Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.

Please verify your account before proceeding.

In this lesson, we will learn how to determine polynomial and rational functions at their ends or as they tend to infinities.

Q1:

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 Karimβs 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 Bassemβs conclusion correct?

Q2:

Consider the graph of the function π¦ = 1 π₯ .

By looking at the graph and substituting a few successively larger values of π₯ into the function, what is the end behavior of the graph as π₯ increases along the positive π₯ -axis?

Similarly, what is the end behavior of the graph as π₯ decreases?

Finally, by interpreting the graph, what is happening to the function when the value of π₯ approaches zero?

Q3:

Consider a function π ( π₯ ) = π π₯ + π π₯ ο , where π , π and π are integers larger than one. Which of the following statements is true?

Donβt have an account? Sign Up