1) A rational function is a quotient of polynomial functions:
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1 Math Sections 4.4 and 4.5 Rational Functions 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 of x close to the number we skip on the domain and explain what you notice Use the Y1 feature of the calculator in VARS, YVARS, FUNCTION, Y1 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 Use the Y1 feature of the calculator in VARS, YVARS, FUNCTION, Y1 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. 1
2 Math Sections 4.4 and 4.5 Analyzing Rational Functions 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 Use the Y1 feature of the calculator in VARS, YVARS, FUNCTION, Y1 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 Use the Y1 feature of the calculator in VARS, YVARS, FUNCTION, Y1 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 Math Sections 4.4 and 4.5 Analyzing Rational Functions 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 Use the Y1 feature of the calculator in VARS, YVARS, FUNCTION, Y1 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 Use the Y1 feature of the calculator in VARS, YVARS, FUNCTION, Y1 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 Math Sections 4.4 and 4.5 Analyzing Rational Functions 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 Use the Y1 feature of the calculator in VARS, YVARS, FUNCTION, Y1 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 Use the Y1 feature of the calculator in VARS, YVARS, FUNCTION, Y1 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 Math Sections 4.4 and 4.5 Analyzing Rational Functions 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 Math Sections 4.4 and 4.5 Rational Functions - Summary 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). Give examples a. Domain - How do you find it? Do you consider the original or the simplified version of the function? b. Vertical Asymptotes - How do you find them? Do you consider the original or the simplified version of the function? c. Hole - In which case there is a hole? How do you find both coordinates of a hole? d. Y-Intercepts - How do you find the Y-intercepts? 6
7 Math Sections 4.4 and 4.5 Rational Functions - Summary e. X-Intercepts - How do you find the X-intercepts? f. Horizontal Asymptotes - How do you find them? Discuss the three possible cases g. Oblique Asymptote - In which case there is an oblique asymptote and how do you find it? 7
8 Math Sections 4.4 and 4.5 Analyzing Rational Functions 9) Analyze the graph of the following rational functions a) RR(xx) = xx+1 xx(xx+4) (bb) RR(xx) = xx2 +xx 12 xx 2 4 8
9 Math Sections 4.4 and 4.5 Analyzing Rational Functions 10) Analyze the graph of the following rational functions a. RR(xx) = xx2 +3xx+2 xx+1 (bb)rr(xx) = xx3 +1 xx 2 +2xx 9
10 Math Sections 4.4 and 4.5 Analyzing Rational Functions 11) Analyze the graph of the following rational functions a. RR(xx) = xx2 3xx 4 xx+2 (bb)rr(xx) = xx2 +xx 12 xx
11 Math Sections 4.4 and 4.5 Rational Functions Need More Practice? 12) Do you need more practice? The students solution manual for odd numbered problems is in my website. 11
12 Math Sections 4.4 and 4.5 Rational Functions Analyzing Graphs 13) 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? 12
13 Math Sections 4.4 and 4.5 Rational Functions Analyzing Tables 14) 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. 15) 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 13
14 Math Sections 4.4 and 4.5 Rational Functions End and Local Behaviors 16) 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) 0-17) 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 - 18) 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 - 14
15 Math Sections 4.4 and 4.5 Constructing Rational Functions 19) Write the equation of a rational function that satisfies the following conditions: x = 5 is the vertical asymptote y = 0 is the horizontal asymptote 20) 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 21) 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 22) 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 23) Write the equation of a rational function that satisfies the following conditions: there is no vertical asymptote the horizontal asymptote is y = -1 24) Write the equation of a rational function that satisfies the following conditions: there is no vertical asymptote there is no horizontal asymptote 15
16 Math Sections 4.4 and 4.5 Applications of Rational Functions 25) Solve the following word problems 16
17 Math Sections 4.4 and 4.5 Rational Functions 26) Solve the following word problems 17
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