Graphing Rational Functions

10 Key excerpts on "Graphing Rational Functions"

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  Precalculus

  • Cynthia Y. Young(Author)
  • 2018(Publication Date)
  • Wiley

    (Publisher)

  2.6.3 Graphing Rational Functions We can now graph rational functions using asymptotes as graphing aids. The following box summarizes the six-step procedure for Graphing Rational Functions. Graphing Rational Functions Let ƒ be a rational function given by ƒ 1 x 2 5 n 1 x 2 d 1 x 2 . Step 1: Find the domain of the rational function ƒ. Step 2: Find the intercept(s). ■■ ■■ y-intercept: evaluate ƒ 1 0 2 . ■■ ■■ x-intercept: solve the equation n 1 x 2 5 0 for x in the domain of ƒ. Step 3: Find any holes. ■■ ■■ Factor the numerator and denominator. ■■ ■■ Divide out common factors. ■■ ■■ A common factor x 2 a corresponds to a hole on the graph of ƒ at x 5 a if the multiplicity of a in the numerator is greater than or equal to the multiplicity of a in the denominator. ■■ ■■ The result is an equivalent rational function R1 x 2 5 p 1 x 2 q 1 x 2 in lowest terms. Step 4: Find any asymptotes. ■■ ■■ Vertical asymptotes: solve q 1 x 2 5 0. ■■ ■■ Compare the degree of the numerator and the degree of the denominator to determine whether either a horizontal or a slant asymptote exists. If one exists, find it. Step 5: Find additional points on the graph of ƒ—particularly near asymptotes. Step 6: Sketch the graph; draw the asymptotes, label the intercept(s) and additional points, and complete the graph with a smooth curve between and beyond the vertical asymptotes. 2.6.3 SKILL Graph rational functions. 2.6.3 CONCEPTUAL Any real number excluded from the domain of a rational function corresponds either to a vertical asymptote or to a hole on the graph. 2.6 Rational Functions 255 STUDY TIP Any real number excluded from the domain of a rational function corresponds to either a vertical asymptote or a hole on its graph. 256 CHAPTER 2 Polynomial and Rational Functions ▼ A N S W E R x y –5 5 5 –5 EXAMPLE 7 Graphing a Rational Function Graph the rational function ƒ 1 x 2 5 x x 2 2 4 . Solution: STEP 1 Find the domain. Set the denominator equal to zero.

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  Precalculus: Mathematics for Calculus, International Metric Edition

  • James Stewart, Lothar Redlin, Saleem Watson(Authors)
  • 2016(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. SECTION 3.6 ■ Rational Functions 301 ■ Graphing Rational Functions We have seen that asymptotes are important when Graphing Rational Functions. In gen- eral, we use the following guidelines to graph rational functions. SKETCHING GRAPHS OF RATIONAL FUNCTIONS 1. Factor. Factor the numerator and denominator. 2. Intercepts. Find the x-intercepts by determining the zeros of the numerator and the y-intercept from the value of the function at x  0. 3. Vertical Asymptotes. Find the vertical asymptotes by determining the zeros of the denominator, and then see whether y S  or y S  on each side of each vertical asymptote by using test values. 4. Horizontal Asymptote. Find the horizontal asymptote (if any), using the procedure described in the box on page 300. 5. Sketch the Graph. Graph the information provided by the first four steps. Then plot as many additional points as needed to fill in the rest of the graph of the function. A fraction is 0 only if its numerator is 0. EXAMPLE 5 ■ Graphing a Rational Function Graph r 1 x 2  2x 2  7x  4 x 2  x  2 , and state the domain and range. SOLUTION We factor the numerator and denominator, find the intercepts and asymp- totes, and sketch the graph. Factor. y  1 2x  1 21 x  4 2 1 x  1 21 x  2 2 x-Intercepts. The x-intercepts are the zeros of the numerator, x  1 2 and x  4. y-Intercept. To find the y-intercept, we substitute x  0 into the original form of the function. r 1 0 2  21 0 2 2  71 0 2  4 1 0 2 2  1 0 2  2  4 2  2 The y-intercept is 2. Vertical asymptotes. The vertical asymptotes occur where the denominator is 0, that is, where the function is undefined. From the factored form we see that the verti- cal asymptotes are the lines x  1 and x  2.

