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To complete the square and determine the vertex and axis of symmetry of the quadratic function y = -x^2 + 8x + 7, we can follow these steps:
Step 1: Rewrite the quadratic equation in the vertex form by grouping the x terms:
y = -(x^2 - 8x) + 7
Step 2: To complete the square, take half of the coefficient of the x term, square it, and add it inside the parentheses:
y = -(x^2 - 8x + (8/2)^2) + 7
= -(x^2 - 8x + 16) + 7
Step 3: Simplify the equation inside the parentheses:
y = -(x - 4)^2 + 7
Step 4: Now, we can see that the vertex form of the quadratic equation is y = a(x - h)^2 + k, where (h, k) represents the vertex. In this case, the vertex is (4, 7), so the h-value represents the x-coordinate of the vertex, and the k-value represents the y-coordinate of the vertex. Therefore, the vertex is (4, 7).
Step 5: The axis of symmetry for a quadratic function is given by the equation x = h, where h represents the x-coordinate of the vertex. Therefore, the axis of symmetry is x = 4.
So, the vertex is (4, 7) and the axis of symmetry is x = 4.