Assignment 10

PARAMETRIC CURVES

by Rives Poe

In this investigation, we will look at parametric curves. A parametric curve in the plane is a pair of functions

x=f(t) and y=g(t)

where the two continuous functions define ordered pairs. The curve will depend on the range of t . We are able to graph this type of equation on Graphing Calculator easily.

We will start our investigation looking at the graph of two parametric equations: x=cos(t) and y=sin(t). We will have the range of t be from 0 to 2pi.

Because the two functions define ordered pairs, we should have expected the curve to be a circle. For example sin(0)=1 and c0s(0)=0, therefore the ordered pair is (1,0), which we can see on the graph above.

Let's try changing the equations to explore other graphs.

If we change the coefficient of t, how do you think the graph will change? (I will not change the range of t however -at least not yet!)

For example, let's look at x=cos(at) and y=sin(bt). If we leave the a and b equal to 1, we will have the same graph as above. What will happen if we set a = 2 and b=2? Any ideas? Let's try it.

It is exactly the same as the graph above. What if the coefficients are different? In the next graph I will let a =1 and b=2. Let's see what happens.

Now, that is different! Before we decide what is happening let's lool at a couple more curves.

For x=cos(at) and y=sin(bt), let a = 2 and b=4:

This graph looks the same as when a=1 and b=2. Since a is half the value of b, the curve crosses one time creating 2 areas.

Let's look at a=2 and b=6. Since a is now 1/3 of b, I will conject that the curve will cross two times, creating three areas. What do you think? Take a look!

I was right! So, now we have determined how to predict what happens when a is less than b. What do you think will occur if b is less than a? Let's start with something small, for example let a = 3 and b= 2.

The curve crosses three times along the y-axis, which equals the value of a.

When a is greater than b, the curve seems to graph on the y-axis. When b is greater than a, the curve seems to graph along the x-axis.

Let's investigate what happens if we multiply the two equations by a and b.

x=a cos(t)

y=b sin(t)

If we set a and b equal to 1, then we will have the same graph as our very first one. What do you think is going to happen when we let a and b equal 2? Take a look:

So, as long as a and b are equal, (in all of the cases above) the two equations will create a circle.

Now try a=2 and b=3:

When a is greater than b, an ellipse is formed. When a is less than b, an ellipse will also be created, but on the y-axis. In the following graph, a=3 and b=4

To investigate more with parametric curves for the equations x=a cos(t) and y=b sin(t), click here.