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  College Algebra

  • R. Gustafson, Jeff Hughes(Authors)
  • 2016(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Section 4.6 Rational Functions 469 Accent on Technology Graphing a Rational Function A graphing calculator can be very useful in helping us draw the graph of a rational function. However, certain precautions must be used or we will get a graph that is not going to give us a good idea of how the function looks. For example, in Figure 4-48(a) the graph of the function from Example 5 is shown. Using the ZOOM Standard window, a good representation of the graph is shown. In Figure 4-48(b) the ZOOM Decimal window is used, and part of the graph is not shown. (a) (b) FIGURE 4-48 Graphing a Rational Function Graph: f sxd 5 3x x 2 2 . We will use the steps outlined earlier to graph the rational function. Step 1: Symmetry We find f s2xd. f s2xd 5 3s2xd 2x 2 2 5 23x 2x 2 2 5 3x x 1 2 . Because f s2xd Þ f sxd and f s2xd Þ 2f sxd, there is no symmetry about the y-axis or the origin. Step 2: Vertical asymptotes We first note that f sxd is in simplest form. We then set the denominator equal to 0 and solve for x. Since the solution is 2, there will be a vertical asymptote at x 5 2. Step 3: y- and x-intercepts We can find the y-intercept by finding f s0d. f s0d 5 3s0d 0 2 2 5 0 22 5 0 The y-intercept is s0, 0d. We can find the x-intercepts by setting the numerator equal to 0 and solving for x: 3x 5 0 x 5 0 The x-intercept is (0, 0). Step 4: Horizontal asymptotes Since the degrees of the numerator and denominator of the polynomials are the same, the line y 5 3 1 5 3 The leading coefficient of the numerator is 3. The leading coefficient of the denominator is 1. is a horizontal asymptote. SOLUTION EXAMPLE 6 Copyright 2017 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).

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  Precalculus

  • Cynthia Y. Young(Author)
  • 2023(Publication Date)
  • Wiley

    (Publisher)

  • Understand that the graph of a rational function can have either a horizontal asymptote or a slant asymptote but not both. • Any real number excluded from the domain of a rational function corresponds to either a vertical asymptote or a hole in the graph. 2.6.1 Domain of Rational Functions 2.6.1 Skill Find the domain of a rational function. 2.6.1 Conceptual Understand that the domain of a rational function is the set of all real numbers except those that correspond to the denominator being equal to zero. So far in this chapter we have discussed polynomial functions. We now turn our attention to rational functions, which are ratios of polynomial functions. Ratios of integers are called rational numbers. Similarly, ratios of polynomial functions are called rational functions. Rational Function A function f (x) is a rational function if f (x) = n(x) ____ d(x) d(x) ≠ 0 where the numerator, n(x), and the denominator, d(x), are polynomial functions. The domain of f (x) is the set of all real numbers x such that d(x) ≠ 0. Note: If d(x) is a constant, then f (x) is a polynomial function. The domain of any polynomial function is the set of all real numbers. When we divide two polynomial functions, the result is a rational function, and we must exclude any values of x that make the denominator equal to zero. Preview to Calculus In Exercises 87–92, refer to the following: In calculus we study the integration of rational functions by partial fractions. a. Factor each polynomial into linear factors. Use complex numbers when necessary. b. Factor each polynomial using only real numbers. 87. f (x) = x 3 + x 2 + x + 1 88. f (x) = x 3 − 6x 2 + 21x − 26 89. f (x) = x 4 + 5x 2 + 4 90. f (x) = x 4 − 2x 2 − 7x 2 + 18x − 18 91. f (x) = x 5 + 10x 4 + 50x 3 + 250x 2 + 625x 92. f (x) = x 5 − 7x 3 + 4x 2 − 20x

