if the vertex is the point (h, k) the vertex form of the quadratic equation is

y = a(x - h)^2 + k, where a is the constant that we need to find

(h, k) = (1, 3) that is h=1, k=3

y = a(x - 1)^2 + 3

the line goes trough (2, 10) thus

10 = a(2 - 1)^2 + 3

a + 3 = 10

a = 10 - 3

the quadratic function is:

y = 7(x - 1)^2 + 3

y = 7x^2 - 14x +10