To determine which equation in vertex form matches the given graph of a parabola that opens downward, we need to analyze the parameters of vertex form, which is:
\[ y = a(x - h)^2 + k \]
In this equation:
- \( (h, k) \) is the vertex of the parabola.
- \( a \) determines the direction and width of the parabola: if \( a < 0 \), the parabola opens downward, and if \( a > 0 \), it opens upward.
Based on your description:
- The parabola opens downward, indicating \( a < 0 \).
- The vertex appears to be at \( (5, 10) \).
Now, let's evaluate the options provided:
-
\( y = -12(x - 5)^2 + 10 \):
- This is a downward opening parabola with vertex \( (5, 10) \) (correct form).
-
\( y = \frac{-1}{2}(x - 5)^2 + 10 \):
- This is also a downward opening parabola with vertex \( (5, 10) \) (correct form but different \( a \)).
-
\( y = -12(x - 5)^2 - 10 \):
- This is a downward opening parabola, but the vertex is at \( (5, -10) \) (incorrect vertex).
-
\( y = 12(x - 5)^2 + 10 \):
- This is an upward opening parabola (incorrect direction).
Given this analysis, the equations that correctly feature a downward-opening parabola with the vertex at \( (5, 10) \) are the first two options.
If you are only looking for one correct response, the best match would be:
\( y = -12(x - 5)^2 + 10 \)
This option maintains the downward trajectory and has the correct vertex.