Question: Which of the following equations defines the following quadratic function ?
Graph shape: a downward-opening parabola in the center-left of the image, with vertex at approximately $(1, 3)$ and axis of symmetry $x = 1$.
y = -1/2 (x - 1)^2 - 3
y = -1/2 (x + 1)^2 - 3
y = -2 (x - 1)^2 + 3
y = -2 (x + 1)^2 + 3
1. **State the problem:** We need to identify which quadratic equation matches a parabola with vertex at $(1, 3)$, opening downward, and axis of symmetry $x=1$.
2. **Recall the vertex form of a quadratic function:** $$y = a(x - h)^2 + k$$ where $(h, k)$ is the vertex and $a$ determines the direction and width of the parabola.
3. **Analyze the vertex:** Given vertex $(1, 3)$, the equation must be of the form $$y = a(x - 1)^2 + 3$$.
4. **Determine the direction:** The parabola opens downward, so $a < 0$.
5. **Check the options:**
- $y = -\frac{1}{2}(x - 1)^2 - 3$ has vertex $(1, -3)$, which is incorrect.
- $y = -\frac{1}{2}(x + 1)^2 - 3$ has vertex $(-1, -3)$, incorrect.
- $y = -2(x - 1)^2 + 3$ has vertex $(1, 3)$ and $a = -2 < 0$, matches all conditions.
- $y = -2(x + 1)^2 + 3$ has vertex $(-1, 3)$, incorrect.
6. **Conclusion:** The correct equation is $$y = -2(x - 1)^2 + 3$$.