Martin McBride, 2020-09-12

Tags parametric equation hyperbola

Categories coordinate systems pure mathematics

A rectangular hyperbola has the parametric equations:

$$ \begin{align} x = c t\newline y = \frac{c}{t} \end{align} $$

Where $c$ 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.

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

$$ \begin{align} x = t\newline y = \frac{1}{t} \end{align} $$

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

t | x | y |
---|---|---|

-4 | -3 | -0.25 |

-2 | -2 | -0.5 |

-1 | -2 | -1 |

-0.5 | -0.5 | -2 |

0 | 0 | undefined |

0.5 | 0.5 | 2 |

1 | 1 | 1 |

2 | 2 | 0.5 |

4 | 4 | 0.25 |

Here are the points plotted on a graph:

This curve is actually a standard *reciprocal curve*, as shown here.

The curve can be drawn by plotting the points and drawing a smooth line through them. Notice that the curve value is undefined for $t = 0$:

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