Question

2. Complete the square to determine the vertex and axis of symmetry of y=−x2+8x+7
.
Please show work

Answer 1

To complete the square for the quadratic equation \( y = -x^2 + 8x + 7 \), we will rearrange it and manipulate it to find the vertex form.

1. **Start with the equation**:
\[
y = -x^2 + 8x + 7
\]

2. **Factor out the coefficient of \( x^2 \)**:
We will factor out \(-1\) from the terms involving \(x\):
\[
y = - (x^2 - 8x) + 7
\]

3. **Complete the square**:
To complete the square inside the parentheses \( x^2 - 8x \), take the coefficient of \( x \) (which is \(-8\)), divide by 2 (which gives \(-4\)), and square it (which gives \(16\)).
Add and subtract this square inside the parentheses:
\[
y = - \left(x^2 - 8x + 16 - 16\right) + 7
\]
Simplifying gives:
\[
y = - \left((x - 4)^2 - 16\right) + 7
\]

4. **Distribute the negative sign and simplify**:
\[
y = - (x - 4)^2 + 16 + 7
\]
\[
y = - (x - 4)^2 + 23
\]

5. **Identify the vertex and axis of symmetry**:
Now that we have the equation in vertex form \( y = a(x - h)^2 + k \), where \((h, k)\) is the vertex, we can see that:
- The vertex is \((4, 23)\).
- The axis of symmetry is the vertical line given by \( x = h = 4 \).

### Conclusion
The vertex of the quadratic equation \( y = -x^2 + 8x + 7 \) is \( (4, 23) \) and the axis of symmetry is \( x = 4 \).