Use what you know about translations of functions to analyze the graph of the function [tex]\( f(x) = (0.5)^{x-5} \)[/tex].

1. Graph the parent function, [tex]\( y = 0.5^x \)[/tex], and the function [tex]\( f(x) = (0.5)^{x-5} \)[/tex] on the same axes.

2. The parent function [tex]\( y = 0.5^x \)[/tex] is [tex]\(\_\_\_\_\_\_\_\_\_\_\)[/tex] across its domain because its base, [tex]\( b \)[/tex], is such that [tex]\(\_\_\_\_\_\_\_\_\_\_\)[/tex].

3. The function [tex]\( f(x) \)[/tex] shifts the parent function 5 units [tex]\(\_\_\_\_\_\_\_\_\_\_\)[/tex].

4. The function [tex]\( f(x) \)[/tex] shifts the parent function [tex]\(\_\_\_\_\_\_\_\_\_\_\)[/tex] units [tex]\(\_\_\_\_\_\_\_\_\_\_\)[/tex].



Answer :

Let's analyze the function [tex]\( f(x) = (0.5)^{x-5} \)[/tex] step-by-step. We'll consider the properties of the parent function [tex]\( y = 0.5^x \)[/tex] and then examine how transformations affect the graph.

1. Behavior of the Parent Function [tex]\( y = 0.5^x \)[/tex]:
- The base of the parent function [tex]\( y = 0.5^x \)[/tex] is [tex]\( 0.5 \)[/tex], which lies between 0 and 1. When the base [tex]\( b \)[/tex] of an exponential function [tex]\( y = b^x \)[/tex] is between 0 and 1, the function is decreasing.
- Therefore, the parent function [tex]\( y = 0.5^x \)[/tex] is decreasing across its domain.

2. Horizontal Shift:
- The function [tex]\( f(x) = (0.5)^{x-5} \)[/tex] is derived from the parent function [tex]\( y = 0.5^x \)[/tex] by replacing [tex]\( x \)[/tex] with [tex]\( x-5 \)[/tex].
- Replacing [tex]\( x \)[/tex] with [tex]\( x-5 \)[/tex] results in a horizontal shift. Specifically, it shifts the graph of the parent function 5 units to the right.

3. Vertical Shift:
- The function [tex]\( f(x) = (0.5)^{x-5} \)[/tex] has no additional constants added or subtracted outside the exponent, which means there is no vertical shift.
- Therefore, the vertical shift is 0 units.

Putting these observations together:

- The parent function [tex]\( y = 0.5^x \)[/tex] is decreasing across its domain because its base, [tex]\( b \)[/tex], is such that [tex]\( 0 < b < 1 \)[/tex].
- The function [tex]\( f \)[/tex] shifts the parent function 5 units to the right.
- The function [tex]\( f \)[/tex] has no vertical shift.

Thus, the complete analysis of the function [tex]\( f(x) = (0.5)^{x-5} \)[/tex] with respect to its transformations is:

The parent function [tex]\( y = 0.5^x \)[/tex] is decreasing across its domain because its base, [tex]\( b \)[/tex], is such that [tex]\( 0 < b < 1 \)[/tex].

The function [tex]\( f \)[/tex] shifts the parent function 5 units to the right.

The function [tex]\( f \)[/tex] has no vertical shift.