Answer:
[tex]y=2^{\left(x-2\right)}+3[/tex]
Step-by-step explanation:
To solve this problem, we need to start with the parent function of the exponential function, which is [tex]f(x)=a^x[/tex], where [tex]a[/tex] is the base. In our problem, [tex]a=2[/tex], so our parent function here is [tex]y=2^x[/tex]. Then, we need to perform some transformations to our parent function. Thus:
1. Vertical shrink:
A vertical shrink is a nonrigid transformation because the graph of the function get a distortion in the shape, so this transformation is as follows:
[tex]g(x)=cf(x)[/tex]
where [tex]c[/tex] in this problem equals 0.25 because:
[tex]y=0.25(2^x) \\ \\ y=\frac{1}{4}(2^x) \\ \\ y=\frac{2^x}{2^2} \\ \\ y=2^{(x-2)}[/tex]
2. Vertical shift:
The graph of the function [tex]y=2^{(x-2)}[/tex] get a vertical shift given by:
[tex]y=2^{(x-2)}+3[/tex]
So the graph is shifted 3 units up. So the result is the graph shown above.
