Stephani Eckelkamp

ÒEvery Rose has its ThornsÉÓ

An Exploration of Polar Equations

 

 

Investigation the polar equation

 

 and 

When a, b, and k = 1                    When B and K = 1

                       

 

 

When a, b, and K =2                             When b and K = 2

 

          

 

 

What changes between the graphs when 1 is replaced by 2?

 

F     The x intercept to the right increases by 2, and the x intercept to the left increases by 4, in relation to the origin.

F     The range does not change

F     Creates a Two leaf rose

F     The x intercept increase 1 to the right, 2 to the left and 2 along the y axis in relation to the origin.

F     The Range and the domain are equal

F     Creates a 4 leaf rose

 

Further exploration of

 

What if a=1 and b and k=2?

 

 

From the above graph it seems that a effects the range of the graph.  What might the graph look like if we set a and B = 2 and k = 1?

 

 

Here we donÕt have a range of 2, but we do see that the graph intersects the y axis at 2 and -2.  When we change the k value from 2 to 1 the graph does not intersect itself like when k = 2. 

 

Can you make a conjecture for how the graph might behave for k = 3?

 

 

When we increase our k value the graph intersects itself and forms 3 Òpetals.Ó  This is called a 3 leaf rose. 

Make a conjecture about K=4 and graph your prediction here

 

Now let us look at how the variables b and k affect the graph of .

 

B = 1 and k = 2

 

All of our axis are crossed at 1 and -1, which is our b value.  LetÕs look at the graph when b = 2 and k = 3 to see if we can get a better idea of how k effects this equation.

 

 

Here we have a 3 leaf rose with petals that are 2 units away from the origin. 

 

To further explore how k effects the graph open this document and change the k values.

 

What happens when K is even?

What happens when K is odd?

                                                                      

 

 

What if we replace cos with sin?

 

Our new equation is

 

 A, b, and k = 1

 

 

What is the difference between the cos and sin graphs with a, b, and k=1?

 

 

                                   What changed between the two graphs?

How can we explain these changes?  Let us look at a few more graphs to see if we can generalize the differences between the sin and cos graphs with the same a, b, and k values.

 

A, b, and k = 2

 

 

Is there a difference between this graph and the previous graph?

If we look at the difference between when all variables are equal to 1 we can see a 90 degree rotation clockwise from cos to sin. 

 

When looking at the graph where all the variables equal 2 we can see a 45 degree rotation counter clockwise from our cos to sin graph.

 

Let us look at what happens when 3 is set for all variables a, b, and k.

 

 

Here we can still see that the sin graph has been created by rotating the cos graph.  Is it still 45 degrees?  Look closely? Is there a way to check your prediction?

 

YE!!!  If it is rotated 45 degrees the maximum distance from the center to a pedal should fall on the line y=x.

 

 

as we can see from the graph above it is not a 45 degree rotation.  Do you have any guesses as to what it may be?

 

We can make a guess that since two petals gave us a 45 degree rotation from cos to sin, perhaps 3 leaves will give us 30 degrees?

How did we go from 45 degrees to 30 degrees?

 

Here is an example of a 3 leaf rose in GSP with the angles of rotation marked

 

 

click here to go to open this file

 

Can you create a GSP file to give an example of the degree of rotation for another n leaf rose?

 

Create your picture here

(make sure you graph your function as a polar equation)

 

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