A polar equation is r = a + b cos(kż).  I will change each of the variables in turn to see how they affect the graph of the equation.   First let's change the k variable.  At one the graph looks like a sideways heart.  The indent is looks to be at (0,0).  When there is a number multiplied to the k value the picture begins to change.  Loops begin to form.  A 2k the picture is a figure 8. A petal seems to be added with each additional value multipled to k.  It ends up looking much like a daisy.

Next the a value will be changed and the other variables will stay constant at one.  The picture becomes more of an oval as the value of a is increased.  The size of the oval increases as well as the indent on the side disappears.  A good further exploration may be to see if it ever changes into a circle or changes to a completely different shape.

When the b value is varied and the a and k values are kept constant at one the loop starts to form another loop inside it.  The larger the b value the larger the loops inside it.

Now just for fun I will vary all of the variables at once.  It becomes quite a picture.


This is quite and involved graph and to properly explore it, it should be broken down into four different graphs.  There is much more to be learned on the subject of Polar Equations.