An Infinite Family of Cubic Curves

Consider the cubic equation for Euler's Elegant Example: $t(y^3-y) = s(x^3-x).$ As long as $$t \ne 0$$, we can divide through by $$t$$. If we then let $$\displaystyle p=\frac{s}{t}$$, we have the parameterized cubic equation: $$y^3-y = p(x^3-x). \label{ParameterCubic}$$

Now for any real number that we assign to the parameter $$p$$, we have an equation whose graph passes through the nine points of the $$3 \times 3$$ grid. You can explore these curves using the applet in Figure 8. You can set particular values of $$p$$ using the slider control, or put the applet into play mode and watch as the $$p$$ cycles through values between $$-5$$ and $$+5$$.

Figure 8: Euler's Elegant Example
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Open a dynamic GeoGebra applet in a new window]

For the particular values $$p=0$$, $$p=1$$, $$p=-1$$, we can factor the parameterized cubic equation into factors of lower order.

 Value of $$p$$ Factorization $$-1$$ $$(y+x)(y^2 - xy + x^2 -1)=0$$ $$\; \; \; \; 0$$ $$y(y-1)(y+1)=0$$ $$\; \; \; \; 1$$ $$(y-x)(y^2 + xy + x^2 -1)=0$$

In the case $$p=0$$, the graph consists of three horizontal lines: $$y=0$$, $$y=1$$ and $$y=-1$$. If $$p=1$$, the graph consists of the line $$y=x$$ and a conic section, namely a skewed ellipse with its major axis on the line $$y=-x$$ and its minor axis on the line $$y=x$$. The case $$p=-1$$ is similar, with the roles of the lines $$y=x$$ and $$y=-x$$ reversed. For all other values of $$p$$, the equation can't be factored over the real numbers and the curve consists of a single continuous line.

There is one more important special case, when $$t=0$$ in the original equation for Euler's elegant example. In this case, we have $$x(x-1)(x+1) = 0$$, so that the graph consists of three vertical lines: $$x=0$$, $$x=1$$ and $$x=-1$$. This is the asymptotic case to which the parameterized cubic equation tends when either $$p \rightarrow +\infty$$ or $$p \rightarrow -\infty$$.