Final answer:
The equation of the parabola in vertex form is y = -(x - 4)^2 + 28, given the x-intercepts of -6 and 14, and a maximum value of 28.
Step-by-step explanation:
The student is asking for the equation of a parabola in vertex form. The given x-intercepts are -6 and 14, which can be used to find the axis of symmetry, and the maximum value of 28 tells us the y-coordinate of the vertex. The vertex form of a quadratic is y = a(x - h)^2 + k, where (h,k) is the vertex of the parabola and 'a' determines the direction and width of the parabola.
Firstly, the axis of symmetry lies halfway between the x-intercepts: (14 - 6)/2 = 4, so the x-coordinate of the vertex (h) is 4. Since it has a maximum value, the parabola opens downwards, implying 'a' is negative. The vertex (h,k) is thus (4,28).
To find 'a', we use one of the x-intercepts. Plugging x = -6 (or x = 14) into the vertex form and using the vertex yielded, we get the equation 0 = a(-6 - 4)^2 + 28. Solving for 'a' gives us a = -1. Now substituting 'a', 'h', and 'k' into the vertex form, we get the final equation y = -(x - 4)^2 + 28.