Math Sections 4.4 and 4.5 Rational Functions. 1) A rational function is a quotient of polynomial functions:

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1 1) A rational function is a quotient of polynomial functions: 2) Explain how you find the domain of a rational function: a) Write a rational function with domain x 3 b) Write a rational function with domain x 5, 5 3) Analyze the graph of ff(xx) = 1 xx a) You are familiar with the graph of this function; sketch it b) What is the domain? c) Evaluate the function for values close to and explain what you notice d) The equation of the vertical asymptote is e) Evaluate the function for x-values that are very large (in absolute value). Explain what you notice f) The equation of the horizontal asymptote is g) Find the x- and y-intercepts h) Describe this local behavior in symbols. i) Describe this end behavior in symbols. j) Graph the function 1

2 4) Analyze the graph of ff(xx) = 2xx+6 xx+1 a) What is the domain? b) Evaluate the function for values close to and explain what you notice c) The equation of the vertical asymptote is d) Evaluate the function for x-values that are very large (in absolute value). Explain what you notice e) The equation of the horizontal asymptote is f) Find the x- and y-intercepts g) Describe this local behavior in symbols. h) Describe this end behavior in symbols. i) Graph the function 2

3 5) Analyze the graph of ff(xx) = xx2 +xx 12 xx+1 a) What is the domain? b) Evaluate the function for values close to and explain what you notice c) The equation of the vertical asymptote is d) Evaluate the function for x-values that are very large (in absolute value). Explain what you notice e) The equation of the horizontal asymptote is f) If there is no horizontal asymptote, find the oblique asymptote. g) Find the x- and y-intercepts h) Describe this local behavior in symbols. i) Describe this end behavior in symbols. j) Graph the function 3

4 6) Analyze the graph of ff(xx) = xx2 1 xx+1 a) What is the domain? b) Evaluate the function for values close to and explain what you notice c) The equation of the vertical asymptote is d) If there is no vertical asymptote, what is going on in this function? e) Evaluate the function for x-values that are very large (in absolute value). Explain what you notice f) The equation of the horizontal asymptote is g) Find the x- and y-intercepts h) Describe this local behavior in symbols. i) Describe this end behavior in symbols. j) Graph the function 4

5 7) Analyze the graph of ff(xx) = (2xx+1)(xx+3) xx(xx+3) a) What is the domain? b) Is there a hole in this graph? If so, what are the coordinates? c) The equation of the vertical asymptote is d) The equation of the horizontal asymptote is e) Find the x- and y-intercepts f) Describe this local behavior in symbols. g) Describe this end behavior in symbols. h) Graph the function 5

6 8) Summary - Reflect on what we have done and summarize procedures for finding each of the following: (read the notes and/or book if necessary) a. Vertical asymptotes b. Horizontal asymptotes. c. Oblique asymptotes d. X-intercepts e. Y-intercepts f. Coordinates of holes 6

7 9) Graph some of the following functions 7

8 10) Graph some of the following functions 8

9 11) The graphs of rational functions are shown below. Analyze the end and local behavior for each one. Write in symbolic form. Write the equations of the asymptotes. What can you say about the degrees of numerator and denominator? Are they equal? If not, which one has larger degree? 9

10 12) Tables of a rational function are shown below. Is the table describing a local or an end behavior? Write in symbolic form and write the equations of the vertical and horizontal asymptotes, if any. Sketch a possible graph. 13) Tables of a rational function are shown below. Is the table describing a local or an end behavior? Write in symbolic form and write the equations of the vertical and horizontal asymptotes, if any. Sketch a possible graph. Table 1 Table 2 Table 3 10

11 14) Sketch a function with the following local and end behavior. Write the equations of the vertical and horizontal asymptotes, if any. as x 2+ (from the right), f(x) as x 2- (from the left), f(x) as x, f(x) 0+ as x -, f(x) 0-15) Sketch at least two graphs of a rational function satisfying the following conditions The vertical asymptote is x = 2 The horizontal asymptote is y = 1 Graph 1 Graph 2 Now for each of the graphs complete the following: For Graph 1 For Graph 1 As x 2+ As x 2- As x As x - As x 2+ As x 2- As x As x - 16) Sketch the graph of a rational function satisfying the following conditions The x-intercepts are 2 and 2 The vertical asymptote is x = 0 The horizontal asymptote is y = -5 Now complete the following: As x 0+ As x 0- As x As x - 11

12 17) Write the equation of a rational function that satisfies the following conditions: x = 5 is the vertical asymptote y = 0 is the horizontal asymptote 18) Write the equation of a rational function that satisfies the following conditions: x = 1 and = -2 are the vertical asymptotes y = 2 is the horizontal asymptote 19) Write the equation of a rational function that satisfies the following conditions: the only x-intercept is x = 1 x = 2 is the vertical asymptote y = 3 is the horizontal asymptote 20) Write the equation of a rational function that satisfies the following conditions: x = -7 is the vertical asymptote the graph has a hole at x = 2 there is no horizontal asymptote 21) Write the equation of a rational function that satisfies the following conditions: there is no vertical asymptote the horizontal asymptote is y = -1 22) Write the equation of a rational function that satisfies the following conditions: there is no vertical asymptote there is no horizontal asymptote 12

13 23) Solve the following word problems 13

14 24) Solve the following word problems 14