To complete the square for the quadratic equation \( y = -x^2 + 8x + 7 \), we start by rearranging the equation to facilitate the process.
-
Factor out the coefficient of the \( x^2 \) term (which is -1 in this case) from the first two terms:
\[ y = -1(x^2 - 8x) + 7 \]
-
Next, we complete the square inside the parentheses. To do this, take the coefficient of \( x \) (which is -8), divide it by 2, and square it:
\[ \left(\frac{-8}{2}\right)^2 = (-4)^2 = 16 \]
-
Now, we add and subtract this square inside the parentheses:
\[ y = -1(x^2 - 8x + 16 - 16) + 7 \]
-
This can be rewritten as:
\[ y = -1((x - 4)^2 - 16) + 7 \]
-
Simplifying further:
\[ y = -1(x - 4)^2 + 16 + 7 \] \[ y = -1(x - 4)^2 + 23 \]
Now, we can identify the vertex of the parabola and the axis of symmetry:
-
The vertex form of the equation \( y = a(x - h)^2 + k \) indicates the vertex is at point \( (h, k) \). In our case, \( h = 4 \) and \( k = 23 \), so the vertex is:
\[ (4, 23) \]
-
The axis of symmetry of the parabola is given by the line \( x = h \). Here, the axis of symmetry is:
\[ x = 4 \]
Thus, the vertex is \( (4, 23) \) and the axis of symmetry is \( x = 4 \).