The parabolic equation in General
Your General objective in these tasks is an equation that gives you for each x-value one y-value, so you can use it to draw a parabola.
- This is a General parabolic equation has the General Form y = a * x^2 + b * x + c.
- Here, * stands for multiplication and ^ for powers.
- a, b, and c are constant factors, of which particularly influenced the shape of the parabola strongly. That's why this form factor is often given in the tasks.
- In such a case, you normally receive two points (x1, y1) and (x2, y2), as well as a value for a. you will now need to determine b and c uniquely.
- For this, you set up a system of linear equations by inserting each point into the General equation. Since you have two Unknowns, you can solve the System with these two equations and the parabola equation to determine.
Parabola equation set up
Parabola equation in vertex form
Often you will be given the vertex - the Minimum or Maximum of the parabola and either a second point or the shape factor a.
- If you have the vertex (xs,ys), you should use in each case, the vertex Form:
- y = a * (x - xs )^2 + ys.
- If you are now in addition to the vertex of the factor a, multiply the bracket:
- y = a * x^2 - 2a * xs * x + a * xs^2 + ys
- As a, xs and ys are known values, you can drag a * xs^2 + ys still together and to obtain the c of the normal form. As 2a * xs corresponds to the b from the normal form.
- If you will be given, however, not a, but a point (x, y), you can use the vertex Form easy to a and put:
- a = (y - ys) / (x - xs)^2
Parabola equation in vertex form
Parabola equation from zero
Another popular type of task, the generation of the parabolic equation is, if you make only two to zero, and the form factor.
- Zero the points where your parabola intersects the x-axis, so y = 0 is. Often you get two pieces of the set: A = (xN1,0) and B = (xN2,0).
- Now you can use with these two and the factor A. the factored Form of the parabola equation:
- y = a(x - xN1)(x - xN2)
- If you multiply out you get:
- y = a * x^2 - a * xN1 * x - a * xN2 * x + a * xN1 * xN2
- As you know xN1 and xN2, you can make the regular parabolic shape.
- The first Term is already there correctly. - a * xN1 * x - a * xN2 * x = (a * xN1 - a * xN2) * x you can combine the second Term b * x together. And a * xN1 * xN2 corresponds to the c from the standard equation.
Parabola equation from zero
Often you don't need to find the parabolic equation, but to get this made. Then it goes to other mathematical challenges. How about the vertex of a parabola find, we will explain in the next tip.
