GraphicMaths

Visualising maths

Parabolas

A parabola is a curve with the parametric equations:

$$ \begin{align} x = a t^2\\ y = 2 a t \end{align} $$

Where $a$ is a positive constant, and $t$ is the independent variable.

We can plot this curve by calculating the values of $x$ and $y$ for various values of $t$, and drawing a smooth curve through them.

Curve for a = 1

Assuming $a = 1$, the parametric equations simplify to:

$$ \begin{align} x = t^2\\ y = 2 t \end{align} $$

The values are shown in the following table, for $t$ in the range -3 to +3:

t x y
-3 9 -6
-2 4 -4
-1 1 -2
0 0 0
1 1 2
2 4 4
3 9 6

The animation below plots the points and then draw a curve through them. Press the Play button to show this: