Find the equation, f(x) = a(x-h)2 + k, for a parabola that passes through the point (1,-7) and has (2,- 4) as its vertex. what is the standard form of the equation?

The coordinates of the vertex are (h,k)  so we can write:-
 f(x) = a(x -2)^2 - 4
now when f(x) = -7 x = 1 so:-

-7 = a(1-2)^2 - 4
-7 = a - 4
a = -3

so the vertex formula is 
f(x) = -3(x - 2)^2 - 4

standard form:-

f(x) = -3(x^2 - 4x + 4)  - 4
       = -3x^2 + 12x - 16

The answer would be quadratic trinomial