Question

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

Answer 1

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

First, let's rearrange the equation by moving the constant term to the right side:
y + x^2 - 8x = 7.

Next, we need to complete the square by adding and subtracting the square of half the coefficient of x. In this case, the coefficient of x is -8, so the square of half the coefficient is (-8/2)^2 = 16.

Adding and subtracting 16 to the equation, we have:
y + x^2 - 8x + 16 - 16 = 7.

Rearranging the terms, we get:
(x^2 - 8x + 16) + y - 16 = 7.

Now, let's factor the square trinomial (x^2 - 8x + 16) as a perfect square:
(x - 4)^2 + y - 16 = 7.

Simplifying further, we have:
(x - 4)^2 + y - 16 - 7 = 0.

(x - 4)^2 + y - 23 = 0.

Comparing this to the form y = a(x - h)^2 + k, we can see that h = 4 and k = 23.

Therefore, the vertex is (h, k) = (4, 23).

The axis of symmetry is a vertical line that passes through the vertex. The equation of the axis of symmetry is x = h.

Therefore, the axis of symmetry is x = 4.