How can you prove that a quadratic function always makes a parabola?

Answer 1

Use the definition of a parabola, as a line connecting all the points equidistant from a point and a straight line. Let (0,a) be the point. Let y=b be the line. Find all points (x,y) that are the same distance from that point as to the line. y-b is the distance to the line. The distance from (x,y) to (0,a) is the square root of the sum (x-0)² + (y-a)² Set those two distances equal, then square both sides to get (y-b)² = x² + (y-a)² Expand, and cancel out the y² terms leaving -2by + b² = x² -2ay + a² Move the y terms to the left side to get (2a-2b)y = x² + a² Then divide both sides by the coefficient of the y term, to get a quadratic equation in standard form.

Similarly start with a quadratic equation and do the above backwards to always get to a parabola. Also use a more generalized form of the quadratic equation with an additional x term and get to a generalized focus point (c,a) instead of the (0,a) point. But the proof seems more simple, understandable and intuitive by first looking at just a quadratic equation with no x term, just an x² term and constant term. Also, it's shorter this way.