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  Algebra & Trig

  • Ron Larson(Author)
  • 2021(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 318 Chapter 4 Rational Functions and Conics GO DIGITAL 4.2 Graphs of Rational Functions Sketch graphs of rational functions. Sketch graphs of rational functions that have slant asymptotes. Use graphs of rational functions to model and solve real-life problems. Sketching the Graph of a Rational Function When graphing simple rational functions, testing for symmetry may be useful. Recall from Section 2.4 that the graph of f (x) = 1x is symmetric with respect to the origin. Figure 4.8 shows that the graph of g(x) = 1x 2 is symmetric with respect to the y-axis. Guidelines for Graphing Rational Functions Let f (x) = N(x)D(x), where N(x) and D(x) are polynomials and D(x) is not the zero polynomial. 1. Simplify f, if possible. List any restrictions on the domain of f that are not implied by the simplified function. 2. Find and plot the y-intercept (if any) by evaluating f (0). 3. Find the zeros of the numerator (if any). Then plot the corresponding x-intercepts. 4. Find the zeros of the denominator (if any). Then sketch the corresponding vertical asymptotes and plot the corresponding holes. 5. Find and sketch the horizontal asymptote (if any) by using the rule for finding the horizontal asymptote of a rational function on page 312. 6. Plot at least one point between and one point beyond each x-intercept and vertical asymptote. 7. Use smooth curves to complete the graph between and beyond the vertical asymptotes. TECHNOLOGY Some graphing utilities have difficulty Graphing Rational Functions with vertical asymptotes. In connected mode, the utility may connect parts of the graph that are not supposed to be connected. For example, the graph in Figure 4.9(a) should consist of two unconnected portions—one to the left of x = 2 and the other to the right of x = 2. Using the graphing utility’s dot mode eliminates this problem.

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  College Algebra

  Building Skills and Modeling Situations

  • Charles P. McKeague, Katherine Yoshiwara, Denny Burzynski(Authors)
  • 2013(Publication Date)
  • XYZ Textbooks

    (Publisher)

  For example: The domain for f ( x) = x − 4 _____ x − 2 , is {x| x ≠ 2} A rational function is any function that can be written in the form f (x) = P(x) ____ Q(x) where P(x) and Q(x) are polynomials and Q(x) ≠ 0. Rational Function DEFINITION EXAMPLE 1 Graphing Rational Functions VIDEO EXAMPLES SECTION 5.5 382 Chapter 5 Polynomials and Rational Functions The domain for g( x) = x 2 + 5 _____ x + 1 , is {x| x ≠ −1} The domain for h( x) = x _____ x 2 − 9 , is {x| x ≠ −3, x ≠ 3} Notice that, for these functions, f (2), g(−1), h(−3), and h (3) are all undefined, and that is why the domains are written as shown. Graph the equation y = x 2 − 9 _____ x − 3 . How is this graph differ- ent from the graph of y = x + 3? SOLUTION We know from the discussion on factoring and reducing to lowest terms that y = x 2 − 9 _____ x − 3 = (x + 3)(x − 3) ___________ x − 3 = x + 3 This relationship is true for all x except x = 3 because the rational expressions with x − 3 in the denominator are undefined when x is 3. However, for all other values of x, the expressions x 2 − 9 _____ x − 3 and x + 3 are equal. Therefore, the graphs of y = x 2 − 9 _____ x − 3 and y = x + 3 will be the same except when x is 3. In the first equation, there is no value of y to correspond to x = 3. In the second equation, y = x + 3 so y is 6 when x is 3. Now you can see the difference in the graphs of the two equations. To show that there is no y value for x = 3 in the graph on the left in Figure 1, we draw an open circle at that point on the line. We say that there is a hole at that point. EXAMPLE 2 FIGURE 1 –10 –8 –6 –2 2 4 6 8 10 –10 –8 –6 –4 –2 4 6 8 10 x y y H11005 x 2 H11002 9 x H11002 3 y H11005 x H11001 3 y H11005 x 2 H11002 9 x H11002 3 –10 –8 –6 –2 2 4 6 8 10 –10 –8 –6 –4 –2 4 6 8 10 x y y H11005 x H11001 3 5.5 Graphing Rational Functions 383 Notice that the two graphs shown in Figure 1 are both graphs of functions.

