[size=150]In this investigation activity, you will investigate the end behaviors of a polynomial function.[br][br]Exploration #1:[br]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.[/size]

[size=150]What pattern do you observe between the value of the degree and the shape of the graph? [br]Specifically, do you notice any differences between the graphs of [u][b]even [/b][/u][u][b]degree[/b] [/u]functions and [u][b]odd degree[/b][/u] functions?[br][br]You should note that the left and right "ends" of an [b]even degree[/b] polynomial both go in the same direction; while the left and right "ends" of an [b]odd degree[/b] polynomial go in opposite directions.[br][br]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. [br][br]Use the applet below to observe what happens to the graph of an [b]EVEN[/b] degree polynomial as we change the value of the Leading Coefficient. [br][br]We will start by looking at the basic even degree function [math]y=x^2[/math]. [br][br][i]Within the applet, you may also change the degree of the function to be a different even degree polynomial to investigate if your conjectures about the leading coefficient are true for other even functions. [br][/i][/size]

[size=150]After exploring the applet, what conjectures can your make about how the value of a leading coefficient for an [b]even degree[/b] polynomial function effects the end behaviors of the graph of the function? [br][br]You should note that when the leading coefficient is [b]positive[/b] both the left and right "ends" of the graph of the even degree polynomial go up (to positive infinity); and when the leading coefficient is [b]negative[/b] both the left and right "ends" of the graph of the even degree polynomial go down (to negative infinity).[br][br]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. [br][br]We will start by looking at the basic odd degree function [math]y=x^3[/math]. [br][i][br]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. [/i][br][/size]

[size=150]What conjectures can your make about the value of a leading coefficient for an odd degree polynomial function? [br][br]You should note that when the leading coefficient is [b]positive[/b] that the left "end" of the graph of the odd degree polynomial goes down (to negative infinity), while the right "end" of the graph of the odd degree polynomial goes up (to positive infinity.[br]And when the leading coefficient is [b]negative[/b] the left "end" of the graph of the odd degree polynomial goes up (to positive infinity), while the right "end" of the graph of the odd degree polynomial goes down (to negative infinity).[/size]