Use what you know about translations of functions to analyze the graph of the function [tex]f(x)=(0.5)^{x-5}+8[/tex]. You may wish to graph it and its parent function, [tex]y=0.5^x[/tex], on the same axes. The parent function [tex]y=0.5^x[/tex] is ______ across its domain because its base, b, is such that ______. The function, [tex]f[/tex], shifts the parent function 8 units ______. The function, [tex]f[/tex], shifts the parent function 5 units ______.
Answer (2)
The parent function y = 0. 5 x is decreasing because its base is between 0 and 1. The function f ( x ) = ( 0.5 ) x − 5 + 8 shifts the parent function 8 units upward and 5 units to the right. Thus, there are both vertical and horizontal transformations applied to the graph of the parent function.
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The parent function y = 0. 5 x is decreasing because 0 < b < 1 .
The function f shifts the parent function 8 units upward.
The function f shifts the parent function 5 units to the right.
Therefore, the transformations are a vertical shift of 8 units up and a horizontal shift of 5 units to the right. .
Explanation
Decreasing Parent Function The parent function is y = 0. 5 x . Since the base b = 0.5 is between 0 and 1 (i.e., 0 < b < 1 ), the function is decreasing across its domain.
Vertical Shift The function f ( x ) = ( 0.5 ) x − 5 + 8 shifts the parent function vertically by 8 units. The '+8' term indicates a vertical shift upwards.
Horizontal Shift The function f ( x ) = ( 0.5 ) x − 5 + 8 shifts the parent function horizontally by 5 units. The 'x-5' term in the exponent indicates a horizontal shift to the right.
Examples
Consider how sound volume decreases as you move away from a speaker. The exponential decay is similar to the parent function, where volume diminishes rapidly at first, then slows down. The transformations we discussed can model scenarios where the initial volume is adjusted (vertical shift) or the distance from the speaker is altered (horizontal shift). Understanding these shifts helps in predicting sound levels at different locations, optimizing speaker placement for even sound distribution, or designing audio systems with specific decay characteristics.
