To determine which equation in vertex form matches the graph of the firework trajectory, we need to analyze the characteristics of a parabola in vertex form, which is given by:
\[ y = a(x - h)^2 + k \]
where \( (h, k) \) is the vertex of the parabola, and \( a \) determines the direction (upward if \( a > 0 \) and downward if \( a < 0 \)) and the width of the parabola.
Since the parabola shown opens downward, the value of \( a \) must be negative.
Reviewing the provided options:
- y = 12(x − 5)² + 10 → Opens upwards (a positive value: 12)
- y = (1/2)(x + 5)² + 10 → Opens upwards (a positive value: 1/2)
- y = −12(x − 5)² + 10 → Opens downwards (a negative value: -12)
- y = −(1/2)(x − 5)² − 10 → Opens downwards (a negative value: -1/2)
Since the graph opens downward, we look for options 3 and 4.
Next, we need to identify the vertex. If the vertex appears to be at the point (5, 10) based on your description, then option 3 is the most appropriate because it maintains the vertex at \( (5, 10) \) (since it follows the form):
\[ y = -12(x - 5)^2 + 10 \]
Thus, the final answer is:
y = -12(x − 5)² + 10.