Question

use what you know about translations of functions to analyze the graph of the function $f(x) = (0.5)^{x - 5} + 8$. you may wish to graph it and its parent function, $y = 0.5^x$, on the same axes.

the parent function $y = 0.5^x$ is $\boldsymbol{\quad\quad}$ across its domain because its base, $b$, is such that $\boldsymbol{\quad\quad}$.

the function, $f$, shifts the parent function 8 units $\boldsymbol{\quad\quad}$.

the function, $f$, shifts the parent function 5 units $\boldsymbol{\quad\quad}$.

Explanation:

1. For the first blank: The parent function \( y = 0.5^x \) is an exponential function with base \( b = 0.5 \). Since \( 0 < b < 1 \), exponential functions with \( 0 < b < 1 \) are decreasing across their domain. So the first blank should be "decreasing".
2. For the second blank: The base \( b = 0.5 \) satisfies \( 0 < b < 1 \), which is the condition for an exponential function to be decreasing. So the second blank should be " \( 0 < b < 1 \)".
3. For the third blank: The general form of a vertical shift for a function \( y = f(x) \) is \( y = f(x)+k \), where \( k>0 \) shifts up and \( k < 0 \) shifts down. In \( f(x)=(0.5)^{x - 5}+8 \), the \( + 8 \) indicates a vertical shift. Since \( 8>0 \), it shifts the parent function 8 units up. So the third blank is "up".
4. For the fourth blank: The general form of a horizontal shift for a function \( y = f(x) \) is \( y = f(x - h) \), where \( h>0 \) shifts right and \( h < 0 \) shifts left. In \( f(x)=(0.5)^{x - 5}+8 \), we have \( x-5\), so \( h = 5>0 \), which means it shifts the parent function 5 units right. So the fourth blank is "right".

Answer:

1. The parent function \( y = 0.5^x \) is \(\boldsymbol{\text{decreasing}}\) across its domain because its base, \( b \), is such that \(\boldsymbol{0 < b < 1}\).
2. The function, \( f \), shifts the parent function 8 units \(\boldsymbol{\text{up}}\).
3. The function, \( f \), shifts the parent function 5 units \(\boldsymbol{\text{right}}\).