Graphs of 4th Degree Polynomial Functions

(please wait while the applet loads...)

Continuing with our discussion of cubic polynomial functions, we'll now look at the graph of a 4th degree polynomial function which is given in factored form g(x)=a(x-p)(x-q)(x-r)(x-s). Again, a will stretch the graph vertically and reflect it vertically if a is negative. p, q, r,and s 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, r, and s.

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

Observe the following:

• Changes in a implement true vertical stretching and/or reflection.
• Behavior at : If a is positive, then the graph goes up on both the left and on the right (in other words, g(x) approaches as x approaches either or ). If a is negative, this behavior is reversed: the graph goes down on both the left and on the right.
• If p, q, r, and s are all different, then the graph will cross the x-axis at each of these x-intercepts.
• If just two of p, q, r, and s are the same, say p=q, then the graph will touch the x-axis at the x-intercept p=q, and will cross the x-axis at r and s. To explore this case further, change the formula in the second function box to a*(x-p)^2*(x-r)(x-s), so the "q" slider will have no effect.
• If two pairs are the same, say p=q and r=s, then the graph will touch the x-axis at the x-intercept p=q, and will also touch the x-axis at r=s. To explore this case further, change the formula in the second function box to a*(x-p)^2*(x-r)^2, so the "q" and "s" sliders will have no effect.
• If three of p, q, r, and s are the same, say p=q=r, then the graph will cross the x-axis at the x-intercept p=q=r (although the graph will have a cubic shape near this x-intercept), and will also cross the x-axis at s. To explore this case further, change the formula in the second function box to a*(x-p)^3*(x-s), so the "q" and "r" sliders will have no effect.
• If p=q=r=s, then the graph will touch the x-axis at the x-intercept p=q=r=s (in this case, g(x)=a(x-p)⁴, so this is just a horizontal shift of x⁴. To explore this case further, change the formula in the second function box to a*(x-p)^4, so the "q", "r", and "s" sliders will have no effect.

This touch/cross behavior observed above is determined at each zero by the multiplicity of the corresponding zero. If k is a zero of g(x), and (x-k)^m appears in the factored form of g(x), then the multiplicity of the zero k is defined to be the power m. Now if m is even, then the graph will touch at k, and if m is odd, then the graph will cross at k.

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