The purpose of this assignment is to investigate Polar Equations, using
the equation below:
The graph of this equation is as follows:
Changing the angle while keeping the rest of the equation constant, we note the following changes:
The major axis rotates 45 degrees when angle is changed by 1/4 pi.
Now let's change only the angle again.
This time the major axis has rotated 180 degrees when the angle is changed by the value of pi.
At this point in the investigation, I would direct my students to make predictions on the rotation if the value of the angle is changed by 1/2 pi and then by 1/3 pi. Then, they would graph those values to determine the accuracy of their predictions. I would then ask them to explain the basis of their predictions, ie. converting pi radians to degrees.
In further exploration of Polar equations, we will investigate various angle values for the following parent equation and its graph:
The parent graph is a hyperbola with vertical and horizontal asymptotes.
As in the last equation, we will change the angle value by 1/4 pi.
Using a pencil or other straight object to represent an asymptote, we can see the its actual rotation of 45 degrees.
Again, as in the first investigation, we will change the angle value by the value of pi.
Note the complete 180 degree rotation of the asymptotes.
At this point in the investigation, I would direct the students to predict the graphs if the angle values were changes by first 1/2 pi and then 1/3 pi and then graph the changed equations.
When the students were finished with explanations of their predictions and their outcomes, discussion of transformations with respect to their graphs would be a good opening for the next investigation.