Presented by Godfried Lawson

With the graphing calculator 2.7 I will investigate the graph of:

* The investigation will focus on how different values of a,
b, and k affect the shape of the graph..

* I will compare the above equation with

* Then I will observe what happen when **cos(k-theta
) i**s replaced with **sin(k-theta
).**

__DEFINITION OF POLAR COORDINATE SYSTEM__

The polar coordinate system is a system of coordinate in which
a point in the plane is identified by its distance r along a ray
from a fixed point (the pole) and by the angle theta
between a fixed line ( the polar axis ) and the ray

Every point P, other than the pole , can be represented by an
ordered pair of real numbers ( r, theta).

Before we observe the graph of :

Let observe the table and the graph of:

Theta in degree |
cos(theta) |
1 + cos(theta) |
1 + cos(2 theta) |
1 + 3cos(2 theta) |

0 | 1 | 2 | 3 | 4 |

30 | .15 | 1.15 | .05 | -1.9 |

60 | -.95 | .05 | 1.81 | 3.44 |

90 | -.45 | .55 | .40 | -.80 |

120 | .81 | 1.81 | 1.32 | 1.98 |

150 | .70 | 1.70 | .98 | .93 |

180 | -.60 | .40 | .72 | .150 |

210 | -.88 | .12 | 1.56 | 2.69 |

240 | .33 | 1.33 | .21 | -1.34 |

270 | .98 | 1.98 | 1.94 | 3.81 |

300 | -.02 | .98 | 0.00 | -2.00 |

330 | -.99 | .01 | 1.97 | 3.90 |

360 | -.28 | .72 | .166 | -1.51 |

Polar graphs can be classified in families. The graphs of cos(theta)
( in light blue ) and sin( theta) ( not shown on the on graph
above ) are the the parents graphs of polar graphs.

Both graphs are circles with a diameter of 1 unit and both pass
through the origin. As with many families of graphs, you can alter
the position and shape of the graph by multiplying the function
by a number or by adding to it. You can also multiply theta by
a number or add a number to it to alter the graph.

Notice that the graph of the equation.
( in dark blue ) is symmetric with respect to the polar
axis, as is its parent graph .

However, the appearance of a small loop inside the outside graph,
which is no a circle, does not resemble the parent graph.

This graph is called a __Limacon.__

Polar equations of the form make graphs that
are called **Rose. K **is a positive integer.

Look at he graph of the equations below to see the pattern in
the amplitudes and in k-theta.

What happen when cosine is replace by sine?

( dark blue )

( green )

( light blue )

(red )

The difference between r = a + bcos (theta) and = a +
bsin (theta) is due to fact that

for any angle in standard position with measure theta , a point
p(x, y) on its terminal side , and ,

the trigonometric functions of theta are as follow;