1. By Concavity

2. By number of roots

Concave up: a > 0

Concave down: a < 0

This below image illustrates types of parabola by concavity:

Other way of classifying parabola is by number of times the parabola intersects with the axis line.

2 roots: > 0

1 root: = 0 if the vertex touches the axis.

0 roots: < 0 if both x and y axis does not touch x or y axis. This could be found out with the help of the formula: b

This below image illustrates types of parabola by number of roots (intersecting with the axis):

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