x^3 + y^3 = 3xy
as a parametric equation.

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Let's examine the equation:

x^3 + y^3 = 3xy

For most applications other than ones such as Graphing Calculator, this equation would be difficult to graph. Most would require the user to first solve for y, and then input that equation. However, if we can first re-write this in parametric form we can graph this equation in most applications, including graphing calculators such as the TI-81 or TI-82.

We can accomplish this by first letting y = tx cross the curve at (0, 0) and at (x, y). Then, substituting, we have

x^3 + t^3*x^3 = 3tx^2
x^3(1+t^3) = 3tx^2

This results in the following parametric equations:

x= (3t)/(1 + t^3)
y = (3t^2)/(1 + t^3)

Now we must find a suitable range for t. Since we are able to graph our original equation with Graphing Calculator, let's compare the graph of the original equation to graphs of the parametric equation to gain insight into what range to use for t. For each of the comparisons, we have the graph of the original equation on the left and the graph of the parametric equation on the right.

We begin with t in the range

-pi < t < pi

Graph Graph

Obviously our range of t is not large enough. In addition to noticing that we do not obtain the entire graph for this range, we see that when graphing the parametric equation we get an asymptotic line in the graph. Also note that the parametric graph shows the curve as complete on the interval [-.1, .1] on the X-axis.

Next we graph with t in the range

-2pi < t < 2pi

Graph Graph

We obtain more of curve for this range. Now let's try t in the range

-3pi < t < 3pi

Graph Graph

Getting closer, but still not the whole curve. Try t in the range

-4pi < t < 4pi

Graph Graph

Almost. Now graph with t in the range

-5pi < t < 5pi

Graph Graph

Still a little short. Further tries will show that with t in the range

-8pi < t < 8pi
we get a fairly good approximation of the graph.

Graph Graph

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Write-up by Robert Childres.