Complete the square to determine the vertex and axis of symmetry of y=-x^2 + 8x + 7
To complete the square for the given quadratic equation y = -x^2 + 8x + 7, follow these steps:
1. Rewrite the equation by moving the constant term to the right side:
y - 7 = -x^2 + 8x
2. To complete the square, take half of the coefficient of x (which is 8/2 = 4), square it (16), and add it to both sides of the equation:
y - 7 + 16 = -x^2 + 8x + 16
y + 9 = -(x^2 - 8x + 16)
3. Factor the trinomial inside the parentheses as a perfect square:
y + 9 = -(x - 4)^2
4. Distribute the negative sign to the perfect square expression:
y + 9 = -1 * (x - 4)^2
5. Rewrite the equation in vertex form by subtracting 9 from both sides:
y = -(x - 4)^2 - 9
Now, the equation is in the form y = a(x - h)^2 + k, where (h, k) represents the vertex of the parabola. From the equation above, we see that the vertex is (4, -9).
The axis of symmetry is given by the x-coordinate of the vertex. Therefore, the equation of the axis of symmetry is x = 4.
Hence, the vertex of the parabola is (4, -9) and the axis of symmetry is x = 4.