Algebra 2 - finding the equation through given vertex and point

Question

What is the equation of the parabola with the vertex (-3,-5) passing through the point (-2,-7)?

Answer 1

Of course, the axis of the parabola was not specified. Since it may point in any direction (except through the two points directly or perpendicular to that through the vertex), the set of solutions is infinite. You will just be rotating axes for the variables, using x ' = cos(θ)*x + sin(θ)*y and y' = sin(θ)*x - cos(θ)*y or something like that, and then solving for the two points. Messy! It won't be a function, for the original two coordinate variables, but will have first and second powers of each and cross-terms. And there will be symmetries, since the second point may lie on either "limb" of the parabola.

-- Cheers, --Mr. d.

Answer 2

If the vertex is (h,k) then y = a (x-h)² + k

So here h = -3, k = -5.

Substitute these in to y. Then you know when x= -2, y = -7, so you can use these to solve for a.