To find the function [tex]\( g(x) \)[/tex] from the given parent function [tex]\( f(x) = x^2 \)[/tex], we need to understand that [tex]\( g(x) \)[/tex] is a vertical compression of [tex]\( f(x) \)[/tex] by a factor of [tex]\( r \)[/tex].
### Step-by-Step Solution
1. Identify the parent function:
The parent function given is [tex]\( f(x) = x^2 \)[/tex].
2. Understand vertical compression:
Vertical compression by a factor of [tex]\( r \)[/tex] means that every [tex]\( y \)[/tex]-value of the function [tex]\( f(x) \)[/tex] is divided by [tex]\( r \)[/tex]. Mathematically, if [tex]\( f(x) \)[/tex] undergoes a vertical compression by a factor of [tex]\( r \)[/tex], the new function [tex]\( g(x) \)[/tex] can be written as:
[tex]\[
g(x) = \frac{f(x)}{r}
\][/tex]
3. Substitute the parent function into the vertical compression formula:
Substitute [tex]\( f(x) = x^2 \)[/tex] into the vertical compression formula:
[tex]\[
g(x) = \frac{x^2}{r}
\][/tex]
4. Simplify the expression:
The expression [tex]\(\frac{x^2}{r}\)[/tex] already represents the function [tex]\( g(x) \)[/tex] in its simplest form.
Therefore, the transformed function [tex]\( g(x) \)[/tex] after applying the vertical compression by a factor of [tex]\( r \)[/tex] to the parent function [tex]\( f(x) = x^2 \)[/tex] is:
[tex]\[
g(x) = \frac{1}{r} x^2
\][/tex]
So, the correct function is [tex]\( g(x) = \frac{1}{r} x^2 \)[/tex].
