps determine the base \( a \) of the exponential function. The curve exhibits growth as it moves from left to right, which is characteristic of exponential functions with bases greater than 1.
### Steps to Find the Base \( a \)
1. Use the point \( (0, 1) \) to establish that \( a^0 = 1 \), confirming that \( a \) is valid.
2. Use the point \( \left(-2, \frac{1}{16}\right) \):
\[
f(-2) = a^{-2} = \frac{1}{16}
\]
This equation can be solved to find \( a \):
\[
a^2 = 16 \quad \Rightarrow \quad a = \sqrt{16} = 4
\]
Thus, the exponential function is \( f(x) = 4^x \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6334a19f-a7da-4de0-bbdc-42373eb8a643%2F99a8ade8-2649-4b20-beba-775342750b38%2F8e4x53o_processed.png&w=3840&q=75)
Transcribed Image Text:#1. Find the exponential function \( f(x) = a^x \) whose graph is given. The function passes through the points \( (0, 1) \) and \( \left(-2, \frac{1}{16}\right) \).
### Graph Description
The graph shows an exponential curve. It crosses the y-axis at the point \( (0, 1) \), indicating that when \( x = 0 \), \( f(x) = 1 \). This point suggests that the base \( a \) raised to the power of 0 is equal to 1, which is true for any non-zero value of \( a \).
Another indicated point on the graph is \( \left(-2, \frac{1}{16}\right) \). This point helps determine the base \( a \) of the exponential function. The curve exhibits growth as it moves from left to right, which is characteristic of exponential functions with bases greater than 1.
### Steps to Find the Base \( a \)
1. Use the point \( (0, 1) \) to establish that \( a^0 = 1 \), confirming that \( a \) is valid.
2. Use the point \( \left(-2, \frac{1}{16}\right) \):
\[
f(-2) = a^{-2} = \frac{1}{16}
\]
This equation can be solved to find \( a \):
\[
a^2 = 16 \quad \Rightarrow \quad a = \sqrt{16} = 4
\]
Thus, the exponential function is \( f(x) = 4^x \).
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