Parametric Curves

By Natalie Streiner

LetÕs focus on the parametric equations

x= a cos (t)

y= b sin (t)

for 0 < t < 2¹

We are going to investigate how various values of a and b change the graph of our equations.

We will start by graphing

x= a cos (t)

y= b sin (t)

when a and b are equal to 1.

Here we see a circle of radius 1.

Now, letÕs see what happens when we increase both a and b at the same time.

The graphs of the parametric equations remain circles with radius=a=b.

What if a increases but b remains 1?

Here we see that the graphs are ellipses, rather than circles, when a>1.

Since sine is the vertical component of the function, each graph intercepts the y-axis at 1 and -1.

Since cosine is the horizontal component of the function, the function intercepts the x-axis at a and –a.

What if b increases but a remains 1?

Once again, the graphs are ellipses, rather than circle.

We see that the function intercepts the y-axis at b and –b.

While all of the functions intercept the x-axis at 1 and -1.

Conclusion

Here we see that when x=a cos(t) the horizontal component of the graph is affected by the value of a. Similarly, when y=b sin(t) the vertical component of the graph is affected by the value of b.

We see that if a=b the graph remains a circle with radius a=b.

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