Mathematical Investigations IV
Name:
Mathematical Investigations IV
Polar Coordinates-Out and Around
Basic Curves – 1ntro
In previous MI courses, you explored basic types of functions and 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 some basic types of polar graphs along with how these parameters affect the graph of a
function in polar coordinates, r = a f(b + ) + k.
A: Circles: r = A cos  and r = A sin 
1.
Graph and label carefully the curve r = A cos() for several different values of A. Label each
graph. Be sure to choose at least one negative value.
Using complete sentences, describe how changing A affects
the graph of r = A cos . What is the radius of the circle?
What are the coordinates of the center? What happens if
A < 0?
2.
Now try r = A sin . Sketch a few graphs and label the value of A for each graph.
What changes when the sine function replaces cosine?
In terms of A, what is the center of the circle?
Polar 3M.1
Rev. F06
Mathematical Investigations IV
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3.
Remember y = ƒ(x + C)? Now consider r = 4 sin( - ) where  = /2, /4, and 2/3. Try
other values of  as you wish. Sketch and label at least two graphs. What is the effect of ?
Why does this make sense?
B.
Limaçons: r = A + B cos  and r = A + B sin 
Case 1: | A | = | B | (The Cardioid)
1.
Sketch the graph of r = A + A cos  for several values of A on the computer. Sketch three of
these below. Be sure to state the equation and to label all intercepts carefully.
Equation:
Equation:
Equation:
2.
How is the curve r = A – A cos  different from r = A + A cos ? Why? Sketch and label one
graph below.
Equation:
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Mathematical Investigations IV
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3.
Investigate r = A + A sin  and sketch and label two below.
Equation:
Equation:
Explain how these graphs differ from those of the form r = A + A cos .
Case 2: | A | > | B | (The Dimpled or Convex Limaçon)
1.
Graph and carefully label the following curve.
r = 4 + 3 cos 
Choose two more combinations of A and B, where 0 < B < A. Graph and label them carefully.
A
 2 .)
(Include one example where
B
Equation:
Equation:
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Mathematical Investigations IV
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Do more graphs as necessary so that you can answer the following questions.
2.
What happens to the dimple as A and B vary?
3.
What happens to the graph if r = A + B sin , with 0 < B < A ? That is, what happens
when the cosine function is replaced by the sine function?
4.
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 them carefully.
Equation:
Equation:
Explain in complete sentences what you observe.
A
A
 2 and where 1 
 2 . (This requires that
B
B
| B | < | A | as stated before.) Limaçons of the first type are called convex. Limaçons of the second
type are called dimpled.
Note: You've graphed functions where
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Mathematical Investigations IV
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Case 3: | A | < | B | (The limaçon with inner loop)
1.
Choose 2 combinations of A and B, where 0 < A < B. Graph and label carefully.
Equation:
Equation:
Sketch a few additional combinations of A and B to understand the pattern.
2.
What is the length of the inner loop in terms of A and B?
3.
One intercept is further from the origin than the others. What is the value of this intercept
in terms of A and B?
4.
Now look at the axis that is not intersected by the inner loop. Two intercepts are the same
size (with one positive and one negative). What are the values of these intercepts in terms
of A and B?
5..
What happens to the graph in the case r = A + B sin , with 0 < A < B?
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Mathematical Investigations IV
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6.
What happens to the graph if A and B have opposite signs?
(e.g. r = 4 – 6cos ). Graph a couple of examples and label them 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 have opposite signs:
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Mathematical Investigations IV
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C.
Rose Leaves: r = A cos (n) and r = A sin (n)
1.
This is yet another category of basic polar curves. Sketch a few curves and label them
carefully. Do more examples as necessary to complete the summary on the following page:
Equation:
Equation:
Equation:
Equation:
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Mathematical Investigations IV
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2.
Summarize your observations about rose leaves in complete sentences by stating the
significance of each of the following:
size of | A |:
n odd:
n even:
sign of n:
If sine is used instead of cosine:
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