Sketch the graph of the function and check the graph with a graphing calculator. Before doing so, describe how the graph of the function can be obtained from the graph of a basic exponential function.

[tex]\[ f(x)=\left(\frac{5}{2}\right)^{x-2} \][/tex]

Select the correct choice below and fill in the answer box to complete your choice.

A. Start with the graph of [tex]\( y=\left(\square\right)^x \)[/tex]. Shift the graph 2 units to the left.

B. Start with the graph of [tex]\( y=\left(\square\right)^x \)[/tex]. Shift the graph 2 units down.

C. Start with the graph of [tex]\( y=\left(\square\right)^x \)[/tex]. Shift the graph 2 units to the right.

D. Start with the graph of [tex]\( y=\left(\square\right)^x \)[/tex]. Shift the graph 2 units up.



Answer :

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.