Complete the square to determine the vertex and axis of symmetry of y=−x2+8x+7 .(4 points)

Answer 1

To complete the square, we need to rewrite the given equation in the form y = a(x-h)^2 + k, where (h,k) is the vertex of the parabola.

First, let's group the terms with x:
y = -x^2 + 8x + 7
= -(x^2 - 8x) + 7

Now, we want to find a constant term to complete the square. To do this, we need to take half of the coefficient of x, square it, and add it to both sides of the equation:
y = -(x^2 - 8x + 16) + 7 + 16
= -(x^2 - 8x + 16) + 23

Note that we added +16 to the right side because we added 16 inside the parentheses, so we need to balance it out. We also added +16 inside the parentheses because (x - 4)^2 = x^2 - 8x + 16.

Now, we can rewrite the equation as:
y = -(x - 4)^2 + 23

Comparing this equation with y = a(x-h)^2 + k, we can see that the vertex is (h,k) = (4, 23).

The axis of symmetry is the vertical line passing through the vertex, so the axis of symmetry is x = 4.