Parametric Equations

By Colleen Garrett

I will begin this investigation by
exploring x=acos(t) and y=bsin(t) for various a and b and for 0£t£2p.

LetÕs see what happens when a=b.

When a and b both equal 2 we get a
circle with center at the origin and radius 2.

When a and b both equal -2 we get
a circle equivalent to the circle when a and b both equal 2

When a and b both equal .5 we get
a circle with center at the origin and radius .5.

gives us the radius of our circle centered at the origin.

What happens when a<b?

We can see here that we have ellipses. gives us the distance from the origin along the horizontal axis and gives us the distance from the origina along the vertical axis. Notice that if then the horizontal axis is the major axis of the ellipse.

What happens when a>b?

We can see here that we have ellipses. gives us the distance from the origin along the horizontal axis and gives us the distance from the origin along the vertical axis. Notice that if then the vertical axis is the major axis of the ellipse.

Now letÕs investigate x=a(cos(t))^{2} and
y=b(sin(t))^{2} for various values of a and b
and for 0£t£2p.

Click here
to watch a movie as a=b go from -5 to 5. This takes care of the case when a=b.

We get a line with slope equal to
-1 which lies in the first quadrant when a and b are greater than 0 and lies in
the third quadrant when a and b are less than 0.

LetÕs now look at what happens
when a<b.

We can see that we get several
lines. –b/a will give us the
slope of the line. a gives us the x-intercept of the line and b gives us the
y-intercept of the line.

What will happen if a>b?

We can again see that we get
several lines. Once again, –b/a will give us the slope of the line. a
gives us the x-intercept of the line and b gives us the y-intercept of the
line.

LetÕs look at x=a(cos(t))^{3 }and
y=b(sin(t))^{3} for various values of a and b and for 0£t£2p.

Click here
to watch a movie as a=b goes from -5 to 5. This takes care of the case when a=b.

We get a diamond shape with curved
sides. As a and b increase the
diamond shape gets larger and as they decrease the shape gets smaller.

What happens if a>b?

We have a and
–a equal to the x-intercepts of the graph and b and –b equal to the
y-intercepts of the graph.

What happens if a<b?

Once again we have a and –a
equal to the x-intercepts of the graph and b and –b equal to the
y-intercepts of the graph.

LetÕs take a look at what happens
if we keep increasing the power.

As the power of our equation
decreases the graph approaches our ellipse!