Mathematical Investigations IV
Name:
Polar Coordinates-Out and Around
Basic Curves – 1
The Circle
In MI-2 you explored how changes in the parameters a, b, h, and k effected the graph of an
arbitrary Cartesian function y = a f(bx + h) + k. Now we are going to explore how these
parameters effect the graph of a function in polar coordinates, r = a f(b + ) + k.
Circles: r = A cos  and r = A sin 
1.
Graph and label carefully the curve r = A cos() for five different values of A. Be sure to
choose at least one negative value.
2.
In complete sentences, describe rhow
= A changing
cos . BeAsure
effects
to reflect
the graph
on the
of effect A has on the
center and radius as well as what happens for A < 0.
The graph is a circle with diameter A and center at (A/2, 0) {polar}. Thus, as A increases, so does the cir
3.
Repeat the above using r = A sin 
r = 3sin()
r = 1.5sin()
r = -2sin()
Description:
The graph of r = Asin() is a circle with a diameter of |A| and center at (A/2, /2),
{polar}. If A < 0, then the graph lies below the x-axis.
4.
For a given value of A (e.g. A = 6), how is the curve r = A sin  different from the curve r
= A cos ?
Since sin(x) = cos(x - /2), we would expect these curves to be 90 o rotations of the cosine
graphs and indeed they are.
Polar 3.1
Rev. F03
Mathematical Investigations IV
Name:
5.
How do the radius and center of the circle depend on A in the case r = A sin ?
See above comments.
6.
To investigate the effect of a phase shift on the curve, use your calculator to graph
r = 5 sin ( - ), where  = /4, /5,3/4. How does the phase shift change the
graph?
r = 5sin( - /4)
r = 5sin( - /5)
r = 5sin( - 3/4)
When graphing in Cartesian coordinates, y = f(x – h) effects a horizontal translation of
h-units. When graphing in polar coordinates, r = f( - ) effects a rotation of -units
about the pole.
C.
Limaçons: r = A + B cos  and r = A + B sin 
Case 1: | A | = | B | (The Cardioid)
Graph and label carefully the curve r = A + A cos  for various values of A.
(If you need some more data, use your calculator and graph additional curves of the form r = A +
A cos )
1.
What happens to the curve graphed above as A increases? decreases?
Polar 3.2
Rev. F03
Mathematical Investigations IV
Name:
2.
How is the curve r = A - A cos  different from r = A + A cos ? Why?
3.
Investigate r = A + A sin  and explain how the graph differs from r = A + A cos .
Polar 3.3
Rev. F03
Mathematical Investigations IV
Name:
Case 2: | A | > | B | (The dimpled limaçon, maybe)
Graph and label carefully the curve
r = 4 + 3 cos 
Choose 2 more combinations of A and B, where 0 < B < A. Graph and label carefully.
Equation: __________________
Equation: __________________
A
 2 , and graph them on
B
your calculator to understand of what happens to the graph as A and B vary.
Choose a few other combinations, with at least one example where
1.
What happens to the dimple as A and B vary?
2.
What happens to the graph in the case r = A + B sin , with 0 < B < A (i.e., when the
cosine function is replaced with the sine function.?
Polar 3.4
Rev. F03
Mathematical Investigations IV
Name:
3.
What happens to the graph in the case where A and B have opposite signs?
(e.g. r = 6 - 4 cos ). Graph a couple of examples and label carefully.
Equation: __________________
Equation: __________________
Explain in complete sentences what you observe:
A
A
 2 and
 2 . Limaçons of the first type are
B
B
called convex. Limaçons of the second type are called dimpled.
Note: Earlier you graphed functions where
Case 3: | A | < | B | (The limaçon with inner loop)
Choose 2 combinations of A and B, where 0 < A < B. Graph and label carefully.
Equation: __________________
Equation: __________________
Polar 3.5
Rev. F03
Mathematical Investigations IV
Name:
Choose a few other combinations and graph them on your calculator to get a good idea of what
happens to the graph as A and B vary.
1.
How does the position of the loop change with A and B?
2.
What happens to the graph in the case r = A + B sin , with 0 < A < B?
3.
What happens to the graph in the case where A and B have opposite signs?
(e.g. r = 4 - 6cos ). Graph a couple of examples and label carefully.
Equation: __________________
Equation: __________________
Summarize your observations about limaçons in complete sentences:
| A | = | B |:
| A | < | B |:
| A | > | B |:
Using sine instead of cosine:
A and B opposite signs:
Polar 3.6
Rev. F03
Mathematical Investigations IV
Name:
D.
Rose Leaves: r = A cos n and r = A sin n
You now have some experience investigating the effect of a parameter change has on a given
curve. Put your experience to work for you with regard to this curve. Here is some room to
graph some curves--label them carefully. Then complete the summary on the following page:
Equation: __________________
Equation: __________________
Equation: __________________
Equation: __________________
Observations:
Polar 3.7
Rev. F03
Mathematical Investigations IV
Name:
Summarize your observations about rose leaves in complete sentences:
size of | A |:
sign of A:
n odd:
n even:
sign of n:
If sine is used instead of cosine:
Polar 3.8
Rev. F03