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Polynomial Graphs: End Behavior (page 1 of 5) Sections: End behavior, Zeroes and their multiplicities, "Flexing", Turnings & "bumps", Graphing When you're graphing (or looking at a graph of) polynomials, it can help to already have an idea of what basic polynomial shapes look like. One of the aspects of this is "end behavior", and it's pretty easy. Look at these graphs:
Copyright © Elizabeth Stapel 2006-2008 All Rights Reserved
As you can see, even-degree polynomials are either "up" on both ends or "down" on both ends, depending on whether the polynomial has, respectively, a positive or negative leading coefficient. On the other hand, odd-degree polynomials have ends that head off in opposite directions. If they start "down" and go "up", they're positive polynomials; if they start "up" and go "down", they're negative polynomials. All even-degree polynomials behave, on their ends, like quadratics, and all odd-degree polynomials behave, on their ends, like cubics.
The important things to consider are the sign and the degree of the leading term. The exponent says that this is a degree-4 polynomial, so the graph will behave roughly like a quadratic: up on both ends or down on both ends. Since the sign on the leading coefficient is negative, the graph will be down on both ends. (The actual value of the negative coefficient is irrelevant for this problem. All I need is the "minus" part of the leading coefficient.) The only graph with both ends down is Graph B.
This polynomial is much too large for me to view in the standard screen on my graphing calculator, so either I can waste a lot of time fiddling with WINDOW options, or I can quickly use my knowledge of end behavior. This function is an odd-degree polynomial, so the ends go off in opposite directions. A positive cubic is down on the left and up on the right, and this will do the same, since the leading coefficient is positive. "Down" on the left and "up" on the right. Top | 1 | 2 | 3 | 4 | 5 | Return to Index Next >>
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Copyright © 2006-2008 Elizabeth Stapel | About | Terms of Use |
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