From
Jim
Wilson’s website: 4. Investigate:

Appears to be
an ellipse
with center (3,0).
The major axis lies on the x-axis.
Points on the x-axis will have theta values of multiples of pi;
as a
result, the *r*-value can be
determined. When . So r = 8. When ,
so r = 2. This means
the Cartesian coordinates for the intersection of the ellipse and the
x-axis
are (-2, 0) and (8,0).
So the ellipse has a major axis of length 10.
The center of the ellipse will lie halfway
between these points, so the center is (3,0)
as
predicted. The minor axis appears to have
length 8, but I do not see quickly how to arrive at this result
algebraically.

In order to
ensure that
this curve is an ellipse, we can change the curve from polar
coordinates to Cartesian
coordinates:

This is
the equation of an ellipse whose center is (3,0)
with
semi-major (and horizontal) axis of length 5.
The ellipse also has a semi-minor (and vertical) axis of length
4 as the
graphical representation suggested.

Appears to be
an
ellipse.
Algebraically, this equation is similar to
the first, only theta has been shifted degrees. This
suggests that the major axis lies along
the line y = x, the major axis has length 10, and the minor axis has
length
8. When ,
reaches it’s minimum
value of 2, so r will be at a maximum of 8.
When ,
reaches it’s maximum
value of 8, so r will be at a minimum value of 2.

Appears to be
a
hyperbola with center (-2, -3) in Cartesian coordinates.
Algebraically, we can transform this equation
to Cartesian coordinates to see if it satisfies a more familiar
definition of a
hyperbola.

The result in
Cartesian
coordinates is a hyperbola most algebra II students would recognize. It has been shifted so that it’s center is at (-2, -2) as the polar graph
indicated.

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