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 function \( y = -x^2 + 8x + 7 \), we follow these steps:

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

2. **Complete the square**:
To complete the square for the expression \( x^2 - 8x \), we find the term that completes the square. We take half of the coefficient of \( x \) (which is \(-8\)), square it, and add/subtract that value inside the parentheses:
\[
\left( \frac{-8}{2} \right)^2 = (-4)^2 = 16
\]
Add and subtract \( 16 \) inside the parentheses:
\[
y = -1(x^2 - 8x + 16 - 16) + 7
\]
This simplifies to:
\[
y = -1((x - 4)^2 - 16) + 7
\]
Distributing the \(-1\) gives:
\[
y = -1(x - 4)^2 + 16 + 7
\]
\[
y = -1(x - 4)^2 + 23
\]

3. **Identify the vertex**:
The vertex form of a quadratic function is given by \( y = a(x - h)^2 + k \), where \((h, k)\) is the vertex. From our completed square:
\[
y = -1(x - 4)^2 + 23
\]
So, the vertex is:
\[
(4, 23)
\]

4. **Determine the axis of symmetry**:
The axis of symmetry is the vertical line that passes through the vertex. For the vertex \((4, 23)\), the axis of symmetry is:
\[
x = 4
\]

**Final Answer**:
- Vertex: \((4, 23)\)
- Axis of symmetry: \(x = 4\)