[tex] (x+3)^2 =4 y[/tex]
That's a parabola, almost in vertex form:
[tex]y = \frac 1 4 (x+3)^2[/tex]
The translation is
[tex] (x',y') = T_{(-7,2)} (x,y) = (x-7, y+2)[/tex]
[tex] x= x'+7, y=y'-2[/tex]
Substituting translates the graph. None of the choices have fractions so we substitute into the original form:
[tex](x'+7+3)^2=4(y'-2)[/tex]
We can drop the primes now and simplify,
[tex](x+10)^2=4(y-2)[/tex]
[tex]x^2 + 20 x + 100 = 4y -8[/tex]
[tex]x^2 + 20 x - 4y + 108 = 0[/tex]
Choice C.
This method generalizes nicely to rotations, reflections and other transformations.
