Question

Complete 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 will rearrange the equation and factor it appropriately.

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

2. **Factor out the coefficient of \( x^2 \)**:
Since the coefficient of \( x^2 \) is negative, we will factor out \(-1\):
\[
y = - (x^2 - 8x) + 7
\]

3. **Complete the square**:
To complete the square for the expression \( x^2 - 8x \):
- Take half of the coefficient of \( x \) (which is -8), giving us \(-4\), and then square it:
\[
(-4)^2 = 16
\]
- Add and subtract this square inside the parentheses:
\[
y = - (x^2 - 8x + 16 - 16) + 7
\]
- Simplify it:
\[
y = - ( (x - 4)^2 - 16 ) + 7
\]
\[
y = - (x - 4)^2 + 16 + 7
\]
\[
y = - (x - 4)^2 + 23
\]

4. **Identify the vertex and axis of symmetry**:
The equation is now in the vertex form \( y = a(x - h)^2 + k \), where \((h, k)\) is the vertex. Here, \( a = -1 \), \( h = 4 \), and \( k = 23 \).

Thus, the vertex is:
\[
(4, 23)
\]

The axis of symmetry is the vertical line that passes through the vertex:
\[
x = 4
\]

In conclusion, the vertex of the parabola is \((4, 23)\), and the axis of symmetry is \(x = 4\).