Lab04: Graphing Parametric Equations

Objectives:

Mathematica can be used to graph curves defined by parametric equations.  In fact, it can be instructive to see the relationship between points on a parametric curve and points on the component curves.  In this lab we will explore the ParametricPlot command and use the Manipulate command to investigate the aspects of graphs of curves defined parametrically.

Example:

Consider the parametric equations and curve defined by

x=(t^3)-2t and y=(t^2)-t.

Step 1: define in the normal rectangular graphs the following equations:

x=t^3-2*t
y=t^2-t

and plot the two functions of t.

Plot[{x,y},{t,-5,5},PlotRange->5,PlotStyle->{Red,Blue}, AxesLabel->{t,"x,y"}]

Note:  These graphs are representations of how the x-coordinate and the y-coordinate will change as t changes.  That is, as the red graph increases or decreases, so will the x-coordinate of the parametric graph.  Likewise, as the blue graph increases or decreases, so will the y-coordinate of the parametric graph.  The following chart illustrates how the parametic will behave for different (approximate) intervals of t.

Step 2: Create a table (the entries are explained in Step 3) :

({{approximate t interval, x-coordinate, y-coordinate, plot (x,y) for t=}, {-∞<t<-1, →, ↓, t=-1}, {-1<t<1/2, ←, ↓, t=1/2}, {1/2<t<1, ←, ↑, t=1}, {1<t<∞, →, ↑, t=2}})

Step 3: Looking along the t-axis for the graph, approximate the t-values where either of the graphs change from increasing to decreasing or vice-versa.  For example, we first note that up to t ≈ -1, x is increasing and y is decreasing, so we indicate this in the table with a ( → ) in the second column and a ( ↓ ) in the third column.  We indicate the interval over which this behavior occurs in the first column, and in the last column we indicate the approximate t value where it would be nice to put a point.  At t ≈ -1 we see that x begins to decrease and y continues to decrease until t ≈ 1/2.  We fill in row two appropreately, and continue this process until we see no more chages for the two graphs.

Step 4: Plot (x, y) for the t-values in the last column of the above table, and label each point on the graph by its corresponding t-value.

({{t, x, y, point}, {-1, 1, 2, (1,2)}, {1/2, -7/8, -1/4, (-7/8,-1/4)}, {1, -1, 0, (-1,0)}, {2, 4, 2, (4,2)}})

Step 5: Starting at one of these t-values, let the corresponding arrows guild you to the next labeled point as t changes.  For example, if we begin at t = -1, we are going in to the interval -1 < t < 1/2, and the arrows say that to get to the point corresponding to t = 1/2 go backward and down simultaneously.  Continue until all t-intervals have been considered.

From these steps, one could skecth the parametric curve.  In Mathematica we can graph this curve as follows:

ParametricPlot[{x,y},{t,-2,2},AxesLabel->{"x","y"}]

Note that we use the above graphs to get “good” values for the parameter t.

Manipulate[
{p1=Plot[{x,y},{t,-2,2},PlotRange->6,PlotStyle->{Red,Blue}, AxesLabel->{t,"x,y"}];
p2=Graphics[{PointSize[Large],Red,Point[{a,x/.{t->a}}]}];
p3=Graphics[{PointSize[Large],Blue,Point[{a,y/.{t->a}}]}];
Show[p1,p2,p3],
p4=ParametricPlot[{x,y},{t,-2,2},AxesLabel->{"x","y"}];
p5=Graphics[{PointSize[Large],Black,Point[{x/.{t->a},y/.{t->a}}]}];
Show[p4,p5]},
{a,-2,2,0.1}
]

Note that in this example, the t-values used were approximate.  The actual “turning point” for the graph near t = 1, is in fact closer to t = 0.9 instead.

Exercises:

In the following, it will be to your benifit not to use the ParametricPlot command to look at the parametric graph of the curve before you finish the sketch by hand.

1.

Consider the parametric curve fiven by x=(t^3)-2t, y=(t^2)-1.

a.  Plot the graphs of x and y in rectangular coordinates.

b.  Fill in the following table as instructed in Step 3.  (There may be more rows than needed.)

({{approximate t interval, x-coordinate, y-coordinate, plot (x,y) for t=}, {□, □, □, □}, {□, □, □, □}, {□, □, □, □}, {□, □, □, □}, {□, □, □, □}})

c.  Fill in the following table as instructed in Step 4.  (There may be more rows than needed.)

({{t, x, y, point}, {□, □, □, □}, {□, □, □, □}, {□, □, □, □}, {□, □, □, □}, {□, □, □, □}})

d.  On a separate sheet of paper, sketch a graph of the parametric curve from the information obtained.  (Check your graph using the ParametricPlot command when you have finished your sketch.  If there is a problem with your sketch, go back and find your mistake.)

e.  How many times is y = 0 in the rectangular graph?
     How do you interpret this in terms of what you see on the parametric graph?

f.  How many times is x = 0 in the rectangular graph?
    How do you interpret this in terms of what you see on the parametric graph?

g.  What is the approximate minimum value of the rectangular graph of y?
     What is the approximate minimum value of the parametric graph?
     Do you think this is simply a coincidence that they are the same?  Explain.

2.

Use the steps outlined above to sketch (on a separate sheet of paper) the graph of the parametric curve given by:

x=(t^3)-2(t^2), y=(t^3)-2t.

(Show all of the steps outlined above.  Check your graph using Mathematica when you have finished your sketch.  If there is a problem with your graph, go back and find your mistake.)

({{approximate t interval, x-coordinate, y-coordinate, plot (x,y) for t=}, {□, □, □, □}, {□, □, □, □}, {□, □, □, □}, {□, □, □, □}, {□, □, □, □}})

({{t, x, y, point}, {□, □, □, □}, {□, □, □, □}, {□, □, □, □}, {□, □, □, □}, {□, □, □, □}})