Assignment 1: Exploration of the cubic equation

by

John Simmons

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Basically this equation is a cubic equation in x and y.

When a = 4 we get the following graph.

This is a graph of the equation above. It seems to show some symmetry about the y-axis. The curve crosses the x-axis at -2, 0, and 2 and it crosses the y-axis at 1, 0, and -1. Looking at other values of a may reveal what effect changes in a will have on the curve.

When a = -4

In this case the extreme curvature of the graph from x=-2 to x=2 has been smoothed .

When a =1

When a is 1 the graph changes to a ellipse with a line segment through the center. It actually appears to be the graph of two relations--one interposed upon the other. In fact by manipulating this equation by hand or with a symbol manipulator, a factored form of the equation can be found (Wilson).

From the two graphs above, we can see that indeed the two forms are equal.

When a =0 this graph seems to be a function, x=f(y)

When a is -1

With the next two graphs we can compare what happens when a= 0.9 and a=1.1. In this situation the graph seems to be rotated 90 degrees then reflected along the x-axis.

Here we have seven equations, with various values of a, all graphed on a single grid.

In this next figure a and b are chosen to be nearly equal, with values of a between -5 and +3 and values of b within 0.1 of a.

Next we see what happens when the equation is set to equal z.

The picture above is three dimensional. Graphing Calculator 3.2 returned this graph in animation.