To solve this question:

Problem Breakdown

We start with the given function:

$f(x) = x^2 + 4$

We are tasked with translating the function 3 units to the right and 5 units upwards.

Transformation Rules:

1. Rightward translation: To shift a function $h$ units to the right, replace $x$ with $(x - h)$. For this case, $h = 3$, so replace $x$ with $(x - 3)$. $(x - 3)^2 + 4$

2. Upward translation: To shift a function $k$ units upwards, add $k$ to the entire function. For this case, $k = 5$, so add 5: $(x - 3)^2 + 4 + 5$

Simplify: $(x - 3)^2 + 9$

Correct Answer:

The correct choice is d. $(x - 3)^2 + 9$.

Would you like more explanation, or do you have further questions?

Related Questions:

1. How do translations affect the vertex of a parabola?
2. What happens when you translate a function to the left instead of right?
3. How does a downward translation differ in terms of the equation?
4. How can you identify the new vertex of a translated function?
5. Can you rewrite this function in vertex form after translation?

Tip:

Always remember that translating a function horizontally involves modifying the $x$-variable directly, while vertical translations add or subtract from the function itself.