Final answer:
The equation of a quadratic function with roots at 2 and 4 and a vertex at (3, -2) is y = 2(x - 3)^2 - 2.
Step-by-step explanation:
To find the equation of the quadratic function in vertex form when given the roots at 2 and 4 and the vertex at (3, -2), we follow a step-by-step process. The vertex form of a quadratic is given by:
y = a(x - h)^2 + k
where (h, k) is the vertex of the parabola. Given the vertex (3, -2), we have:
y = a(x - 3)^2 - 2
Since the roots are 2 and 4, the quadratic function will be zero at these points. Substituting x = 2:
0 = a(2 - 3)^2 - 2
0 = a(1)^2 - 2
a = 2
Now substituting x = 4:
0 = a(4 - 3)^2 - 2
0 = a(1)^2 - 2
Again, a = 2
So, the equation of the quadratic function in vertex form is:
y = 2(x - 3)^2 - 2