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  Precalculus

  Building Concepts and Connections 2E

  • Revathi Narasimhan(Author)
  • 2016(Publication Date)
  • XYZ Textbooks

    (Publisher)

  Using Technology Graphing calculators can be helpful for Graphing Rational Functions. Figure 3 shows the graph of f ( x ) = 1 ____ x − 1 in a standard window. Recent calculator models will skip over the asymptote, and will not connect the two parts of the graph. Keystroke Appendix: Sections 7 and 8 Figure 3 y Graph of f ( x ) = Vertical asymptote : x = 1 x x – 1 1 Figure 2 Note A case in which p ( x ) and q ( x ) have common factors is given in Example 7. Example 2 3.7 Rational Functions 285 b. The numerator and denominator of f ( x ) = x + 2 _____ x 2 − 1 have no common factors, so we can set the denominator equal to zero and find the vertical asymptote(s). In this case, the denominator can be factored, so we will apply the Zero Product Rule: x 2 − 1 = 0 ⇒ ( x + 1)( x − 1) = 0 ⇒ x = 1 , − 1 This function has two vertical asymptotes: the line x = 1 and the line x = − 1. c. Once again, the numerator and denominator have no common factors. The vertical asymptote is the line x = − 1 _ 2 , because x = − 1 _ 2 is the solution of the equation 2 x + 1 = 0. Check It Out 2 Find all vertical asymptotes of f ( x ) = 3 x _____ x 2 − 9 . End Behavior of Rational Functions and Horizontal Asymptotes Just as we did with polynomial functions, we can also examine the end behavior of rational functions. We will use this information later to help sketch complete graphs of rational functions. We can examine what happens to the values of a rational function r ( x ) as | x | gets large. This is the same as determining the end behavior of the rational function. For example, as x → ∞ , f ( x ) = 1 ____ x − 1 → 0 , because the denominator becomes large in magnitude, but the numerator stays constant at 1. Similarly, as x → −∞ , f ( x ) = 1 ____ x − 1 → 0. These are instances of the LARGE-small principle. When such a behavior occurs, we say that y = 0 is a horizontal asymptote of the function f ( x ) = 1 ____ x − 1 .

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  College Algebra

  • James Stewart, Lothar Redlin, Saleem Watson, , James Stewart, Lothar Redlin, Saleem Watson(Authors)
  • 2015(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  11. (a) What is a rational function? (b) What does it mean to say that x  a is a vertical asymp-tote of y  f 1 x 2 ? (c) What does it mean to say that y  b is a horizontal asymptote of y  f 1 x 2 ? 12. (a) How do you find vertical asymptotes of rational functions? (b) Let s be the rational function s 1 x 2  a n x n  a n  1 x n  1  . . .  a 1 x  a 0 b m x m  b m  1 x m  1  . . .  b 1 x  b 0 How do you find the horizontal asymptote of s ? (c) Find the vertical and horizontal asymptotes of f 1 x 2  5 x 2  3 x 2  4 13. (a) Under what circumstances does a rational function have a slant asymptote? (b) How do you determine the end behavior of a rational function? 14. (a) Explain how to solve a polynomial inequality. (b) What are the cut points of a rational function? Explain how to solve a rational inequality. (c) Solve the inequality x 2  9  8 x . ■ CONCEPT CHECK ANSWERS TO THE CONCEPT CHECK CAN BE FOUND AT THE BACK OF THE BOOK. Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 356 CHAPTER 3 ■ Polynomial and Rational Functions 1–4 ■ Graphs of Quadratic Functions A quadratic function is given. (a) Express the function in standard form. (b) Graph the function. 1. f 1 x 2  x 2  6 x  2 2. f 1 x 2  2 x 2  8 x  4 3. f 1 x 2  1  10 x  x 2 4. g 1 x 2   2 x 2  12 x 5–6 ■ Maximum and Minimum Values Find the maximum or minimum value of the quadratic function. 5. f 1 x 2   x 2  3 x  1 6. f 1 x 2  3 x 2  18 x  5 7. Height of a Stone A stone is thrown upward from the top of a building.

