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.