for e > 1, e = 1, and e < 1.
First notice, that when p or e = 0, r = 0, thus we eliminated these possibilities from the investigation. We can also disregard negative values for p since both cosine and sine oscillate between 1 and -1. (For the same reason we can ignore negative values for e.)
Therefore we began our investigation allowing p to vary, and e=1. Consider the case when p = 1 and e = 1.
Notice that the first cosine graph is just the negative of the second
cosine graph. Now look at the sine graphs for p = 1 and e = 1.
The same relationship occurs for the two sine functions. Furthermore,
the first sine function is actually the inverse of the first cosine function
and the second sine function is the inverse of the second cosine function.
In other words,
Thus, we restricted our investigation to just one of the functions, since
they are all either negatives, inverses, or negative inverses of each other.
We continued the investigation with
Using this function, consider p = 5 , 10, 15, 20, and e = 1.
Examining the graphs, one notices that as p increases, the parabola expands.
So, as p decreases, let's see what happens to the parabola. Letting p =
0.75, 0.5, 0.25, 0.1 and e = 1, we obtain the following graphs.
Thus as expected, as p decreases, the parabola contracts.
When e > 1, the graph becomes a hyperbola, and as p increases the
graph again expands.
Next examine the graphs as p decreases. Setting p = 1, 0.75, 0.5, 0.25 and e = 1.5, produces the following pictures.
Again as expected, when p decreases, the graphs of the hyperbolas contract.
When e < 1, the graph becomes an ellipse and as p increases the graph
again expands (look closely at the range of r). Consider the graphs
as p decreases, p= 1, 0.75, 0.5, 0.25 and
e = 0.5. We obtained the following pictures.
Once again for e < 1, we have an ellipse, and as p decreases the ellipse