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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:

even-degree polynomials

with a positive
leading coefficient

with a negative
leading coefficient

positive leading coefficient

negative leading coefficient

positive leading coefficient

negative leading coefficient

     Copyright Elizabeth Stapel 2005-2011 All Rights Reserved

Odd degree polynomials

with a positive
leading coefficient

with a negative
leading coefficient

positive leading coefficient

negative leading coefficient

positive leading coefficient

negative leading coefficient

As you can see, even-degree polynomials are either "up" on both ends (entering and then leaving the graphing "box" through the "top") or "down" on both ends (entering and then leaving through the "bottom"), 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" (entering the graphing "box" through the "bottom") and go "up" (leaving the graphing "box" through the "top"), 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.

  • Which of the following could be the graph of a polynomial
    whose leading term is
    "3x4"?
       
  • Graph A: down on the left, up on the right

    Graph B: down on both ends

     

     

    Graph C: up on the left, down on the right

    Graph D: up on both ends

    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, 3 in this case, is actually irrelevant for this problem. All I need is the "minus" part of the leading coefficient.)

    Clearly Graphs A and C represent odd-degree polynomials, since their two ends head off in opposite directions. Graph D shows both ends passing through the top of the graphing box, just like a positive quadratic would. The only graph with both ends down is:

      Graph B

  • Describe the end behavior of  f(x) = 3x7 + 5x + 1004
  • 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, just like every cubic I've ever graphed. A positive cubic enters the graph at the bottom, down on the left, and exits the graph at the top, up on the right. Since the leading coefficient of this odd-degree polynomial is positive, then its end-behavior is going to mimic a positive cubic.

      "Down" on the left and "up" on the right.

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Cite this article as:

Stapel, Elizabeth. "Polynomial Graphs: End Behavior." Purplemath. Available from
    http://www.purplemath.com/modules/polyendshtm. Accessed
 

 



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