In this write-up, I examine the graph of the following parametric equation:

for different values of the coefficients *a *and* b*.
The graphs will be explored for values of *t* greater than
or equal to zero and less than or equal to 2 pi.

The graph of the parametric equation that results for *a*
= 1 and *b* = 1:

becomes the graph of the unit circle shown below:

When the value of *b* equals zero, the parametric equation
becomes

for which the graph is the line segment with endpoints (-a,
0) and (a, 0). Similarly, when *a* equals zero, the equation
is

for which the graph is the line segment with endpoints (0,
-b) and (0, b). The graph when *a* equals zero and *b*
equals zero is just the origin (0, 0).

If we leave the value of *b* equal to one and vary the
coefficient *a*, the equation becomes

For values of *a* > 1, the graph is an ellipse with
a horizontal major axis of length 2**a*. The figure below
shows the graphs for *a* = 1.5 (purple graph), *a* =
3 (red graph), and *a* = 10 (blue graph).

The graph for *a* = 1 is a circle as shown previously.
When 0 < *a* < 1, the graph is an ellipse with a vertical
major axis of length 2. The figure below shows the graphs for
*a* = .2 (purple graph), *a* = .5 (red graph), and *a*
= .9 (blue graph).

The graph when *a* equals zero is the line segment with
endpoints (0, -1) and (0, 1). The graphs for values of *a*
< 0 are identical to the graphs of the absolute value of *a*.

If we leave the value of *a* equal to one and vary the
coefficient *b*, the equation becomes

For values of *b* > 1, the graph becomes an ellipse
with a vertical major axis of length 2**b*. The figure below
shows graphs for *b* = 2 (purple graph), *b* = 5 (red
graph), and *b* = 8 (blue graph).

The graph for *b* = 1 is a circle as shown previously.
When 0 < *b* < 1, the graph is an ellipse with a horizontal
major axis of length 2. The figure below shows graphs for *b*
= .2 (purple graph), *b* = .5 (red graph), and *b* =
.8 (blue graph).

The graph when *b* equals zero is the line segment with
endpoints (-1, 0) and (1, 0). The graphs for values of *b*
< 0 are identical to the graphs of the absolute value of *b*.

When both *a* and *b* are varied, the graph is an
ellipse with the length of the horizontal axis equal to 2**a*
and the length of the vertical axis equal to 2**b*. For values
of *a* > *b*, the horizontal axis is the major axis.
For values of *a* < *b*, the vertical axis is the
major axis. For values of *a* = *b*, the ellipse is
a circle with radius *a* = *b*.

The figure below shows graphs for *a* = 3, *b* =
2 (purple graph); *a* = 10, *b* = 4 (red graph); and
*a* = 5, *b* = 5 (blue graph); and *a* = 4, *b*
= 10 (green graph).