Parametric Graphs

We wish to take a look at the parametric curve defined by the equations y=cos(t), x=sin(t).

The parametric equations define the graph of a circle with a radius of 1. We might conjecture that x=2sint,y=2cost would define a circle (again with center 0,0) with a radius of 2. Lets check that.

So in general it appears that parametric equations of the form x=rsint,y=rcost define circular graphs with center at (0,0) and radius r. Lets prove that.

Of course we may move the center of the circle by the addition of constant terms to our equations.

Lets turn our attention now to equations of the form x=asint,y=bcost where a and b are not equal.Lets look at an example.

Looking at equations where a and b are unequal but both even numbers appears to give us the graph of an ellipse. As with the graph when a and b are equal the center of this ellipse is at (0,0) and the maximum value for x and y correspond to the coefficients of the variable. Lets try odd coefficients.

This produces the same result. Now lets try coefficients that are both positive where one is odd and the other one is even. What result do you expect?

Is this what you thought? May we prove that equations of this form will always yield an elliptical graph in much the same way that we proved the circular result ?

The next exploration we would try here is with changing the coefficient of t, that is, equations like x=cos2t,y=sint2t.

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