Thomas Earl Ricks
Assignment # 11
Investigate with different values of p:
For k > 1, k =1, k < 1.
Note: The parameter k is called the “eccentricity” of these conics. It is usually called “e” but for many software programs e is a constant and cannot be set as a variable.
We will begin our investigations by looking at all four graphs in unison for a set value of p, and varying the eccentricity parameter k. Then we will investigate one graph in more depth.
Setting p = 1 and k = 1 yields:
Notice that each appears to be a parabola, with the sine equations being up and down facing parabolas, and the cosine graphs left and right.
Dropping k < 1 yields ellipse shapes of various kinds.
For p = 1 and k = .7 we get:
Dropping k even further has the ellipses approach a circle.
For p =1 and k = .3 we get:
Thus as k approaches zero, the graphs not only shrink in size, but approximate a circle more and more. Let us zoom in to observe the more circular shape:
What would you predict to happen if we raised k above 1?
If you guessed a hyperbola, you are right!
We will observe just the first graph as it will be confusing to interpret four overlapping hyperbolas all at once.
If p = 1 and k = 1.3 we get:
As we increase k, the hyperbola becomes more and more “steep” as in the example below where k = 2
We could observe the change in the shape of the graphs as k changes from below 1 to above one in the graph below:
Or if you wish to see a movie as the value of k varies between .1 and 2, click here. The graph with k =1 is given in purple as a reference graph.
Now let us investigate what happens to a specific graph as p is varied. Can you create a rough guess of what you think will happen? Will it shift, expand, contract?
Let us one polar equation and vary p (holding k constant at k = 1):
We will graph various values for p. When we do this we observe:
It appears that the p values changes the “steepness” of the graph, at least for a parabola. What about for an ellipse? Thus for a value of k < 1, say k = .5, we vary the p value for several graphs and obtain:
It appears that the right side of the ellipse goes through the p value on the x-axis.
Can you explore hyperbolas and determine what varying p does to them?
For more information on eccentricity, click here.
For more information on how to derive these polar equations from the rectangular conic formulas, click here.