Question

Compare the square to determine the vertex and axis of symmetry of y =-x^2 +8x +7

Answer 1

To find the vertex and axis of symmetry for the quadratic equation \( y = -x^2 + 8x + 7 \), we can rewrite it in vertex form using the method of completing the square.

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

2. **Factor out -1 from the \(x^2\) and \(x\) terms**:
\[
y = - (x^2 - 8x) + 7
\]

3. **Complete the square**:
- Take the coefficient of \(x\) (which is \(-8\)), halve it to get \(-4\), and square it to get \(16\).
- Add and subtract \(16\) inside the parentheses:
\[
y = - (x^2 - 8x + 16 - 16) + 7
\]
\[
y = - ((x-4)^2 - 16) + 7
\]
\[
y = - (x-4)^2 + 16 + 7
\]
\[
y = - (x-4)^2 + 23
\]

4. **Identify the vertex**:
From the vertex form \(y = a(x - h)^2 + k\), the vertex \((h, k)\) can be read as:
- \(h = 4\)
- \(k = 23\)

Thus, the vertex is \((4, 23)\).

5. **Determine the axis of symmetry**:
The axis of symmetry is given by the vertical line \(x = h\), where \(h\) is the x-coordinate of the vertex:
\[
\text{Axis of symmetry: } x = 4
\]

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