Finding the quadratic function when vertex and a point are given

Given a vertex (1,-2) and a point (4,5) on a parabola, find the equation for such parabola.
First, we recall a function f(x)=a(x-h)²+k, which will give us vertex (h,k). So the function we are looking for should look like f(x)=a(x-1)²-2, and since the parabola passes through (4,5), we have 5=a(4-1)²-2, which means 5=9a-2, so a = (7/9) and f(x) = (7/9)(x-1)²-2.