During this investigation, we will examine the graphs of polar equations,
as certain coefficients are varied. The following equations are the equations
that were investigated. Click on any equation to jump to the investigation
of that equation.
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This pattern continues as k gets larger. When k is odd, there are the same number of loops as k. When k is even, the number of loops is twice the value of k. When k is negative, the graphs are rotated 180°. This is expected because sine is an odd function. When fractional values of k were examined, only one pattern of significance developed. When the value of k = 1/2, the graph looped back onto itself, and the two endpoints of the graph met each other. When k was greater than 1/2, the endpoints were tangeant to the graph, while when k was less than 1/2, the two endpoints are not tangeant to the graph at all.
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When we vary a in 
, the size of the graphs
changes. For example, when k = 1, the radius of the circle grows as we increase
a. If we change the sign of a, it rotates the graph of a 180°. If we
were to decrease the value of a from 1 to a fractional value, the graph
would shrink (not pictured.)
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As the graphs of sine and cosine are related, we expect the graphs of the first two equations to behave in a similiar manner. As the following graphs demonstrate, the graphs do behave similarly, there is only a 90° rotation when k is integral. This is expected because of the relationship between sine and cosine. One difference is that if negative value is used for k, the result is the same graph as if the positive value had been used. This occurs because cosine is an even function.
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When the k values that are substituted into the equation
, are fractional
the result is not a simple rotation of the graph the we obtained for the
first equation. The graphs below show that the tangeancy of an endpoint
does occur, but not at the same values as when 2asin(kt) was graphed.
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When the a coefficient is changed from 1, the same results occur that
occurred with the first equation. If a >1, it exapands the graph, if
0<a<1, it shrinks the graph, if a =0, there is no graph, if -1<a<0,
there is a rotation of 180° and the graph shrinks, if a =-1, the graph
is rotated 180° and if a<-1, the graph is rotated and expanded.
When a constant is added to the first equation, the third equation is
obtained. The graphs of the two equations should behave in a similar manner.
When k is negative, the graph is still rotated by 180°. Changing a still
has the same affects on the graph, rotating, shrinking or expanding.
Although these similarities exist, there are also several differences. When
a=b=k=1, the graph becomes becomes a limacon. When k =2, we no longer have
four congruent loops, but two sets of two congruent loops, the pattern of
two sets of congruent loops continues for even integral values of k. When
k=3, loops within loops are obtained, this pattern also continues for odd
integral values of k.
| a=1 b=1 k=1 |
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a=1 b=-1 k=1 |
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| a=1 b=1 k=2 |
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a=1 b=-1 k=2 |
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| a=1 b=1 k=3 |
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a=1 b=-1 k=3 |
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| a=1 b=1 k=4 |
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a=1 b=-1 k=4 |
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When the graphs are examined that have a negative value for value of b,
it appears that nothing happens if k is odd and that the graph is rotated
by pi/k if k is even. In the graphs below the blue is the graph when b is
positive, while red is the graph when b is negative. The plots are the same
for odd k values.
 k=2 |
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a=1 k=4 |
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Upon further examination, by graphing the function from 0 to Pi, when k
is an odd, we get the complete graph, when k is even we get two halves of
a graph. This would appear to show that when k is an odd integer, the effect
a of negative b is to trace the same graph, with a different starting point.
This also appears to show that when k is an even integer, the effect of
negative b is to flip the graph about the r axis when t = 0 ( or the x-axis
in an x-y coordinate plane).
| a=1 k=1 |
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a=1 k=2 |
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k=3 |
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a=1 k=4 |
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If the value of b was chosen between zero and one, it had the affect
of changing ratio of the sizes of the loops. The smaller b was, the closer
together in size that loops became.
| a=1 b=1/2 k=1 |
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a=1 b=1/2 k=2 |
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a=1 b=1/2 k=3 |
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b=1/4 k=1 |
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b=1/4 k=2 |
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b=1/4 k=3 |
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As the value of k was increased, the resulting shapes began to look like
flowers. With the ease of modern computers, some new "flowers"
are ctreated.
The behavior of the fourth equation should be predictable based upon the
behavior of the second and third equations. It behaves as expected, when
k is fractional values, the graph still does not completely loop. Negative
values of k still have no effect on the graph, while negative values of
a rotate the graph 180°.
a=1/2 b=1 k=1 |
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b=-1 k=1 |
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b=1 k=2 |
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b=-1 k=2 |
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b=1 k=3 |
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b=-1 k=3 |
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b=1 k=4 |
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b=-1 k=4 |
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When the graphs are examined that have a negative value for value of b,
it appears that nothing happens if k is odd and that the graph is rotated
by pi/k if k is even. In the graphs below the blue is the graph when b is
positive, while red is the graph when b is negative. The plots are the same
for odd k values.
k=2 |
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k=4 |
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| a=1 k=1 |
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a=1 k=2 |
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k=3 |
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a=1 k=4 |
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| a=1 b=1/2 k=1 |
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a=1 b=1/2 k=2 |
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a=1 b=1/2 k=3 |
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| a=1 b=1/4 k=1 |
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a=1 b=1/4 k=2 |
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a=1 b=1/4 k=3 |
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When the graphs of this equation are examined, they draw "spokes".The
effect of changing k is to draw a different number of spokes. When k = 2.
there are four "spokes", when k = 3, there are three "spokes".
The pattern of 2k spokes for even values of k and k spokes for odd values
of k continues for integral values of k. The effect of changing either a
and b is to rotate the spokes around the origin, the larger the value for
a or b, the more verticalat least one of the spokes becomes. If both a and
b are changed together, or when c is changed, the spokes become wider or
narrower. The following graphs demonstrate these behaviors.
| a=2 b=2 c=2 k=2 |
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| a=1 b=1 c=1 k=2 |
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| a=1 b=6 c=1 k=2 |
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| a=6 b=1 c=1/2 k=3 |
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| a=1 b=1 c=1/2 k=4 |
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| a=1 b=1 c=1/2 k=5 |
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