Mathematical Investigations IV
Name:
Mathematical Investigations IV
Polar Coordinates-Out and Around
Graphing Polar Functions
1. Graph the polar function: r  2sin( )
a. Fill in the table and carefully graph each point. Connect with a smooth
r

0
curve. Plot additional values, as needed, between those listed. Be Careful!!

6

4

3

2
2
3
3
4
5
6

1
2
7
6
5
4
4
3
3
2
5
3
7
4
11
6
2
b. Describe what happens to r as theta goes from 0 to  and
from  to 2.
c. What does the graph appear to be? Find its equation in rectangular coordinates.
Polar 2.1
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Mathematical Investigations IV
Name:
d. Describe in words: What do you think the graph of r  sin( ) would look like? (Then
check your guess on your calculator and show the graph.)
2. Graph the polar function: r  1  2 cos( )
a. Fill in the table and carefully graph each point. Connect with a smooth curve.
r

0

6

4

3

2
2
3
3
4
2
2
5
6
4

7
6
5
4
4
3
3
2
5
3
7
4
b.
11
6
For what values of  is the radius negative?
What happened to the graph of each point with a negative radius?
Polar 2.2
2
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Mathematical Investigations IV
Name:
Recall that the conversions between (x, y) and (r, ) are:
r 2  x 2  y 2 , tan( )  xy ; and x  r cos( ) , y  r sin( )
c.
The rectangular equation for the graph is much more complicated:
r  1  2cos
 r  r 1  2cos  
2
r 2  r  2r cos 
We can now substitute partway:
x2  y 2  r  2x
That r is still problematic. Isolate r and then square both sides.
x2  y 2  2x  r
x
2
 y2  2x   r 2
2
We can now replace r2. (Replacing r by itself requires  x 2  y 2 . Not good.)
x
2
 y 2  2x   x2  y 2
2
This gives us the rectangular form which is a relation, not a function, of x. In polar form,
however, r  1  2 cos( ) is a function r of .
Explain why r is a function of .
Some Problems to Ponder:
3. Given a circle with center (–5, 0) as shown:
a. Write an equation for the circle, in rectangular form.
b. Algebraically, convert the equation from
rectangular form to polar form.
4. Consider the graphs of two concentric circles.
Polar 2.3
a. Write an equation, in rectangular form,
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Mathematical Investigations IV
Name:
for the graph of the each circle.
b. Write an equation, in polar form,
for the graph of each circle.
c. Algebraically, convert the equation from rectangular form to polar form for the inner
circle. What do you notice about this polar equation compared to your equation above?
Explain.
d. Find the polar coordinates of the points where the tangent line at the top of the smaller circle
intersects the larger circle.
e. Write a polar inequality and a rectangular inequality for the region between the two circles.
f. Find the area of the region between the two circles.
Polar 2.4
Spr 07