What is the equations of the quadratic function with a vertex (4, -4) and a point at (3, -6)?

What is the equations of the quadratic function with a vertex (4, -4) and a point at (3, -6)?  There's just one such equation.  Let's begin with the standard equation of a vertical parabola whose vertex is (h,k):

y-k = a(x-h).    

The equivalent equation of a horiz. polynomial is x-h = a(y-k)^2.

Plot the given points and try to determine from your graph whether the parabola in question is vertical or horizontal.

You could, of course, just begin substituting the coordinates of the given points and determine whether doing so leads to a proper equation of a parabola.

Let's actually try this for the parabola with vertex (4,-4) that passes thru the point (3,-6):

y-[-4] = a(x-4)^2
Let's now substitute the coordinates of the point (3,-6):

-6+4 = a(3-4)^2

This becomes -2 = a.  Thus, from the given two points, we end up with the equation of a vertical parabola

y+4 = -2(x-4)^2

Let's check our work here.  First, focus on the vertex:  It's (4,-4).  Let's substitute these coordinates into y+4 = -2(x-4)^2.  Is the equation then true?

-4+4= -2(4-4)^2            Is 0 = to 0?  Yes.  What does that tell us?

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