Question

Complete the square to determine the vertex and axis of symmetry of y=−x^2+8x+7
.

Answer 1

To complete the square and find the vertex and axis of symmetry for the quadratic equation \( y = -x^2 + 8x + 7 \), we can follow these steps:

1. **Rearrange the equation** so that we can focus on the \( x \) terms:
\[
y = - (x^2 - 8x) + 7
\]

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

3. **Distribute the negative sign** and simplify:
\[
y = - (x - 4)^2 + 16 + 7
\]
\[
y = - (x - 4)^2 + 23
\]

Now we have the vertex form of the quadratic equation, which is:
\[
y = - (x - 4)^2 + 23
\]

From this, we can identify the vertex and the axis of symmetry:

- **Vertex**: The vertex is at the point \( (h, k) \), where \( h = 4 \) and \( k = 23 \). Thus, the vertex is \( (4, 23) \).

- **Axis of symmetry**: The axis of symmetry is the vertical line that passes through the vertex, given by \( x = h \). Therefore, the axis of symmetry is:
\[
x = 4
\]

In summary:
- The vertex of the parabola is \( (4, 23) \).
- The axis of symmetry is \( x = 4 \).