To determine which equation in vertex form matches the given graph of a downward-opening parabola, we need to find the vertex and the direction of the parabola.
-
Vertex form of a parabola: The general form is \( y = a(x - h)^2 + k \), where \((h, k)\) is the vertex. If \(a\) is negative, the parabola opens downward.
-
Identifying the vertex: Since the graph opens downward, we want an equation with a negative coefficient \(a\). The graph likely has its vertex at the point (5, 10), as this appears to be where the highest point of the parabola is located based on your description.
-
Evaluating the options:
- The first option \(y = -\frac{1}{2}(x - 5)^2 + 10\) has vertex \((5, 10)\) and opens downward.
- The second option \(y = \frac{1}{2}(x + 5)^2 + 10\) opens upward and does not match.
- The third option \(y = -\frac{1}{2}(x - 5)^2 - 10\) has its vertex at (5, -10) and opens downward.
- The fourth option \(y = \frac{1}{2}(x - 5)^2 + 10\) opens upward, which does not match the downward trajectory.
Based on this analysis, the equation that matches the graph of the firework's trajectory (which opens downward and has a vertex at (5, 10)) is:
y = -1/2(x - 5)^2 + 10.
1 answer