To complete the square for the equation y = -x^2 + 8x + 7, we need to rewrite the equation in the form y = a(x - h)^2 + k.
First, we group the x terms:
y = -(x^2 - 8x) + 7
Next, we need to add and subtract the square of half the coefficient of x in the expression x^2 - 8x. Since the coefficient of x is -8, half of it is -4, and the square of -4 is 16. We add and subtract 16 inside the parentheses:
y = -(x^2 - 8x + 16 - 16) + 7
Now, we can rewrite the expression inside the parentheses as a perfect square:
y = -(x^2 - 8x + 16) + 16 + 7
Next, we simplify the expression -x^2 + 8x + 16 by rewriting it as -(x - 4)^2:
y = -(x - 4)^2 + 23
Now the equation is in the desired form y = a(x - h)^2 + k, where a = -1, h = 4, and k = 23.
The vertex of the parabola is given by the values (h, k), so the vertex is (4, 23).
The axis of symmetry is the vertical line that passes through the vertex, so the axis of symmetry is x = 4.
Complete the square to determine the vertex and axis of symmetry of y=−x2+8x+7
(4 points)
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