The graph shows the trajectory of a firework. Which equation in vertex form matches this graph?

Question

(5, 10)
(1 point)
Responses

y=−1/2(x−5)2+10

y=1/2(x+5)2+10

y=−1/2(x−5)2−10

y=1/2(x−5)2+10

Answer 1

To determine which equation in vertex form matches the graph of the firework's trajectory, we consider the vertex form of a quadratic equation, which is given by:

\[ y = a(x - h)^2 + k \]

where \((h, k)\) is the vertex of the parabola.

In this case, we know that the vertex of the graph is at the point \((5, 10)\). This means \(h = 5\) and \(k = 10\). Thus the equation can be rewritten as:

\[ y = a(x - 5)^2 + 10 \]

Now we need to identify the value of \(a\). Since we're dealing with a firework, it likely has a parabolic trajectory that opens downwards (as it goes up and then comes down), which means \(a\) should be negative.

Let's examine the response options given:

1. y = -1/2(x - 5)^2 + 10 (This has the correct vertex and a negative \(a\))
2. y = 1/2(x + 5)^2 + 10 (Incorrect vertex \(h\) and positive \(a\))
3. y = -1/2(x - 5)^2 - 10 (Correct vertex \(h\) but incorrect \(k\) and negative \(a\))
4. y = 1/2(x - 5)^2 + 10 (Correct vertex \(h\) but positive \(a\))

Given the analysis of the potential equations, the equation that matches the graph with the vertex \((5, 10)\) and opens downwards is:

y = -1/2(x - 5)^2 + 10

So the correct response is:

y = -1/2(x - 5)^2 + 10