David wants to investigate the end behavior of various polynomials. He decides to plot polynomials with increasing degrees (increasing powers of the leading coefficient) and look at their end behavior. He plots the following graphs.
David notices that there is a similarity in the end behavior of polynomials with an even degree and that of polynomials with an odd degree.
Initially, David concludes that all polynomials with an odd degree are strictly increasing: they enter from the bottom left and exit from the top right. He decides that all polynomials with an even degree contain exactly one turning point, enter from the top left, and exit from the top right.
His friend Jacob shows him the graph of , which has degree three.
Use this example to determine if David’s statement is correct.
Jacob concludes that polynomials with an odd degree always enter and leave from diagonally opposite quadrants and that polynomials with an even degree enter and leave from horizontally adjacent quadrants. Is Jacob’s conclusion correct?
Consider the graph of the function .
By looking at the graph and substituting a few successively larger values of into the function, what is the end behavior of the graph as increases along the positive -axis?
Similarly, what is the end behavior of the graph as decreases?
Finally, by interpreting the graph, what is happening to the function when the value of approaches zero?
Consider a function , where , and are integers larger than one. Which of the following statements is true?