Through the assistance of Graphing Calculator 3.2, we investigate the different variances when graphing Polar equations.
Explore the equation r = a + b cos (k q) such that 0 £ q £ 2 p
Since there are three variables a, b, and k to explore, there are many cases to explore.
1. When a and b are equal, and k is an integer, this is referred to
as the “n-leaf rose.”
Let us graph such that :
a = b = 2, k = 1 (red)
a = b = 4, k = 1 (green)
Notice that when a=b , a and b are scalar factors for the “n-leaf rose”. Also, when k=1, the roots of r = a + b cos (k q) are 0 and a+b.
Next, we observe the effect of k on the equation.
Below are the graphs of:
a = b = 2, k = 3 (red)
a = b = 4, k = 5 (green)
2. Let’s now look at the graph of a=2, b=8, and k=1.
3. What happens when a > b and k is an integer?
To see, let’s investigate the graph of a = 5, b= 2, and k= 10.
The “leaves” are merging towards a circle form.
The leaves come into a point on the circle centered on the origin with the radius a-b.
The tips of the leaves work out to a point on the circle centered at the origin with the radius a+b.
The function oscillates between these two circles k times to produce k “leaves”.
Once k becomes large enough, other characteristics can be explored.
Look at a=5, b= 4, and k=1000
This graph has many different characteristics that could be explored. Observe that the center is not filled.
Notice that there is a five leaf rose in the center as well as the outer leaves (if you will) are also five.
This seems to be related to the “a” value.
When k=2000 is tried, notice that the internal number of leaves becomes 10.
4. What about when a = b and k is NOT an integer?
Let a = b = 5, and k = 3 (green)
and a = b = 5, and k = 3.4 (red)
When a < b and k is not an integer, observe a similar transformation taking place as we witness when a=b and k is not an integer.
6. How about when a > b and k is not an integer.
When a > b and k is not an integer, what do you think is observed this time?
Š As a, b, and k vary there seems to be many relationship issues that can be discussed. The number of leaves and the relationship that both a and b have seem to be related.
Š It becomes more interesting when the values are no longer integers, that is when all the changes and predictions change.
This looks like an investigation that can be shared with high school students allowing them to draw quite a few different and interesting conclusions as well as what has been observed.