Use the image to answer the question.

Question

A coordinate plane shows a parabola that opens downward. The x-axis ranges from 0 to 15 in increments of 5. The y-axis ranges from 0 to 15 in increments of 5.

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

(1 point)
Responses

y=−12(x−5)2+10
y equals negative Start Fraction 1 over 2 End Fraction left parenthesis x minus 5 right parenthesis squared plus 10

y=12(x+5)2+10
y equals Start Fraction 1 over 2 End Fraction left parenthesis x plus 5 right parenthesis squared plus 10

y=−12(x−5)2−10
y equals negative Start Fraction 1 over 2 End Fraction left parenthesis x minus 5 right parenthesis squared minus 10

y=12(x−5)2+10
y equals Start Fraction 1 over 2 End Fraction left parenthesis x minus 5 right parenthesis squared plus 10
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Answer 1

To determine which equation in vertex form matches the graph of the parabola that opens downward, we can analyze the characteristics of vertex form and the graph.

The vertex form of a parabola 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 \(a > 0\), downward if \(a < 0\)) and the width of the parabola.

Given the information:

1. The parabola opens downward, which means \(a\) must be negative.
2. From your descriptions, the vertex appears to be at \( (5, 10) \).

Using this information, we can evaluate the options:

1. Option: \(y = -12(x - 5)^2 + 10\)

   • Vertex at (5, 10); this opens downward (correct).

2. Option: \(y = \frac{-1}{2}(x - 5)^2 + 10\)

   • Vertex at (5, 10); this also opens downward (correct but not the steepest).

3. Option: \(y = -12(x - 5)^2 - 10\)

   • Vertex at (5, -10); this does not match.

4. Option: \(y = \frac{1}{2}(x - 5)^2 + 10\)

   • Vertex at (5, 10); this opens upward (incorrect).

Based on this analysis, the correct equation, which matches a downward opening parabola with a vertex at (5, 10), is:

y = -12(x - 5)^2 + 10.