Assignment #11 – Polar Equations

Jackie Gammaro

This investigation involves
polar equations. For this
write-up, I am going to investigate some different polar equations and the
differences in their graphs.

The three polar equation
investigated are:

#1

#2

#3

#1. First lets look at graph of .

Click on the link below to
investigate the graph in Graphing Calculator 3.5.

From
the graph we can estimate:

1)
The graph has a
period of 2 and an amplitude of 3, this comes from the trig function 3cosq, which also has an amplitude of 3 and a period of
2.

2)
The minimum
value looks to be 2, while the maximum value is 8. We can prove this by taking the derivative of the function,
setting it equal to zero then solving for theta. Then solve for r, for all values of theta.

Let

Let .

Let

#2. Now lets look at the graph of .

Click on the link below to
investigate the graph in Graphing Calculator 3.5.

One should notice this
function is quite similar in both equation form and graphically to the polar
equation: . It would be
advantageous to compare graphs simultaneously on one set of axis, which sure
enough Graphing Calculator allows.

Here
is a snapshot of the two functions on one coordinate plane.

Click on the link below to
investigate the graph in Graphing Calculator 3.5.

One graph is the other graph
shifted p/4 units. The graphs have the same period of 2p and an amplitude of 3. The minimum and maximum values of each graph are the same
but not at the same values for theta.
Again, we can take the derivative to find min and max values of the function.

Let
, then .

Let

Let .

Let

#3. Lets investigate now.

To get a better look at the
graph and investigate independently click on the link below to view the curve
in Graphing Calculator 3.5.

To determine the period of
the graph, I look at values of q where there is an
asymptote, values of theta where sin q - cos q = 1.

From the graph, one can see
at q
= 0, 2p there are vertical asymptotes.

Also, notice the period of
the function is 2p.

Again to find intervals of
increase and decrease, minimum and maximum values, take the derivative of the
function, set it equal to zero and solve for q.

Let *rÕ*= 0, q = ,

Maxima at , *n **ë** Z*

Minima at

This lesson is great for a
Calculus class, i.e., using the derivative to have students find max, min, and
intervals of increase and decrease.
Also, students have the opportunity to review trig functions and the
idea of period, amplitude, and values of sine and cosine along the unit
circle. Students will also be able
to explore basic function ideas, e.g. asymptotes.