Question

Write the equation of the quadratic function with roots 6 and 10 and a vertex at (8, 2).

Answer 1

6 is a root implies (x-6) is a factor.
10 is a root implies (x-10) is a factor.
Therefore the factors of the quadratic function are: (x-6)(x-10)
The more general equation is: a * (x-6)(x-10) where 'a' is a constant.
y = a * (x-6)(x-10) = a(x^2 - 16x + 60)
(8,2) is a point on the parabola.
Put x = 8 and y = 2 and solve for 'a'

Answer 2

Another way to do this problem is:
The equation of a parabola in the vertex form is: y = a(x-h)^2 + k
where (h,k) is the vertex and 'a' is a constant.
Here, (h,k) is (8,2).
y = a(x - 8)^2 + 2
x = 6 is a root. That means when x = 6, y = 0. Substitute:
0 = a(6 - 8)^2 + 2
0 = 4a + 2
4a = -2
a = -1/2
y = -1/2 * (x - 8)^2 + 2
y = -1/2 * (x^2 - 16x + 64) + 2
y = -1/2x^2 + 8x - 32 + 2
y = -1/2 x^2 + 8x - 30 is the equation of the quadratic function.