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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 minus 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 plus 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

y=12(x+5)2+10

1 answer

To determine which equation in vertex form matches the graph of the downward-opening parabola, we need to analyze some key characteristics:

  1. Vertex Form of a Quadratic Equation: The vertex form is \( y = a(x-h)^2 + k \), where \( (h, k) \) is the vertex. The value of \( a \) determines the direction of opening; if \( a < 0 \), the parabola opens downward.

  2. Determine Vertex from the Graph: The coordinates of the vertex can be derived from the graph. If the vertex appears to be at \( (5, 10) \), then \( h = 5 \) and \( k = 10 \).

  3. Direction of Opening: Since the parabola opens downward, \( a \) must be negative.

Based on this information, we can evaluate the response options:

  1. Option 1: \( y = -\frac{1}{2}(x-5)^2 - 10 \) (opposite direction for vertex)
  2. Option 2: \( y = -\frac{1}{2}(x-5)^2 + 10 \) (correct vertex, opening downward)
  3. Option 3: \( y = \frac{1}{2}(x-5)^2 + 10 \) (opening upward)
  4. Option 4: \( y = \frac{1}{2}(x+5)^2 + 10 \) (opening upward)

Given that the vertex seems to be \( (5, 10) \) and the parabola opens downward, the correct equation that matches the graph is:

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