To find the vertex form of the equation of a parabola that opens downward, we can use the vertex form equation:
\[ y = a(x-h)^2 + k \]
where \((h, k)\) is the vertex of the parabola, and \(a\) is a coefficient that determines the direction and the width of the parabola.
From the information given, you have a point (-3, -1) that lies on the parabola. We need to find the vertex, which is typically the highest point for a downward-opening parabola.
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Identify the vertex: Since the vertex is not explicitly given in your description, we need to estimate it based on the shape and the fact that the parabola opens downward. Let's assume that the vertex is the highest point within the range given.
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Determine the vertex and open direction: If we assume that the vertex (h, k) is slightly to the right of -3 along the x-axis and has a y-coordinate greater than -1 (since the curve is upward next to -3), let’s say the vertex is (-2, 2) as a hypothetical place for a vertex just for demonstration purposes, but the exact vertex would depend on the plot.
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Determine the "a" value: Since we are given that the parabola opens downward, we can assume the value of a < 0. To find 'a', we can use the point (-3, -1) we know is on the parabola.
Let’s assume the vertex is (-2, 2) for calculation. Therefore, from our vertex form:
\[ y = a(x + 2)^2 + 2 \]
Now plug in the point (-3, -1):
\[ -1 = a(-3 + 2)^2 + 2 \] \[ -1 = a(-1)^2 + 2 \] \[ -1 = a + 2 \] \[ a = -3 \]
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Write the equation: The final equation in vertex form is:
\[ y = -3(x + 2)^2 + 2 \]
Given the specific format in your question, modify it accordingly:
\[ y = -3(x - (-2))^2 + 2 \]
So the answers you need to fill in are:
- \(h = -2\)
- \(k = 2\)
Thus:
\[ y = -3 (x + 2)^2 + 2 \]
You can also plug the appropriate values in the blanks you provided:
- a = -3
- h = -2
- k = 2
Use these to complete your required vertex form:
- Final answer in the format presented in your question:
\(y = -3(x - (-2))^2 + 2\) or in your fill format:
\(y = -2(x -(-2))^2 + 2\) as needed based on understanding of the vertex. Double check your initial points and vertex position for final correction based on graph interpretation.