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
the function, f, shifts the parent function 8 units
the function, f, shifts the parent function 5 units

Explanation:

Step1: Analyze horizontal shift

For an exponential function \( y = b^{x - h}+k \), the horizontal shift is determined by \( h \). In \( f(x)=(0.5)^{x - 5}+8 \), comparing to \( y = 0.5^{x} \), the horizontal shift is \( h = 5 \). But first, check the base \( b = 0.5 \), which is between \( 0 \) and \( 1 \), so the parent function \( y = 0.5^{x} \) is a decreasing function. Wait, the first blank: the parent function \( y = 0.5^{x} \) is decreasing across its domain because \( 0 < b=0.5< 1 \).

Step2: Analyze horizontal shift of \( f(x) \)

The function \( f(x)=(0.5)^{x - 5}+8 \) has a horizontal shift. The form \( y = b^{x - h} \) shifts the parent function \( y = b^{x} \) \( h \) units to the right. Here \( h = 5 \), but wait, the second part: "The function, \( f \), shifts the parent function 8 units" – no, wait, the vertical shift is \( k = 8 \), and horizontal shift is 5 units. Wait, let's re - examine:

First blank: The parent function \( y = 0.5^{x} \) is decreasing across its domain because its base \( b = 0.5 \) is such that \( 0 < b<1 \).

Second: Wait, no, the first sentence: "The parent function \( y = 0.5^{x} \) is \(\underline{}\) across its domain because its base, \( b \), is such that \(\underline{}\)". Since \( 0 < 0.5<1 \), the exponential function is decreasing. So first blank: decreasing.

Third: "The function, \( f \), shifts the parent function 8 units" – no, vertical shift. Wait, the function \( f(x)=(0.5)^{x - 5}+8 \): the horizontal shift is 5 units to the right (because of \( x - 5 \)) and vertical shift 8 units up. But the second statement: "The function, \( f \), shifts the parent function 8 units" – vertical shift, so 8 units up. And "The function, \( f \), shifts the parent function 5 units" – 5 units to the right.

Wait, let's correct:

1. The parent function \( y = 0.5^{x} \) is decreasing across its domain because its base \( b = 0.5 \) satisfies \( 0 < b<1 \).

1. The function \( f(x)=(0.5)^{x - 5}+8 \) shifts the parent function 5 units to the right (horizontal shift, due to \( x - 5 \)) and 8 units up (vertical shift, due to \( + 8 \)).

Assuming the blanks:

First blank: decreasing (because \( 0 < 0.5<1 \), exponential function with \( 0 < b<1 \) is decreasing).

Second (about 8 units): The function \( f \) shifts the parent function 8 units up (vertical shift).

Third (about 5 units): The function \( f \) shifts the parent function 5 units to the right (horizontal shift).

Answer:

First blank: decreasing; Second (8 units): up; Third (5 units): to the right (assuming the blanks are for "decreasing", "up", "to the right" respectively, based on function transformation rules \( f(x)=b^{x - h}+k \), where \( h = 5 \) (right shift), \( k = 8 \) (up shift), and \( 0 < b = 0.5<1 \) (decreasing function)).