Definition: A parabola is the U-shaped curve that is the graph of a quadratic function; it is symmetric about a line and has a single turning point, the vertex.
What does a parabola look like?
It is a smooth curve that opens either up or down. It has one vertex, one axis of symmetry, and at most two x-intercepts. The graph never levels off and never repeats.
What shapes the curve?
In y = ax2 + bx + c, the sign of a gives the direction. A large |a|, such as 4, makes the curve narrow, while a small |a|, such as 0.25, makes it wide.
Example. Compare y = 4x2 and y = 0.25x2 at x = 2. The first gives 16, the second gives 1. So the first parabola rises much faster and looks thinner.
Where do parabolas appear?
Thrown objects, bridge cables under certain loads, and satellite dishes all follow this shape. A typical mistake is to draw a parabola as a V with straight sides; plot at least five points, including the vertex and its mirror images, to see the real curve. Another is to expect the curve to end: both arms continue forever.