Question

omplete the square to determine the vertex and axis of symmetry of y=−x2+8x+7
.(4 points)

Answer 1

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

1. **Factor out the coefficient of \(x^2\)**: Since the coefficient of \(x^2\) is \(-1\), we'll factor that out from the first two terms:
\[
y = -1(x^2 - 8x) + 7
\]

2. **Complete the square**: To complete the square inside the parentheses, we take half of the coefficient of \(x\) (which is \(-8\)), square it, and add/subtract that value inside the parentheses.

Half of \(-8\) is \(-4\), and squaring \(-4\) gives \(16\). We then add and subtract \(16\) inside the parentheses:
\[
y = -1(x^2 - 8x + 16 - 16) + 7
\]
Rearranging this gives:
\[
y = -1((x - 4)^2 - 16) + 7
\]

3. **Distribute and simplify**:
\[
y = -1(x - 4)^2 + 16 + 7
\]
\[
y = -1(x - 4)^2 + 23
\]

4. **Identify the vertex**: The vertex form of a parabola is given by \(y = a(x - h)^2 + k\), where \((h, k)\) is the vertex. From our equation:
\[
h = 4, \quad k = 23
\]
Therefore, the vertex is \((4, 23)\).

5. **Find the axis of symmetry**: The axis of symmetry is the vertical line that passes through the vertex. The equation is given by:
\[
x = h
\]
Thus, the axis of symmetry is:
\[
x = 4
\]

### Summary
- **Vertex**: \((4, 23)\)
- **Axis of Symmetry**: \(x = 4\)