Complete the square to determine the vertex and axis of symmetry of y=−x^2+8x+7.

Answer 1

To complete the square for the given quadratic function y = -x^2 + 8x + 7, we need to manipulate the quadratic equation and rewrite it in the form y = a(x - h)^2 + k, where (h, k) represents the vertex of the parabola.

First, let's rewrite the equation:
y = -x^2 + 8x + 7

To complete the square, we need to add and subtract (-8/2)^2 = 16 to the right side of the equation, since (-8/2)^2 = 16. However, since we added 16 to the right side, we also need to subtract 16 from the right side to ensure the equation remains balanced.

y + 16 - 16 = -x^2 + 8x + 7

Rearranging the quadratic terms:
y = -(x^2 - 8x) + 7 + 16 - 16

Now, we can rewrite the quadratic term as a perfect square:
y = -(x^2 - 8x + 16) + 7 + 16 - 16

Simplifying the equation inside the parentheses:
y = -(x - 4)^2 + 7

The equation is now in the desired form y = a(x - h)^2 + k.

Comparing this to the general equation form, we can identify the values of a, h, and k. We find:
a = -1, h = 4, k = 7.

Therefore, the vertex of the parabola is at the point (4, 7), and the axis of symmetry is the vertical line x = 4.