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Worksheet: End Behavior of Polynomial and Rational Functions Graphs

Q1:

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 𝑦 = βˆ’ ( π‘₯ βˆ’ 1 ) ( π‘₯ + 1 ) ( π‘₯ + 2 ) , which has degree three.

Use this example to determine if David’s statement is correct.

  • AIt is incorrect.
  • BIt 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?

  • AYes
  • BNo

Q2:

Consider the graph of the function 𝑦 = 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?

  • AThe value of 𝑦 approaches negative infinity as π‘₯ increases.
  • BThe value of 𝑦 approaches infinity as π‘₯ increases.
  • CThe value of 𝑦 approaches zero as the value of π‘₯ increases.

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

  • AThe value of 𝑦 approaches zero.
  • BThe value of 𝑦 approaches βˆ’ ∞ .
  • CThe value of 𝑦 approaches ∞ .

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

  • AThe value of 𝑦 approaches negative infinity when π‘₯ gets closer to zero from the negative direction and approaches positive infinity when π‘₯ gets closer to zero from the positive direction.
  • BThe value of 𝑦 approaches negative infinity when π‘₯ gets closer to zero from the negative direction or from the positive direction.
  • CThe value of 𝑦 approaches positive infinity when π‘₯ gets closer to zero from the negative direction or from the positive direction.
  • DThe value of 𝑦 approaches positive infinity when π‘₯ gets closer to zero from the negative direction and approaches negative infinity when π‘₯ gets closer to zero from the positive direction.

Q3:

Consider a function 𝑓 ( π‘₯ ) = π‘Ž π‘₯ + 𝑏 π‘₯  , where π‘Ž , 𝑏 and 𝑛 are integers larger than one. Which of the following statements is true?

  • A The ends of the graph will both tend in different directions.
  • B If 𝑛 is odd, the ends of the graph both tend in the positive direction.
  • C The end behavior cannot be determined without more information.
  • DIf 𝑛 is even, the ends of the graph both tend in the positive direction.