Assignment 10 Write-Up
Parametric Equations: Ellipses
In doing many of the investigations from the assignment, my interest was in the
problems involving ellipses. So in this write-up I will discuss two of these problems—#3,
and #9—and explain how I might use a combination of these investigations in a
I began with Problem 3, for which I graphed the parametric equation
x acos t x sin t
(which is, incidentally, the same as the graph of This
bsin t y cos t
produced the graph of an ellipse (with center at the origin) for which the horizontal axis
has length 2a and the vertical axis has length 2b. For example, the graph of
y 3sin t
is an ellipse with a horizontal axis of length of 4 and vertical axis of length 6. That is, the
ellipse intersects the x-axis at -2 and 2, and has y-intercepts y = -3 and y = 3.
It is interesting and informative to investigate the graph for various values of a
and b. For example, when a = b, the graph is a circle (which is just an ellipse whose
x cos t
horizontal and vertical axes are congruent). Below is the graph of a
y sin t
circle with radius n.
Varying values of a can be graphed by using n instead. See GCF Graph 3 for
these dynamic graphs.
x acos t
For Problem 9, I began again with the ellipse . In order to allow both
x cos t
a and b to vary, I wrote the equation in the form This strategy has
y 1)sin t
its limitations in that b depends on a, but the strategy served its purpose (a and b are
x n(cos t) 2
different, and they vary). Added to this was the graph of shown
1)(sin t) 2
in red below.
This red segment has endpoints at (a, 0) and (0, b), an x- and y-intercept of the ellipse.
x n(cos t) 3
The next equation is which produces a graph with 4
1)(sin t) 3
curves, shown in green below, along with the ellipse and the segment. These curves also
have endpoints at the x- and y-intercepts of the ellipse.
There is a dynamic version of these graphs at Graph 1 here.
I continued this investigation to try to discern the effect of the exponents in the
x a(cos t) n
previous 2 equations. So I graphed to see what I could find.
y b(sin t) n
x 2(cos t)
Specifically, I compared the graphs of the following equations: (blue),
y 3(sin t)
x 2(cos t) 2
x 2(cos t) 3
x 2(cos t) 7
(red), (green), (purple), and
y 3(sin t) 2
y 3(sin t) 3
y 3(sin t) 7
x 2(cos t) n
(black). They are shown below, but the graph is dynamic in this
y 3(sin t) n
Graphing Calculator File (Graph 2).
When n > 2, as the value of n increases, the concavity of the curve increases as
well. Though the axes appear to be asymptotes for large exponents, the curves in fact do
intersect them. This can be seen by zooming in on the graph. Another interesting
observation is that when n is even, there is only a single curve (in the first quadrant), but
when n is odd, there are four identical curves, one in each quadrant.
When 0 < n < 2, as the value of n decreases, the curve moves away from the axes
and eventually degenerates (at n = 0) to become a point with coordinates (2, 3). Negative
values of n produce similar curves—they are simply reflections about the point (2, 3) of
their positive counterparts.
Ideas for classroom use:
This investigation would be an excellent way to demonstrate alternative ways of
graphing ellipses (that is, using parametric equations) since most students only learn the
x 2 y2
equation of an ellipse in the form 2 2 1. Students’ previous knowledge of ellipses
would be very useful here, however, as they examined where the “a” and “b” show up in
the two forms of equation (standard and parametric).
A lesson using this investigation could begin with a review of the structure of an
ellipse (definition, equation, etc.) and proceed with students’ exploration of the
parametric equations that generate ellipses. The teacher could have students investigate
what happens to the graph with various values of a and b (as I have done above).
Important observations include determining what happens when a = b (and
understanding why), as well as seeing the effect of changing the exponents (as in
Problem 9). It would also be important that students be able to predict what certain
graphs would look like and write equations of ellipses given certain stipulations. For
example, a problem presented in class might be to write the equation of an ellipse, in both
standard and parametric form, with a horizontal axis of length 11 and vertical axis of