Question: A quadratic function $y = f(x)$ is plotted on a graph and the vertex of the resulting parabola is $(4, 5)$. 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 $(4, 5)$.
We want to find the vertex of $g(x) = -f(x) + 3$.
2. **Recall the vertex form and transformations:**
The vertex of $f(x)$ is $(h, k) = (4, 5)$.
The function $g(x) = -f(x) + 3$ reflects $f(x)$ over the x-axis and then shifts it up by 3 units.
3. **Apply the transformations to the vertex:**
- Reflection over x-axis changes the $y$-coordinate from $k$ to $-k$.
- Vertical shift up by 3 adds 3 to the $y$-coordinate.
So, the new vertex $V_g$ is:
$$V_g = (h, -k + 3) = (4, -5 + 3)$$
4. **Calculate the new vertex:**
$$V_g = (4, -2)$$
**Final answer:** The vertex of $g(x)$ is $(4, -2)$.