Assignment 10

by

Allison McNeece

For this assignment we will be looking at parametric equations.

A **parametric curve** in the plane is a pair of functions. **x=f(t) **and** y=f(t)** where **f(t)** is a function of time.

The two continuous functions define ordered pairs **(x,y)** and are called **parametric equations** of a curve.

The extent of the curve will depend on the range of **t**.

For example let's look at the following equations:

Allowing t to range from 0 to π we get: |
But if we allow t to rangle from 0 to 2π then we get: |

What if we changed up these equations a bit?

Let's graph the following with different values for a and b and 0 < t < 2π

with a=1 and b=1 |
with a=2 and b=1 |
with a=1 and b=2 |

So this gives us the intuition that if **a>b** then our graph will be stretched out horizontally but if **b>a** then our graph will be stretched out vertically.

What if we manipulated the values of what we are taking the sine and cosine of?

Let's first place an **n** infront of the **t** in the cosine function and see what the graph looks like for different values of **n**:

(a=b=1 and 0<t<2π)

n is odd |
n is even |
---|---|

n=1 |
n=2 |

n=3 |
n=4 |

n=5 |
n=6 |

n=7 |
n=8 |

n=9 |
n=10 |

So we find that there is a difference when **n** is even or odd. Well let's think about this, what do we know about cosine?

cos(π) = -1

cos(2π) = 1

Let's try this again but let's place an **n** infront of the **t** in the sine funtion:

(a=b=1 and 0<t<2π)

n is odd |
n is even |
---|---|

n=1 |
n=2 |

n=3 |
n=4 |

n=5 |
n=6 |

n=7 |
n=8 |

n=9 |
n=10 |

We don't see the difference between even and odd values that we saw when looking at the cosine values.

What do we know about the sine function?

sin(π) = 0

sin(2π) = 0