Assignment 10

Parametric Curves

A parametric curve in the plane is a pair of functions:

x=f(t)

y=g(t)

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

This first graph is one of the simpler graphs. The center is (0,0) and the radius is 1.

There are many different alterations that we can make to the graph. I will begin by changing coefficient in front of the sine function.

Notice that this change elongates the graph and it now has the characteristics of an ellipse with center still at (0,0) and a major vertical axis with vertices at (0,2) and (0,-2) and co-vertices at (1,0) and (-1,0).

We would expect a change in the coefficient for the cosine to have a similar effect, but on the horizontal axis.

So, the center of this ellipse is (0,0). It has a major horizontal axis with vertices at (2,0) and (-2,0) and co-vertices at (0,1) and (0,-1).

When the sine is changed from to , we see the graph is changed to a bowtie curve.

When we increase the coefficient to a 3, there are now three loops.

Again, multiplying by a larger number for the sine, increases the amount of loops to the number we multiplied by.

When the same alteration is made for cosine, we get a parabola.

However, multiplying by a 3 for cosine, we get a similar pattern with 3 loops.

Multiplying by a 4 we see a similar pattern to what happened when the coefficient was a 2.

Conclusion from this investigation: When the coefficient is changed in front of the sine function, it will create the same amount of loops as what the coefficient is. When the coefficient is changed for the cosine function, we will get a continuous graph for odd coefficients and a discontinuous graph for even coefficients.

Another exploration that is interesting is when rational numbers between 0 and 1 are the multipliers.

Click HERE to see what happens when we make this change in the sine function.

Chlick HERE to see what happens when we make this change in the cosine function.