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Quadratic function

A function is quadratic or a second degree polynomial, if it can be written in form:

f(x)=a*x^2+b*x+c

where a is not zero.

The graph of a quadratic function is a parabola. So all formulas and properties associated with parabolas apply.

The significance of the constants

The significance of the constant a
The further away from 0, positive or negative, the value of a is, the steeper the parabolas legs get.

a < 0: The cup faces down.
a > 0: The cup faces up.
a = 0: It’s not a parabola but a straight line

The significance of the constant b
b = 0: The vertex of the parabola is on the y-axis in the point (0,c)
b ≠ 0: The vertex of the parabola is not on the y-axis.

The significance of the constant C
c is the point of intersection between the parabola and the y-axis (because y=c when x=0 in the quadratic function).

The vertex of the parabola

The x- and y-coordinate of the vertex of the parabola is given by:

x=frac{-b}{2*a}

y=frac{-D}{4*a}

where D is the discriminant given by D=b^2-4*a*c

Points of intersection with the x-axis, the roots

Since the parabola will intersect the x-axis, when y=0, you can find the intersection points by solving the quadratic equation:

0=a*x^2+b*x+c

The first thing you do is to calculate the discriminant:

D=b^2-4*a*c

If D is smaller than 0, there is no roots, that is, the parabola doesn’t intersect the x-axis.
If D is equal 0, the parabola intersects the x-axis in one point.
If D is greater than 0, the parabola intersects the x-axis in two points.


The points of intersection are given by:
x=frac{-b±sqrt{D}}{2*a}

Calculate the point of intersection with the x-axis

Please enter the constant a, b and c from a function of the form:
f(x)=a*x^2+b*x+c

a: 
b: 
c: 

Finding the function with three points given

It’s possible to find the function of a parabola, if three of its points are given:
(x_1,y_1)(x_2,y_2)(x_3,y_3)

There are two ways to do that:

You can either solve three equations with tree unknowns. You get the three equations simply by substituting the three different values of x and y into the quadratic function: y=a*x^2+b*x+c.


y_1=a*x_1^2+b*x_1+c

y_2=a*x_2^2+b*x_2+c

y_3=a*x_3^2+b*x_3+c


Or you can use the formula:

y=frac{(x-x_2)*(x-x_3)}{(x_1-x_2)*(x_1-x_3)}*y_1+frac{(x-x_1)*(x-x_3)}{(x_2-x_1)*(x_2-x_3)}*y_2+frac{(x-x_1)*(x-x_2)}{(x_3-x_1)*(x_3-x_2)}*y_3