## 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: