Complete the square to determine the vertex and axis of symmetry of y= −x^2+ 8x+7.
To complete the square, we need to rewrite the equation in the form y = a(x - h)^2 + k. Let's do that:
y = -x^2 + 8x + 7
First, we take -1 as a common factor from the terms involving x:
y = -(x^2 - 8x) + 7
Next, we need to add the square of half the coefficient of x to the equation while also subtracting it to maintain the equality. Half of 8 is 4, and the square of 4 is 16:
y = -(x^2 - 8x + 16 - 16) + 7
We can rewrite the equation now as:
y = -(x^2 - 8x + 16) + 7 - (-16)
y = -(x^2 - 8x + 16) + 7 + 16
y = -(x - 4)^2 + 23
Now, the vertex of the parabola is (h, k). So in this case, the vertex is (4, 23). The axis of symmetry is a vertical line through the vertex, so the axis of symmetry for this parabola is x = 4.