To sketch the graph of the function [tex]\( f(x) = \left(\frac{5}{2}\right)^{x-2} \)[/tex], begin by recognizing how transformations affect basic exponential functions.
The basic exponential function in question is:
[tex]\[ y = \left(\frac{5}{2}\right)^x \][/tex]
To obtain the graph of [tex]\( f(x) = \left(\frac{5}{2}\right)^{x-2} \)[/tex], consider the transformation applied to the exponent.
1. The function [tex]\( \left(\frac{5}{2}\right)^{x-2} \)[/tex] indicates a horizontal shift of the basic exponential graph [tex]\( \left(\frac{5}{2}\right)^x \)[/tex].
2. Specifically, the term [tex]\( x-2 \)[/tex] means that you should shift the graph 2 units to the right.
Therefore, starting with the graph of [tex]\( y = \left(\frac{5}{2}\right)^x \)[/tex], we shift the entire graph 2 units to the right.
Given the choices:
A. Start with the graph of [tex]\( y = \left(\frac{5}{2}\right)^x \)[/tex]. Shift the graph 2 units to the left.
B. Start with the graph of [tex]\( y = \left(\frac{5}{2}\right)^x \)[/tex]. Shift the graph 2 units down.
C. Start with the graph of [tex]\( y = \left(\frac{5}{2}\right)^x \)[/tex]. Shift the graph 2 units to the right.
D. Start with the graph of [tex]\( y = \left(\frac{5}{2}\right)^x \)[/tex]. Shift the graph 2 units up.
The correct choice is:
C. Start with the graph of [tex]\( y = \left(\frac{5}{2}\right)^x \)[/tex]. Shift the graph 2 units to the right.
