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.