Parametric Curves
Parametic curve in the plane is a pair of functions
where the two continuous functions define ordered pairs (x,y). The two
equations are usually called the Parametric equations of a curve. The extent
of the curve will depend on the range of t.
If we consider the following sets of curves for
and select values of a, b, and k in an appropriate range for t we are able
to see a pattern evolve from the values we choose. The graph will rotate
around the point (a,b), as we vary the value of k. Then as we continue to
change the value of k and leave a and b the same, we begin to see that k
is the slope of the line.
The one thing that does need to be considered is range for t. The range
for t needs to cover at least 2 pie. If we cover more than 2 pie then at
some point the graph should be retracing itself.
Using the work from #3, I wanted to be able to write a parametric equation
of a line through (7,5) with slope 3. I knew that my equation needed to
look like the following
The problem that I had was deciding on the range for t considering that
I need to increase my x and y max.'s and min.'s. The original settings for
the max.'s and min.'s was 4. I decided to see a good picture of the graph
that would increase these setting to 12. This was 3 times the original settings.
So to get my graph to cover the whole graph, I needed to graph a range of
t that was 3 (2(pie)) or 6 pie. I then had to decided where to start the
t interval at. I decided on 3(-pie) for a starting point and pie for a stopping
point. This should give me that interval that I needed. The graph came out
as planed.
Parametric equations are interesting. I look at their graphs as being combinations
of Polar and Cartesian graphing systems. They revolve like the polar system
yet you can still consider x and y coordinates like the Cartesian system.