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.
