A function is quadratic or a second degree polynomial, if it can be written in form:
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:
where D is the discriminant given by
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:
The first thing you do is to calculate the discriminant:
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:
Finding the function with three points given
It’s possible to find the function of a parabola, if three of its points are given:
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: .
Or you can use the formula: