Question

Complete the square to determine the vertex and axis of symmetry of y=−x^2+8x+7 (4 points)

Answer 1

shorter
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Answer 2

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

1. **Factor out the coefficient of \( x^2 \)** (which is \(-1\) in this case) from the first two terms:

\[
y = - (x^2 - 8x) + 7
\]

2. **Complete the square** inside the parentheses. To do this, we take the coefficient of \( x \) (which is \(-8\)), divide it by 2 to get \(-4\), and square it to get \(16\). We then add and subtract this value inside the parentheses:

\[
y = - \left( x^2 - 8x + 16 - 16 \right) + 7
\]

Simplifying this, we have:

\[
y = - \left( (x - 4)^2 - 16 \right) + 7
\]

3. **Distribute the negative sign**:

\[
y = - (x - 4)^2 + 16 + 7
\]

Combine like terms:

\[
y = - (x - 4)^2 + 23
\]

Now, the function is in the vertex form \( y = a(x - h)^2 + k \), where \( (h, k) \) is the vertex. From our equation \( y = - (x - 4)^2 + 23 \), we can identify:

- The vertex \( (h, k) = (4, 23) \).
- The axis of symmetry is given by the line \( x = h \), which, in this case, is \( x = 4 \).

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

Answer 3

Sure! Here’s the shorter version:

To complete the square for \( y = -x^2 + 8x + 7 \):

1. Factor out \(-1\):
\[
y = - (x^2 - 8x) + 7
\]

2. Complete the square:
\[
y = - \left( (x - 4)^2 - 16 \right) + 7 \implies y = - (x - 4)^2 + 23
\]

**Vertex**: \( (4, 23) \)

**Axis of symmetry**: \( x = 4 \)