Question

Complete the square to determine the vertex and axis of symmetry of y=−x2+8x+7
.

Answer 1

shorter

Answer 2

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

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

2. **Complete the square** for the expression in parentheses. To do this, take half of the coefficient of \( x \) (which is -8), square it, and add and subtract this inside the parentheses:
\[
\text{Half of } -8 = -4 \quad \text{and } (-4)^2 = 16
\]
Now rewrite the equation:
\[
y = -1(x^2 - 8x + 16 - 16) + 7
\]
This simplifies to:
\[
y = -1((x - 4)^2 - 16) + 7
\]

3. **Distribute the -1**:
\[
y = -1(x - 4)^2 + 16 + 7
\]
Simplifying further gives:
\[
y = -1(x - 4)^2 + 23
\]

Now we can identify the vertex and the axis of symmetry:

- The vertex form of the equation is \( y = a(x - h)^2 + k \), where \( (h, k) \) is the vertex. Here, \( h = 4 \) and \( k = 23 \). Thus, the vertex is:
\[
(4, 23)
\]

- The axis of symmetry can be found from the vertex, which is the vertical line \( x = h \):
\[
x = 4
\]

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