Project 5 - Exploring Polar Equations
Name:_____________________________
This project is due by Friday, June 3 at 4pm. No late projects will be accepted.
In this project you will explore polar graphs and note how changing the equation will change the look
of the graph. You will also make a colorful polar graph. Your calculator needs to be set to the
following:
DEGREE and POLAR Mode
WINDOW Settings: Scroll down to enter all values.
Part I:
(1) Graph each of the following on your calculator. Pay attention to the differences and similarities
among the graphs. Notice what changing signs and/or constants do to the graphs. It will be
helpful to make quick small sketches of each graph to make it easier to identify patterns.
(a) r = 1 + sin
(e) r = 3 + 3 cos
(b) r = 1  sin
(f) r = 3  3 cos
(c) r = 2 + 2 sin
(g) r = 4 + 4 cos
(d) r = 2  2 sin
(h) r = 4  4 cos
(2) Describe the general shape of these graphs:
(3) Consider the general forms: r = a ± a sin and r = a ± a cos.
(a) What does the constant a determine in each graph? (Hint: Did you notice the location
of the intercepts of each graph?)
(b) What do the cosine and the sine determine in the graphs?
Part II:
(1) Graph each of the following on your calculator. Pay attention to the differences and
similarities among the graphs, especially to the sizes of the loops. It will be helpful to make
quick small sketches of each graph to make it easier to identify patterns.
(a) r = 1 + 3sin
(e) r = 1 + 4cos
(b) r = 3 + 5sin
(f) r = 2 + 3cos
(c) r = 2  5sin
(g) r = 1  5cos
(d) r = 3  5sin
(h) r = 4  5cos
(2) Describe the general shape of these graphs:
(3) Consider the general forms: r = a ± b sin and
r = a ± b cos.
(a) What does a + b determine in the graph?
(b) What does b  a determine in the graph?
Part III:
(1) Graph each of the following on your calculator. Pay attention to the differences and
similarities among the graphs. It will be helpful to make quick small sketches of each graph
to make it easier to identify patterns.
(a) r = 3cos(2 )
(d) r = 5cos(4 )
(b) r = 4cos(2 )
(e) r = 5cos(5 )
(c) r = 4cos(3 )
(f) r = 6cos(6 )
(2) Describe the general shape of these graphs:
(3) Consider the general forms: r = a cos (b ).
(a) What does the constant a determine in the graph?
(b) How is b related to the number of petals if b is even? What about if b is
odd?
Part IV: Use the table feature of your calculator to help you generate the points you need for
the following graphs. You will graph each of the 3 graphs on the SAME polar paper. Color as
indicated. You can borrow colored pencils at the information counter in the Learning Center.
Input your equation, go to 2nd TBLSET (see the screenshot), and then go to 2nd TABLE.
GRAPH 1: Fill in the
table, plot each point,
connect with a smooth
curve, and fill in area
GREEN.
r = 10cos(4)

10
0
15
30
45
60
75
90
105
120
135
150
165
180
195
210
225
240
255
270
285
300
315
330
345
360
GRAPH 1: Fill in the
table, plot each point,
connect with a smooth
curve, and fill in area
BLUE OR RED.
r = 9sin(2)

0
0
15
30
45
60
75
90
105
120
135
150
165
180
195
210
225
240
255
270
285
300
315
330
345
360
GRAPH 3: Fill in the
table, plot each point,
and connect with a
smooth YELLOW
curve.
r = 10

10
0
15
30
45
60
75
90
105
120
135
150
165
180
195
210
225
240
255
270
285
300
315
330
345
360