Yes, your conversion of the quadratic equation to vertex form and the subsequent steps to find the vertex and the value of x = 6 are correct.
To convert the given quadratic equation, y = -2x^2 + 2x + 2, to vertex form, you followed these steps:
1. Rewrite the equation by factoring out -2 from the x^2 and x terms: y = -2(x^2 - x - 1).
2. Complete the square for the expression inside the parentheses. You take the coefficient of the x term (in this case, -1), divide it by 2, and square it: (-1/2)^2 = 1/4.
3. Add this value inside the parentheses and subtract its equivalent value outside the parentheses to maintain the equality: -2(x^2 - x + 1/4 - 1/4 - 1).
4. Simplify the expression inside the parentheses: -2((x - 1/2)^2 - 5/4).
5. Distribute the -2 to both terms inside the parentheses: -2(x - 1/2)^2 + 5/2.
6. Now, the equation is in the vertex form: y = -2(x - 1/2)^2 + 5/2.
To find the vertex, you use the formula (h, k), where h is the x-coordinate and k is the y-coordinate. In this case, the vertex is (1/2, 5/2).
To find the value of y when x = 6, you substitute x = 6 into the vertex form equation:
y = -2(6 - 1/2)^2 + 5/2.
Now, you can evaluate this expression to find the value of y.