Recall that we define
a point (x,y) on the plane as x units to the right of the origin
and y units to the left of the origin. Below is a graph of the
This works great for lines
and parabolas, but now we are going to graph more complex figures
using polar coordinates.
of Polar Coordinates:
Now a point will be given as such: (r, theta). This r is the distance
from the origin to the point and the second coordinate, theta,
is the angle that the vector from the origin to the point makes
with the positive x-axis. To convert cartesian equations to polar
equations use the following rules:
into Polar Equations
going to be exploring the above polar equation for different values
of a, b and k and see how the values affect the graph.
I : A=B and k is an integer that will vary:
seems to be the pattern here? As k increases, the number of "leaves
in the flower increases"? If we increase k=4, we should get
look at k=10 and see if we the pattern still works.
all the above graphs, A=B=1 and k varied. What if A=B=5,
would k still be the numbe of leaves?
you compare the above equation to the equation earlier equation
with four leaves, you will see that as you increase a=b,
the graph becomes "leaves" become larger.
When A=B, then k= the number of "leaves in the flower."
II : A>B and k is an integer that will vary:
graph looks very similar to the above when k=1 and a=b,
let's keep investigating and see if a>b has an impact
on the graph.
the leaves are not meeting in the middle, let's keeping investigating:
we can see, when a>b, the leaves do not meet at the
origin as they did when a=b. In all the above polar
equations for part II, a=3 and b=2. Let's see what happens when
a=4 and b=2.
In comparing the two above equations we see that as the difference
from a to b gets larger, the leaves meet further
and further away from the origin.
III : A<B and k is an integer that will vary:
us better compare how a and b are affecting the
graphs, I will draw three graphs on each picture, the new graphs
when a<b will be in blue.
blue curve has one loop at the origin when k=1. When we increase
k=2, let's see what happens:
we see two extra loops when k=2. Let's compare the above graph
with a couple more that we have already seen:
do we expect for k=10, when a<b? From the above patterns,
the graph will look just like the graph when a=b, but there
will be 10 extra loops inside the larger loops or "leaves".
IV : Investigate
b=1 and k varies
b varies and k=1
can continue investigating polar equations in Graphing Calculator
3.2 or TI-83 and similar calculators.
would be to change cos to sin and investigate as we have just