Graphs of Cubic Polynomial Functions

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Consider the general cubic polynomial function g(x)=ax³+bx²+cx+d. The first applet below illustrates how changes in the coefficients a, b, c, and d will affect the graph. The red graph is f(x)=x³, and the green graph is g(x)=ax³+bx²+cx+d. Use the sliders to explore the effect of changing the values of a, b, c, and d.

MultiGraph applet written by David Eck (http://math.hws.edu/javamath/index.html)

Notice that changing the coefficient d certainly causes a vertical shift, but the roles of a, b, and c are unclear. Changes in a cause some kind of vertical stretching and/or reflection (but not true vertical stretching or reflection), and b and c seem to affect the amount that the curve "bends", and sometimes affect shifting as well.

Behavior at : In spite of the observations above, we can still see that if a is positive, then the graph goes down on the left and up on the right (in other words, g(x) approaches as x approaches , and g(x) approaches as x approaches ). If a is negative, this behavior is reversed: the graph goes up on the left and down on the right.

If the cubic polynomial can be written in factored form, i.e., g(x)=a(x-p)(x-q)(x-r), then it is much easier to see what the graph should look like. Again, a will stretch the graph vertically and reflect it vertically if a is negative. p, q, and r are the zeros of g(x), so the graph will have an x-intercept at each of these values. Use the sliders to explore the effect of changing the values of a, p, q, and r.

Observe the following:

• Now changes in a implement true vertical stretching and/or reflection.
• If p, q, and r are all different, then the graph will cross the x-axis at each of these x-intercepts.
• If two of p, q, and r are the same, say p=q, then the graph will just touch the x-axis at the x-intercept p=q, and will cross the x-axis at r. To explore this case further, change the formula in the second function box to a*(x-p)^2*(x-r), so the "q" slider will have no effect.
• If p=q=r, then the graph will cross the x-axis at the x-intercept p=q=r (in this case, g(x)=a(x-p)³, so this is just a horizontal shift of x³. To explore this case further, change the formula in the second function box to a*(x-p)^3, so the "q" and "r" sliders will have no effect.

NEXT: Discussion of 4th degree polynomial functions

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