Question

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 quadratic equation \( y = -x^2 + 8x + 7 \), we start by rearranging the equation to facilitate the process.

1. Factor out the coefficient of the \( x^2 \) term (which is -1 in this case) from the first two terms:

\[
y = -1(x^2 - 8x) + 7
\]

2. Next, we complete the square inside the parentheses. To do this, take the coefficient of \( x \) (which is -8), divide it by 2, and square it:

\[
\left(\frac{-8}{2}\right)^2 = (-4)^2 = 16
\]

3. Now, we add and subtract this square inside the parentheses:

\[
y = -1(x^2 - 8x + 16 - 16) + 7
\]

4. This can be rewritten as:

\[
y = -1((x - 4)^2 - 16) + 7
\]

5. Simplifying further:

\[
y = -1(x - 4)^2 + 16 + 7
\]
\[
y = -1(x - 4)^2 + 23
\]

Now, we can identify the vertex of the parabola and the axis of symmetry:

- The vertex form of the equation \( y = a(x - h)^2 + k \) indicates the vertex is at point \( (h, k) \). In our case, \( h = 4 \) and \( k = 23 \), so the vertex is:

\[
(4, 23)
\]

- The axis of symmetry of the parabola is given by the line \( x = h \). Here, the axis of symmetry is:

\[
x = 4
\]

Thus, the vertex is \( (4, 23) \) and the axis of symmetry is \( x = 4 \).