Question

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

Answer 1

To complete the square and find the vertex and axis of symmetry for the quadratic function \( y = -x^2 + 8x + 7 \), we can follow these steps:

1. **Factor out the leading coefficient from the first two terms** (if it's not 1):
\[
y = -1(x^2 - 8x) + 7
\]

2. **Complete the square for the expression inside the parentheses**:
- Take the coefficient of \( x \) (which is \(-8\)), halve it to get \(-4\), and then square it, resulting in \(16\).
- Add and subtract \(16\) inside the parentheses:
\[
y = -1(x^2 - 8x + 16 - 16) + 7
\]
Simplifying this, we have:
\[
y = -1((x - 4)^2 - 16) + 7
\]
\[
y = -1(x - 4)^2 + 16 + 7
\]
\[
y = -1(x - 4)^2 + 23
\]

3. **Identify the vertex**:
The completed square form is \( y = -1(x - 4)^2 + 23 \), which shows:
- The vertex is at \( (4, 23) \).

4. **Find the axis of symmetry**:
The axis of symmetry is the vertical line that passes through the vertex. The equation for the axis of symmetry is:
\[
x = 4
\]

In summary, for the quadratic equation \( y = -x^2 + 8x + 7 \):
- The vertex is \( (4, 23) \).
- The axis of symmetry is \( x = 4 \).