Question

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

Answer 1

To complete the square for the quadratic equation \( y = -x^2 + 8x + 7 \), we can follow these steps.

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

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

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

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

3. **Add and subtract this square inside the parentheses**:

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

Simplifying this gives:

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

4. **Distribute the negative sign and simplify**:

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

Now we have the equation in vertex form \( y = a(x - h)^2 + k \), where \( (h, k) \) is the vertex of the parabola.

5. **Identify the vertex and axis of symmetry**:

- The vertex \( (h, k) \) is \( (4, 23) \).
- The axis of symmetry is the line \( x = h \), which is \( x = 4 \).

### Final Results:
- Vertex: \( (4, 23) \)
- Axis of Symmetry: \( x = 4 \)