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). (b). .
2. Sketch the graph of .
3. Sketch the graph of .

Key Concepts

• For a polynomial function , the end behavior is determined by the leading term . If , approaches or at each end.
• For a rational function , the end behavior is determined by the relationship between the degree of and the degree of • degree of  the degree of :  then the line is a horizontal asymptote for • degree of  the degree of :  then the line is a horizontal asymptote, where and are the leading coefficients of and , respectively
• degree of  the degree of :  then approaches or at each end.
• When sketching a graph, you need all the information from Lesson 17

#### Videos and Practice Problems of Selected Topics

• Limits at infinity and asymptotes
1. Infinite limits and asymptotes (4:13) Using Desmos to analyze graphs of functions and analyze asymptotes using limits.
2. Limits at infinity of rational functions (4:06) For , find and .
• Sketching the graph of a polynomial function
1. Curve sketching with calculus: polynomial (20:30) Sketch the graph of using derivatives.
2. Analyzing a function with its derivative (9:41) Sketch the graph of using derivatives.
• Sketching the graph of a rational function
1.  Curve sketching with calculus: rational function

Graph .

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