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Parabola

The equation of a parabola is given by the quadratic function:

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

The ABC of parabolas

Parabola - The values of a, b and c of the quadratic function
Quadratic function with the values a, b and c depicted in a coordinate system. Try to drag the red dots and see what happens when the values of a, b and c change.

The values a, b and c has influence on how the parabola looks like.


a:

The more the value of a increases, positive or negative, the steeper the parabola becomes.

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

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.

c:

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

Finding the function of a parabola with a and the roots given

It’s possible to determine the function of a parabola when a and the roots x_1 and x_2 (the solutions of the function when y=0) are given:

y=a*(x-x_1)*(x-x_2)

Finding the function of parabola with a and the roots given

First root (x1): 
Second root (x2): 
a: