Final answer:
The equation of the parabola in vertex form that passes through the point (8, 3) and has a vertex at (4, -1) is y = (1/4)(x - 4)^2 - 1.
Step-by-step explanation:
The equation of a parabola in vertex form is given by y = a(x - h)^2 + k, where (h, k) represents the vertex of the parabola.
Given that the vertex is at (4, -1), we can substitute these values into the equation to get y = a(x - 4)^2 - 1.
Now, we need to find the value of 'a'. We can substitute the coordinates of the point (8, 3) into the equation and solve for 'a'. Substituting the x-coordinate 8 and the y-coordinate 3, we get the equation 3 = a(8 - 4)^2 - 1.
Simplifying the equation gives us 3 = 16a - 1. Adding 1 to both sides gives us 4 = 16a. Dividing both sides by 16 gives us a = 1/4.
Substituting the value of 'a' back into the equation, we get the final equation of the parabola: y = (1/4)(x - 4)^2 - 1.