To determine the correct equation in vertex form that matches the graph of the parabola representing the firework's trajectory, we need to analyze the characteristics based on the information provided:
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Direction of Opening: The parabola opens downward, which means the coefficient of the squared term (the leading coefficient) should be negative.
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Vertex: The vertex form of a parabola is given by the equation \( y = a(x - h)^2 + k \), where \((h, k)\) represents the vertex. Since the graph is shown on a coordinate plane, we need to find the coordinates of the vertex from the graph. Based on typical vertex placements, if the vertex appears to be at \((5, 10)\), then \(h = 5\) and \(k = 10\).
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Coefficient: Since the parabola opens downward, the value of \(a\) must be negative. We can infer the value of \(a\) from the shape of the parabola, but without an exact observation of how steep or wide the parabola is on the graph, we can make an educated guess based on the responses given.
Considering these factors, let's evaluate the provided equations:
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\(y = -12(x - 5)^2 - 10\): This opens downward, has a vertex at (5, -10) — not matching.
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\(y = \frac{1}{2}(x - 5)^2 - 10\): This opens upward and has a vertex at (5, -10) — not matching.
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\(y = -\frac{1}{2}(x - 5)^2 + 10\): This opens downward and has a vertex at (5, 10) — this matches.
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\(y = \frac{1}{2}(x + 5)^2 + 10\): This opens upward with a vertex at (-5, 10) — not matching.
Based on the analysis, the correct equation in vertex form that matches the graph of the firework's trajectory is:
\[ y = -\frac{1}{2}(x - 5)^2 + 10\]
1 answer