Learning Outcomes

  1. Understand the end behavior of polynomial functions.
  2. Find horizontal asymptotes and vertical asymptotes of rational functions. 
  3. Sketch the graph of a polynomial or a rational function. 

Textbook 

  • Chapter 4.6 ¬†Limits at Infinity and Asymptotes ¬†(this lesson is usually covered in two lectures)

Textbook Assignment

  • p. 436: ¬† ¬†271, 273, 274, 279, 281,¬†298

WeBWork Assignment

  • Limits-Infinite
  • Application-Asymptotes
  • Application-Shape of Graphs

Exit problems of the session 

  1. Find the horizontal and vertical asymptotes of the following rational functions.                        (a).   y=\dfrac{1}{x^2+x^3}  (b).    y=\dfrac{x^2-x^3}{x+2x^3}.  
  2. Sketch the graph of  f(x)=4x^3-3x^2.
  3. Sketch the graph of  f(x)=\dfrac{x}{x^2-4}.

 

Key Concepts

  • For a polynomial function p(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0, the end behavior is determined by the leading term a_nx^n. If n\neq 0, p(x) approaches \infty or -\infty at each end.¬†
  • For a rational function f(x)=\dfrac{p(x)}{q(x)}, the end behavior is determined by the relationship between the degree of p(x) and the degree of q(x):¬†
    • degree of p(x) < ¬†the degree of q(x): ¬†then the line y=0 is a horizontal asymptote for f(x)
    • degree of p(x) = ¬†the degree of q(x): ¬†then the line y=\dfrac{a_n}{b_n} is a horizontal asymptote, where a_n and b_n are the leading coefficients of p(x) and q(x), respectively
    • degree of p(x) > ¬†the degree of q(x): ¬†then f(x) approaches \infty or -\infty at each end.¬†
  • When sketching a graph, you need all the information from Lesson 17.¬†

 

Videos and Practice Problems of Selected Topics

  • Sketching the graph of a rational function
    1.  Curve sketching with calculus: rational function

              Graph y=\dfrac{x-1}{x^2}.

      1. \rhd Part I (10:00): Find the domain, intercepts, symmetry relations, asymptotes, intervals of increase/decrease, and local extrema.
      2. \rhd Part II (8:04): Find the intervals of concavity and skech the graph of the function.