End Behavior of Polynomials

Directions

In this activity, you will explore different parts of functions which help determine the end behavior of a graph. You will interact with the given graphs, and look for similarities to come up with your conjecture which you will type into the text box.

Exploration #1

With the applet below, explore the what happens to the graph as you change the degree of the polynomial function. Try to observe any patterns between the relationship of the degree and the graph.

Exploration #1: Applet

Exploration #1: Question

What pattern do you observe between the value of the degree and the shape of the graph? Specifically, do you notice any differences between the graphs of even degree functions and odd degree functions?

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Exploration #1 Summary

In the last example, we learned that our graph depends on if the degree of the function is even or odd. This is the first requirement to help us determine what we call the "End Behavior" of a function, or how the function behaves as we look to the left and right sides of the x axis. In the next Exploration, we will learn the second requirement to help us determine the end behavior of a graph of a polynomial function by looking at the Leading Coefficient of the Polynomial.

Exploration #2: Even Degree Polynomials

Use the applet below to observe what happens to the graph of an EVEN degree polynomial as we change the value of the Leading Coefficient. We will start by looking at the basic odd degree function

. Within the applet, you may also change the degree of the function to be a different odd degree polynomial to investigate if your conjectures about the leading coefficient are true for other odd functions.

Testing Lead Coefficient (Even Degree)

Exploration #2: Question

After exploring the applet, what conjectures can your make about the value of a leading coefficient for an even degree polynomial function? Specifically, explain the difference in what happens graphically when the leading coefficient is positive vs negative.

Exploration #3: Odd Degree Polynomials

Use the applet below to observe what happens to the graph of an ODD degree polynomial as we change the value of the Leading Coefficient. We will start by looking at the basic odd degree function

Testing Lead Coefficient (ODD Degree)

Exploration #3: Question

After exploring the applet, what conjectures can your make about the value of a leading coefficient for an odd degree polynomial function? Specifically, explain the difference in what happens graphically when the leading coefficient is positive vs negative.

Summary:

In this area, please give a brief summary of what you learned from the 3 Explorations above, and specifically describe how the degree and leading coefficient effect the graph / end behavior of a polynomial. Feel free to go back to the applets and re-investigate as needed.