Question: A quadratic function $y = f(x)$ is plotted on a graph and the vertex of the resulting parabola is $(3, 6)$. What is the vertex of the function defined as $g(x) = f(-x) - 3$?
1. **State the problem:** We have a quadratic function $f(x)$ with vertex at $(3, 6)$.
2. **Recall vertex form:** A quadratic function can be written as $f(x) = a(x - h)^2 + k$ where $(h, k)$ is the vertex.
3. **Given vertex:** Here, $h = 3$ and $k = 6$, so $f(x) = a(x - 3)^2 + 6$ for some $a$.
4. **Define $g(x)$:** $g(x) = f(-x) - 3$.
5. **Substitute $f(-x)$:**
$$g(x) = a(-x - 3)^2 + 6 - 3 = a(-x - 3)^2 + 3$$
6. **Simplify inside the square:**
$$-x - 3 = -(x + 3)$$
7. **Square the expression:**
$$(-x - 3)^2 = (x + 3)^2$$
8. **Rewrite $g(x)$:**
$$g(x) = a(x + 3)^2 + 3$$
9. **Identify vertex of $g(x)$:** The vertex form is $a(x - h')^2 + k'$, so here $h' = -3$ and $k' = 3$.
10. **Final answer:** The vertex of $g(x)$ is $(-3, 3)$.