### Parametric Curves

#### Audrey V. Simmons

Let’s look at several graphs where x and y are expressed in terms of parameter t.

We will be evaluating the changes that occur in the graphs when we vary the coefficients  of a and b in the following parametric equations:

x = a cos (t)

y = b sin (t)

Let   0< t <    for all the graphs investigated.

# PART I

Let b = 1 and vary a.

When a =1 and b=1 we have our unit circle.  ()

When a and b are different we have an ellipse.  ()

 t Cos (t) Sin (t) 0 1 0 0 1 0 1

Since the cosine of zero is one, our values for x will be a factor of a.

When a = -3 the graph looks the same as when a = 3.  The orientation or direction of the graphs is different.

## PART II

This time we will let a=1 and vary b for all t such that   0 < t <  .

The purple graph is still our unit circle.  However, this time our ellipse is elongated vertically instead of horizontally.

## PART III

What happens when we vary both a and b?

Since a=b, the graph is of a circle with radius equal to a and b.  If a or b had been negative, the graph would still be a circle of radius equal to the absolute value of a or b.

When a and b are different values our graph is different than the circle.  It is now the ellipse.

When a=2, -2 the ellipse crosses the x axis at (2,0) (-2,0).  When

a=3, -3, the ellipse crosses the x axis at (3,0) (-3,0).

When b =2, -2  the graph crosses the y axis at (0,2) and (0,-2).  When

b = 5, -5, the graph crosses the y axis at (0,5) and (0,-5).

Click HERE to see an example of changing graph for a parametric equation where t has a changing coefficient.

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