Set up parabola equation - how it works

In math problems you get very different specifications from which you should then set up a parabola equation. We'll explain how it works.

The parabola equation in general

Your general goal in these tasks is to set up an equation that gives you a y value for each x value so that you can use it to draw a parabola.

This general parabola equation has the general form y = a * x ^ 2 + b * x + c.
* Stands for multiplication and ^ for a power.
a, b and c are constant factors, of which especially a strongly influences the shape of the parabola. That is why this form factor is often given in tasks.
In such a case you usually get two points (x1, y1) and (x2, y2) and a value for a. You now have to uniquely determine b and c.
To do this, you set up a linear system of equations by inserting a point in the general equation. Since you have two unknowns, you can solve the system with these two equations and thus determine the parabola equation.

You are often given the vertex - the minimum or maximum of the parabola - and either a second point or the form factor a.

If you have the vertex (xs, ys), you should definitely use the vertex shape:
y = a * (x - xs) ^ 2 + ys.
If you now have the factor a in addition to the vertex, multiply the bracket:
y = a * x ^ 2 - 2a * xs * x + a * xs ^ 2 + ys
Since a, xs and ys are known values, you can still combine a * xs ^ 2 + ys and thus get the c of the normal form. Similarly, -2a * xs corresponds to b from the normal form.
If, on the other hand, you are given a point (x, y) instead of a, simply change the vertex shape to a and insert:
a = (y - ys) / (x- xs) ^ 2

Another popular type of task is generating the parabola equation if you only have two zeros and the form factor.

Zeros are the points at which your parabola intersects the x-axis, i.e. y = 0. You often get two of them: A = (xN1.0) and B = (xN2.0).
Now you can use the factorized form of the parabola equation with these two and the factor a:
y = a (x - xN1) (x - xN2)
If you multiply that, you get:
y = a * x ^ 2 - a * xN1 * x - a * xN2 * x + a * xN1 * xN2
Since you know xN1 and xN2, you can use it to directly form the regular parabola shape.
The first term is already there correctly. - a * xN1 * x - a * xN2 * x = (- a * xN1 - a * xN2) * x you can summarize for the second term b * x. And a * xN1 * xN2 corresponds to the c from the regular equation.