Final answer:
The vertex of the quadratic function f(x) = 3x^2 + 6x + 6 is found using the vertex formula. The x-coordinate of the vertex is -1, calculated by -b/2a, and the y-coordinate is 3, obtained by evaluating the function at x = -1. Hence, the vertex is (-1, 3).
Step-by-step explanation:
The vertex of the quadratic function f(x) = 3x^2 + 6x + 6 can be found using the vertex formula.
The vertex formula for a quadratic function f(x) = ax^2 + bx + c is given by the point (-b/2a, f(-b/2a)).
For the function in question, a = 3 and b = 6.
So the x-coordinate of the vertex is -b/2a = -6/(2×3) which simplifies to -1. To find the y-coordinate, substitute x = -1 into the original function:
f(-1) = 3(-1)^2 + 6(-1) + 6 = 3 - 6 + 6 = 3.
Therefore, the vertex of the quadratic function f(x) = 3x^2 + 6x + 6 is (-1, 3).