Question

Complete the square to determine the vertex and axis of symmetry of y=−x2+8x+7
.(4 points)IN WORD FORM

Answer 1

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

1. **Rearrange the equation**: Start with the quadratic in the standard form:
\[
y = -x^2 + 8x + 7
\]

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

3. **Complete the square**: We need to take the \( x^2 - 8x \) part and complete the square. To do this, take half of the coefficient of \( x \) (which is \(-8\)), square it, and add and subtract that value inside the parentheses. Half of \(-8\) is \(-4\), and squaring it gives \(16\):
\[
y = - (x^2 - 8x + 16 - 16) + 7
\]
This simplifies to:
\[
y = - ((x - 4)^2 - 16) + 7
\]
\[
y = - (x - 4)^2 + 16 + 7
\]
\[
y = - (x - 4)^2 + 23
\]

4. **Identify the vertex**: The vertex form of a quadratic equation is given by \( y = a(x - h)^2 + k \), where \((h, k)\) is the vertex. From our completed square, we can see that \( h = 4 \) and \( k = 23 \). Therefore, the vertex is at the point \( (4, 23) \).

5. **Determine the axis of symmetry**: The axis of symmetry can be found from the \( x \)-coordinate of the vertex. This means that the axis of symmetry is the vertical line given by the equation \( x = 4 \).

In conclusion, the ***vertex*** of the quadratic function \( y = -x^2 + 8x + 7 \) is at the point ***four, twenty-three***, and the ***axis of symmetry*** is the vertical line ***x equals four***.