Final answer:
The equation of the quadratic function in vertex form that passes through given points is f(x) = 1.875(x + 2)^2 - 1.
Step-by-step explanation:
To find the equation of the quadratic function in vertex form, which passes through the points (-3,-1), (-1,-1), and (2,29), we will use the vertex form equation f(x) = a(x - h)^2 + k, where (h, k) is the vertex, and a is a constant.
Since the function has the same y-value (-1) for two different x-values (-3 and -1), we know that the axis of symmetry is midway between these x-values, which is at x = -2.
Therefore, the vertex of the parabola is (-2, -1), giving us h = -2 and k = -1. We can then use the point (2, 29) to solve for a. Substituting x = 2 and f(x) = 29 into the vertex form, we get:
29 = a(2 - (-2))^2 + (-1)
29 = a(4)^2 - 1
30 = 16a
a = 30/16
a = 1.875
Thus, the quadratic function in vertex form is:
f(x) = 1.875(x + 2)^2 - 1