Lab 5: Implicit Curves

0. Objective:

In this lab we will explore how Mathematica can be used to generate graphs of implicit functions.

1. Example:

Consider the family of curves given implicitly by

(x^2)+(y^2)+c (x^2)(y^2)=1

where c is a constant.  Notice that c = 0 is a circle, centered at the origin, with radius 1.  To see the graph in Mathematica we can define a function of two variables and one parameter as follows:

f[x_,y_,c_]=x^2+y^2+c*x^2*y^2

and then use the ContourPlot command:

ContourPlot[f[x,y,0]==1,{x,-2,2},{y,-2,2}]

We can see the graph for a bunch of c values by using the Manipulate command.  In the following, the PlotPoints option is used to “smooth” the graph.

Manipulate[
ContourPlot[f[x,y,c]==1,{x,-2,2},{y,-2,2},PlotPoints->100],
{c,-5,5,0.1}
]

Notice how the graph changes as c changes.

Name:
PIN:

Task:

Consider the following family of functions:

(y^2)-2 (x^2)(x+8)-c((((y+1))^2)(y+9)-(x^2))=0.

(1) Define a function, like above, and create a Manipulate command in order to see the graph for several values of c.  You may have to “play” with the x and y ranges to get a good graph.

(2) Look in the Documentation Center to see how to graph the curves with c = 0 and c = 2 on the same graph.  Sketch this graph on the back of this page and determine how many times the two curves intersect.  (You may need to “zoom in” to find all of them.)

(3) Now add the curves with c = 5 and c = 10 to your graphs in part (2).  What do you notice?  What about other values of c?