To determine which equation in vertex form matches the graph of the firework's trajectory, you should look for key characteristics in the equations such as the vertex and the direction of the parabola (whether it opens upwards or downwards).
- Vertex Form: The vertex form of a quadratic equation is \( y = a(x - h)^2 + k \), where the vertex is at the point \( (h, k) \).
- Direction: The value of \( a \) determines the direction the parabola opens. If \( a > 0 \), it opens upwards. If \( a < 0 \), it opens downwards.
- Vertex Identification: Identify the vertex from the equations given.
- \( y = -\frac{1}{2}(x - 5)^2 - 10 \) => vertex \( (5, -10) \) (opens down)
- \( y = \frac{1}{2}(x - 5)^2 + 10 \) => vertex \( (5, 10) \) (opens up)
- \( y = \frac{1}{2}(x + 5)^2 + 10 \) => vertex \( (-5, 10) \) (opens up)
- \( y = -\frac{1}{2}(x - 5)^2 + 10 \) => vertex \( (5, 10) \) (opens down)
Check the following:
- If the graph shows a maximum point, the parabola likely opens down (look for equations with \( a < 0 \)).
- If the graph shows a minimum point, the parabola likely opens up (look for equations with \( a > 0 \)).
- The vertex coordinates should match the vertex shown in the graph.
Please refer to the graph to make a decision based on the above points. If the firework's peak is at \( (5, 10) \) and it goes downwards, then the correct equation would be \( y = -\frac{1}{2}(x - 5)^2 + 10 \). If it reaches a point at \( (5, -10) \) and then rises, the correct equation is \( y = -\frac{1}{2}(x - 5)^2 - 10 \).
Provide the specific features of the graph for a definitive answer.