A parametric equation in the plane is a pair of functions where the two continuous functions define an ordered pair (x,y). The two equations are usually called the parametric equations of a curve. The extent of the curve will depend on the range of t.

Part I

Let's take a look at the parametric equation , with the interval . The graph of this equation can be seen in Figure 1 below.

This graph was created by Graphing Calculator 3.2.

Figure 1

Let's take a look at the parametric equation , with the interval . The graph of this equation can be seen in Figure 1 below.

This graph was created by Graphing Calculator 3.2.

Figure 1

Keep in mind that the curve above is
over the interval .
That is, if the interval was then only the bottom half of the circle
would be the curve. See Figure 2 below.

This graph was created by Graphing Calculator 3.2.

Figure 2

This graph was created by Graphing Calculator 3.2.

Figure 2

If the interval is , then
the curve would start at (1,0) and create 2½
revolutions and end at (-1,0). The list below shows the values of x and
y that correspond to multiples
of π/4 for the
parameter t.

t | x = cos t | y = sin t |

0 | 1 | 0 |

π/4 | 1/√2 | 1/√2 |

π/2 | 0 | 1 |

3π/4 | -1/√2 | 1/√2 |

π | -1 | 0 |

5π/4 | -1/√2 | -1/√2 |

3π/2 | 0 | -1 |

7π/4 | 1/√2 | -1/√2 |

2π | 1 | 0 |

9π/4 | 1/√2 | 1/√2 |

5π/2 | 0 | 1 |

11π/4 | -1/√2 | 1/√2 |

3π | -1 | 0 |

13π/4 | -1/√2 | -1/√2 |

7π/2 | 0 | -1 |

15π/4 | 1/√2 | -1/√2 |

4π | 1 | 0 |

17π/4 | 1/√2 | 1/√2 |

9π/4 | 0 | 1 |

19π/4 | -1/√2 | 1/√2 |

5π | -1 | 0 |

Part II

Let's take a look at the parametric equation , with the interval . Let b = 1, and let a vary.

This graph was created by Graphing Calculator 3.2.

Figure 3

Since a
doesn't equal b the result is
an ellipse. The list
below shows the values of x
and
y that correspond to multiples
of π/4 for the
parameter t.

What is interesting to note is that for a = 2 & 3 the curve starts at (1,0) and goes counter clock-wise, where the curve for a = -4 starts at (-4,0) and proceeds clock-wise. Also, the equation of an ellipse is . Multiplying the equation by a^{2}b^{2}
we get the
equation . If a
= b then we get the equation x^{2}
+y^{2} = a^{2} which is the
equation for the circle.

t | x = cos t | x = 2cos t | x = 3cos t | x = -4cos t | y = sin t |

0 | 1 | 2 | 3 | -4 | 0 |

π/4 | 1/√2 | 2/√2 | 3/√2 | -4/√2 | 1/√2 |

π/2 | 0 | 0 | 0 | 0 | 1 |

3π/4 | -1/√2 | -2/√2 | -3/√2 | 4/√2 | 1/√2 |

π | -1 | -2 | -3 | 4 | 0 |

5π/4 | -1/√2 | -2/√2 | -3/√2 | 4/√2 | -1/√2 |

3π/2 | 0 | 0 | 0 | 0 | -1 |

7π/4 | 1/√2 | 2/√2 | 3/√2 | -4/√2 | -1/√2 |

2π | 1 | 1 | 3 | -4 | 0 |

What is interesting to note is that for a = 2 & 3 the curve starts at (1,0) and goes counter clock-wise, where the curve for a = -4 starts at (-4,0) and proceeds clock-wise. Also, the equation of an ellipse is . Multiplying the equation by a

Part III

Let's take another look at the parametric equation , with the interval . This time let a = 1, and let b vary.

This graph was created by Graphing
Calculator 3.2.

Figure 4

Figure 4

Again, since a doesn't equal b the result is an ellipse. The list below shows the values of x and y that correspond to multiples of π/4 for the parameter t.

t | x = cos t | y = sin t | y = 2sin t | y = -3sin t | y = 4sin t |

0 | 1 | 0 | 0 | 0 | 0 |

π/4 | 1/√2 | 1/√2 | 2/√2 | -3/√2 | 4/√2 |

π/2 | 0 | 1 | 2 | -3 | 4 |

3π/4 | -1/√2 | 1/√2 | 2/√2 | -3/√2 | 4/√2 |

π | -1 | 0 | 0 | 0 | 0 |

5π/4 | -1/√2 | -1/√2 | -2/√2 | 3/√2 | -4/√2 |

3π/2 | 0 | -1 | -2 | 3 | -4 |

7π/4 | 1/√2 | -1/√2 | -2/√2 | 3/√2 | -4/√2 |

2π | 1 | 0 | 0 | 0 | 0 |

What is interesting to note is that for b = 2 & 4 the curve starts at (1,0) and goes counter clock-wise, where the curve for b = -3 also starts at (1,0) but proceeds clock-wise.

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