2.6: Rational Functions and Their Graphs

Transcription

1 2.6: Rational Functions and Their Graphs Rational Functions are quotients of polynomial functions. The of a rational expression is all real numbers except those that cause the to equal. Example 1 (like HW #1-8) Find the domain of ( ) You-Try #1 (like HW #1-8) Find the domain of ( ) Arrow Notation o As approaches from the right, ( ) approaches infinity. o As approaches from the left, ( ) approaches negative infinity. o As approaches infinity, ( ) approaches zero. 2.6: Rational Functions and Their Graphs Page 1 of 9

2 Vertical Asymptotes An is a line that the graph of ( ). A graph can touch or cross a asymptote. The line is a if ( ) increases or decreases without bound as approaches. o o If is a zero of but not a zero of, then is a vertical asymptote. Example 2 (like HW #21-28) Find the vertical asymptotes, if any, of ( ) You-Try #2 (like HW #21-28) Find the vertical asymptotes, if any, of ( ) Holes A is a point that is part of the of a function, but does cause an. If is a zero of and a zero of, then there is a hole at. Holes generally are distinguishable on a graphing calculator graph. 2.6: Rational Functions and Their Graphs Page 2 of 9

3 Example of a Hole Horizontal Asymptotes The line is a if ( ) approaches as increases or decreases without bound. o OR o Identifying Horizontal Asymptotes Only the degree term of the top and bottom matter. Let equal the degree of, the numerator. Let equal the degree of, the denominator. If, then the -axis ( ) is a horizontal asymptote. If, then the line is the horizontal asymptote. If, then ( ) does have a horizontal asymptote. Example 3 (like HW #29-33) Find the horizontal asymptote, if any, of each function ( ) ( ) ( ) 2.6: Rational Functions and Their Graphs Page 3 of 9

4 You-Try #3 (like HW #29-33) Find the horizontal asymptote, if any, of each function ( ) ( ) ( ) Graphing Rational Functions 1. Identify any asymptotes (numbers that are zeros of but not zeros of ). Draw a dashed line for each. 2. Identify any (x-values are numbers that are of both and. 3. Identify any asymptotes by examining the terms. Draw a dashed line if one exists. 4. Use the feature on your graphing calculator to get other to graph. Plot these on your graph. 5. Draw a curve through the points, but not touching the. If there was a identified in step 2, put an at that -value. 6. Check your graph with a graphing calculator. Remember that it does properly display asymptotes and holes. 2.6: Rational Functions and Their Graphs Page 4 of 9

5 Example 4 (like HW #37-58) Graph ( ) You-Try #4 (like HW #37-58) Graph ( ) 2.6: Rational Functions and Their Graphs Page 5 of 9

6 You-Try #5 (like HW #37-58) Graph ( ) You-Try #6 (like HW #37-58) Graph ( ) 2.6: Rational Functions and Their Graphs Page 6 of 9

7 Slant Asymptotes A is a line of the form that the graph of a function approaches as The graph of ( ) has a slant asymptote if the of the numerator is exactly than the of the denominator. Find the equation of the slant asymptote by division (synthetic or long), and the remainder. Example 7 (like HW #59-66) Find the slant asymptote and graph ( ) 2.6: Rational Functions and Their Graphs Page 7 of 9

8 You-Try #7 (like HW #59-66) Find the slant asymptote and graph ( ) Applications of Rational Functions The average cost of producing an item ( ) Chemical concentrations over time ( ) Used in numerous science and engineering fields to approximate or model complex situations. 2.6: Rational Functions and Their Graphs Page 8 of 9

9 Example 8 (page 332 #70) The rational function ( ) describes the cost, ( ), in millions of dollars, to inoculate of the population against a particular strain of the flu. a) Find and interpret ( ), ( ), ( ), ( ), and ( ). b) What is the equation of the vertical asymptote? What does this mean in terms of the variables of the function? c) Graph the function. 2.6: Rational Functions and Their Graphs Page 9 of 9