Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN: 9780134463216
Author: Robert F. Blitzer
Publisher: PEARSON
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Question

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 \( f(x) = 4^x \).
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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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