What is a fixed point that lies on the axis of symmetry of a parabola but not on the parabola?

Answer 1

There would be an infinite number of such points. One way to find them is as follows:

Let's take a generic parabola:

y = ax2 + bx + c

Grab it's derivative with respect to x:

dy/dx = 2ax + b

Let that equal zero, and solve for x:

0 = 2ax + b

2ax = -b

x = -b/2a

Now we know the x-coordinate of all points that lie on it's line of symmetry. The only one that we don't want being the one that falls on the curve itself. We can find it's y-coordinate by plugging our x-coordinate into the original parabola

y = ax2 + bx + c

y = a(-b/2a)2 + b(-b/2a) + c

y = a(b2/4a2) - b2/2a + c

y = b2/4a - b2/2a + c

y = b2/4a - 2b2/4a + c

y = -b2/4a + c

So there we have it. Given a parabola in the format ax2 + bx + c, all points that lie on it's axis of symmetry have an x-coordinate of -b/2a, and a y-coordinate other than -b2/4a + c