To determine the function rule for $g(x)$ based on the given information, let's analyze the problem step-by-step.

General Form of a Translated Quadratic Function:

The function rule for a translated quadratic is written as:

$g(x) = a(x - h)^2 + k$

Here:

• $a$ determines the vertical stretch or compression.
• $h$ determines the horizontal translation.
• $k$ determines the vertical translation.

Step 1: Identify $h$ and $k$

The vertex of $g(x)$ is given as $(8, 0)$. In the form $a(x - h)^2 + k$, the vertex is $(h, k)$.

Thus: $h = 8 \quad \text{and} \quad k = 0$

So the equation becomes: $g(x) = a(x - 8)^2$

Step 2: Determine $a$

Since the problem doesn't provide additional points or information about stretching, we assume $a = 1$, which corresponds to no vertical stretch or compression (same as $f(x) = x^2$).

Final Equation:

$g(x) = (x - 8)^2$

This is the function rule for $g(x)$.

---

Let me know if you'd like to verify with another point or need further details!

Related Questions:

1. How does changing $a$ affect the graph of $g(x)$?
2. What happens to the graph if $k$ is nonzero?
3. How would the vertex change if the equation included a negative $h$?
4. Can you sketch a graph of $g(x) = (x - 8)^2$?
5. How is $g(x)$ related to the parent function $f(x) = x^2$?

Tip:

Always check if any additional points are given, as they are needed to calculate the value of $a$.