Parabola Question?

Question

Find an equation of a parabola (in general form Ax^2 + Bx + C) with vertical axis, vertex at (2,1), and passing
through (1,3). What what is A + B?

I know that y=a(x+h)^2 + k and (h,k) is the vertex but I dont know how to find A and B and how the general form relates to the standard form (y=a(x+h)^2 + k).

Thanks
Chris

Answer 1

First, let's put the equation in standard form.

(x-h)^2 = 4p(y-k) where p is the focal length.

Substituting in the vertex gives

(x-2)^2 = 4p(y-1).

Now we can also plug in the given point, (1,3), to solve for p.

(1-2)^2 = 4p(3-1) giving p=1/8.

So the equation is (x-2)^2 = 4(1/8)(y-1).

Now we can expand this and simplify to get the equation as y=2x^2-8x+9.

Therefore a+b=2+(-8) = -6.

Answer 2

y=a(x-2)^2+1
3=a+1 so a= 2 and The equation is y= 2(x-2)^2+1
=2x^2-8x+9 soA=2 and B=-8 and A+B=-6