y = a(x - 8)(x + 1)In the quadratic equation above, a is a nonzero constant. The graph of the equation in the xy-plane is a parabola with vertex (h, k). Which of the following is equal to k, in terms of a ? (A) (-a / 4) - 49a(B) (-49a / 2) + 8a(C) (-49a / 4) - 8a(D) (-98a / 4) - 8a
Question
y = a(x - 8)(x + 1)In the quadratic equation above, a is a nonzero constant. The graph of the equation in the xy-plane is a parabola with vertex (h, k). Which of the following is equal to k, in terms of a ? (A) (-a / 4) - 49a(B) (-49a / 2) + 8a(C) (-49a / 4) - 8a(D) (-98a / 4) - 8a
Solution
The vertex form of a quadratic equation is given by y = a(x - h)² + k, where (h, k) is the vertex of the parabola.
In the given equation y = a(x - 8)(x + 1), we can find the x-coordinate of the vertex by taking the average of the roots of the equation, which are 8 and -1. So, h = (8 - 1) / 2 = 7 / 2 = 3.5.
Substitute x = 3.5 into the equation to find the y-coordinate of the vertex, which is k.
y = a(3.5 - 8)(3.5 + 1) = a(-4.5)(4.5) = a(-20.25) = -20.25a
So, k = -20.25a, which is not in the given options.
However, if we simplify -20.25a, we get -81a/4, which is still not in the given options.
Therefore, there seems to be a mistake in the question or the options provided.
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