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In this lesson, we will learn how to determine polynomial and rational functions at their ends or as they tend to infinities.
Karim 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.
Karim 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, Karim 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 Bassem shows him the graph of
, which has degree three.
Use this example to determine if Karim’s statement is correct.
Bassem 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 Bassem’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
Similarly, what is the end behavior of the graph as
Finally, by interpreting the graph, what is happening to the function when the value of
Consider a function
are integers larger than one.
Which of the following statements is true?
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