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 will format it in the vertex form \( y = a(x - h)^2 + k \), where \( (h, k) \) is the vertex of the parabola.

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

2. **Factor out the coefficient of \( x^2 \) from the first two terms:**
\[
y = -(x^2 - 8x) + 7
\]

3. **Complete the square:**
- Take the coefficient of \( x \) (which is \(-8\)), halve it (\(-4\)), and then square it (\(16\)).
- Add and subtract this square inside the parentheses.

\[
y = -\left(x^2 - 8x + 16 - 16\right) + 7
\]
\[
y = -\left((x - 4)^2 - 16\right) + 7
\]

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

5. **Write in vertex form:**
\[
y = -1(x - 4)^2 + 23
\]

Now, we can identify the vertex and the axis of symmetry.

- **Vertex (h, k):**
The vertex is \( (h, k) = (4, 23) \).

- **Axis of Symmetry:**
The axis of symmetry is the vertical line that passes through the vertex, given by the line \( x = h \):
\[
x = 4
\]

### Summary:
- The vertex of the parabola is \( (4, 23) \).
- The axis of symmetry is \( x = 4 \).