To determine the function \( g(x) \) after translating \( f(x) = x^2 \) 9 units up and 4 units to the right, let's follow these steps:
1. Translation 9 Units Up:
- When a function \( f(x) \) is translated \( k \) units up, we add \( k \) to the function. If we add 9 units up, the function becomes:
[tex]\[
f(x) + 9 \quad \implies \quad x^2 + 9
\][/tex]
2. Translation 4 Units to the Right:
- When a function \( f(x) \) is translated \( h \) units to the right, we replace \( x \) with \( (x - h) \). In this case, we replace \( x \) with \( (x - 4) \):
[tex]\[
f(x - 4) \quad \implies \quad (x - 4)^2
\][/tex]
3. Combining the Translations:
- First, apply the horizontal shift: \( f(x) \rightarrow f(x-4) = (x-4)^2 \)
- Then, apply the vertical shift: \( (x-4)^2 + 9 \)
Finally, the function \( g(x) \) is:
[tex]\[
g(x) = (x - 4)^2 + 9
\][/tex]
Therefore, the correct representation of \( g(x) \) is:
[tex]\[
g(x) = (x-4)^2 + 9
\][/tex]
Among the given options, the correct choice is:
[tex]\[
\boxed{g(x) = (x - 4)^2 + 9}
\][/tex]
