Mathematical Investigations IV
Name:
Mathematical Investigations IV
Polar Coordinates-Out and Around
More Polar Stuff
1. Graph each of the following for 0    2 , WITHOUT using a calculator.
(You may check your results with a calculator.) Label your scale in each graph.
a. Be sure to label important points or features. (max r points, intercepts, etc.)
b. Show the "starting" point where  = 0. (Label with "S".)
c. Number the arcs and show the "direction" of the graph as  goes from 0 to 2 with arrows.
a. r  3
b. r  4 cos( )
c. r  4 sin(3 )
d. r  2 cos(  4 )
e. r  6 sin(  2 )
f. r  3cos(2 )
Polar 4.1
S09
Mathematical Investigations IV
Name:
2. Given the graphs of three limaçons, label the polar coordinates of each of the axis intercepts
and find the coordinates of any points where the graph crosses the pole.
a. r  12  6cos( )
c. r  6  12cos( )
b. r  9  9cos( )
Polar vs. Rectangular forms of functions
3.
Change each polar function into rectangular form.
a. r = 6
d. r  2cos( )  4sin( )
4.
b.  
5
6
c. r   2  sec( )
e. r  tan( )  sec( )
f. r 
12
4sin( )  7cos( )
Change each rectangular function to polar form r  f ( ) .
a. y 
3
3
x
b. x 2  y 2  2 x 2  y 2  2x
Polar 4.2
S09
Mathematical Investigations IV
5.
Name:
Change each ordered pair of rectangular coordinates into polar coordinates in FOUR
WAYS: First as (Positive Radius, Positive Angle), then (Pos., Neg.), (Neg., Pos.), and
then (Neg., Neg.).
a. (6, – 6) use radians
6.
b. (5 3,  5) use radians
c. ( –6, 8) to nearest degree
Find a polar equation for each of the following "standard" polar graphs.
30°
a.
b.
c.
6
45°
d.
e.
f.
Polar 4.3
S09
Mathematical Investigations IV
Name:
7.
The graph of r  sin( 23  ) was attempted at the left below. A better attempt is shown in the
middle, and the actual graph is shown to the right. Explain the most likely cause of the
incomplete graphs. That is, what changed on your calculator to expand the graphs?
8.
Sketch the graph of r  cos( 45  ) on your calculator. What domain is required to give a
complete graph?
9.
The graph on the left is a rectangular graph, y  f (x) . Find its equation. In your equation,
replace y by r and replace x by . Now sketch the resulting polar function, r  f ( ) .
Polar 4.4
S09
Mathematical Investigations IV
10.
Name:
The graph on the left is a polar graph, r  f ( ) . Find its equation. Convert it to a
rectangular graph y = f (x) by replacing r by y and  by x. Sketch the graph of y = ƒ(x).
(3,45°)
(–1,225°)
11.
Consider the graph of a function y = ƒ(x) given on the left below. Now try to sketch the
graph of the corresponding polar function created by replacing y by r and x by . (Think:
As x increases, does ƒ increase or decrease? Thus, rotating as r increases, does r increase
or decrease? So does your graph move farther from or closer to the origin?) Do not simply
find the equation of the given graph!
Polar 4.5
S09
Mathematical Investigations IV
12.
Name:
Let ƒ(x) = 2 – 4sin(x). Sketch the graph of this function.
a. What is the maximum value of ƒ?
b. What is the minimum value of ƒ?
c. For what x is ƒ(x) < 0?
d. If we now consider the function r = ƒ() created by replacing y by r and replacing x by
, what will be the maximum and minimum values of r?
e. For what values of  will r be negative? What happens to the polar graph when r is
negative?
13.
Consider the polar graph of the equation r = a + b sin() where a > 0 and b > 0.
a. Under what conditions on a and b will r > 0 for all  so that there will be no inner
loop? What does this tell you about the corresponding function y = a + b sin(x)?
b. What conditions on a and b will give r < 0 for some values of  so that there will be an
inner loop? What does this tell you about the corresponding function y = a + b sin(x)?
Polar 4.6
S09
Mathematical Investigations IV
14.
Name:
Sketch the graphs of both r = 3 – 2cos and r = 2.
a. Find the values of  where 3 – 2cos = 0.
b. Find the points of intersection of the two
graphs.
15.
Sketch the graphs of both r = 2 – 2sin and r = 2sin. How many points of intersection
are there? Find them. What is interesting here that didn't happen in the previous problem?
16.
We've already looked at the graphs of rose leaves of the form r = sin(n ). What would
change if we considered graphs with equations of the form r = 1 + sin(n )? Sketch several
of these graphs for positive integer values of n. Describe how the graphs differ from the
standard rose leaves.
Polar 4.7
S09