The vertex form of the equation of a parabola is
f(x) = a( x − h)^2 + k
where (h,k) is the vertex of the parabola. In this case, we are given that (h,k) = (5,0). Hence,
f(x) = a( x − 5)^2 + 0
=a( x − 5)^2
Since we also know the parabola passes through the point (7,−2), we can solve for a because we know that f(7) = −2.
a( 7 − 5)^2 = -2
a(2)^2 = -2
4a = -2
a = -1/2
Thus, the given parabola has equation
f(x) = -1/2(x − 5)^2