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  College Algebra

  Building Concepts and Connections 2E

  • Revathi Narasimhan(Author)
  • 2019(Publication Date)
  • XYZ Textbooks

    (Publisher)

  298 Chapter 3 Polynomial and Rational Functions In Exercises 27–46, sketch the graph of the rational function. Indicate any vertical and horizontal asymptote ( s ) , and all intercepts. 27. f ( x ) = 1 ______ x − 2 28. f ( x ) = 1 ______ x + 3 29. f ( x ) = − 12 ______ x + 6 30. f ( x ) = − 10 ______ x + 2 31. f ( x ) = 8 ______ 4 − x 32. f ( x ) = 12 ______ 3 − x 33. f ( x ) = 3 ________ ( x + 1) 2 34. h ( x ) = − 9 ________ ( x − 3) 2 35. g ( x ) = 3 − x _____ x + 4 36. g ( x ) = x + 5 ______ x − 2 37. h ( x ) = − 2 x _____________ ( x − 1)( x + 4) 38. f ( x ) = x _____________ ( x − 3)( x − 1) 39. f ( x ) = 3 x 2 __________ x 2 − x − 2 40. f ( x ) = − 4 x 2 __________ x 2 − x − 6 41. f ( x ) = x − 1 ____________ 2 x 2 − 5 x − 3 42. f ( x ) = x − 2 ___________ 2 x 2 + x − 3 43. f ( x ) = x 2 + x − 6 __________ x 2 − 1 44. f ( x ) = x 2 + 3 x + 2 ___________ x 2 − 9 45. h ( x ) = 1 _______ x 2 + 1 46. h ( x ) = 2 _______ x 2 + 4 In Exercises 47–58, sketch a graph of the rational function. Indicate all intercepts and asymptotes. 47. g ( x ) = x 2 ______ x + 4 48. g ( x ) = x 2 ______ x − 2 49. h ( x ) = − x 2 ______ x − 3 50. h ( x ) = − x 2 ______ x + 1 51. h ( x ) = 4 − x 2 _______ x 52. h ( x ) = x 2 − 9 _______ x 53. h ( x ) = x 2 + x + 1 __________ x − 1 54. h ( x ) = x 2 + 2 x + 1 ___________ x + 3 55. h ( x ) = 3 x 2 + 5 x − 2 ____________ x + 1 56. h ( x ) = 2 x 2 + 11 x + 5 ____________ x − 3 57. h ( x ) = x 3 + 1 ________ x 2 + 3 x 58. h ( x ) = x 3 − 1 ________ x 2 − 2 x In Exercises 59–64, sketch a graph of the rational function involving common factors and find all asymptotes and intercepts. Indicate them on the graph. 59. f ( x ) = 3 x + 9 _______ x 2 − 9 60. f ( x ) = 2 x − 4 _______ x 2 − 4 61. f ( x ) = x 2 + x − 2 ___________ x 2 + 2 x − 3 62. f ( x ) = 2 x 2 − 5 x + 2 ____________ x 2 − 5 x + 6 63. f ( x ) = x 2 + 3 x − 10 ____________ x − 2 64. f ( x ) = x 2 + 2 x + 1 ___________ x + 1 Applications 65.

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  eBook - PDF

  College Algebra

  • Cynthia Y. Young(Author)
  • 2021(Publication Date)
  • Wiley

    (Publisher)

  f (x) = (x − 1)(5x − 2) ___________ (2x + 1)(3x − 2) 21. f (x) = 6 x 5 − 4 x 2 + 5 ____________ 6 x 2 + 5x − 4 22. f (x) = 6 x 2 + 3x + 1 ___________ 3 x 2 − 5x − 2 23. f (x) = 1 _ 3 x 2 + 1 _ 3 x − 1 _ 4 ___________ x 2 + 1 _ 9 24. f (x) = 1 __ 10 ( x 2 − 2x + 3 __ 10 ) ______________ 2x − 1 25. f (x) = (0.2x − 3.1)(1.2x + 4.5) ____________________ 0.7(x − 0.5)(0.2x + 0.3) 26. f (x) = 0.8 x 4 − 1 ________ x 2 − 0.25 Section 4.6 Exercises Section 4.6 Summary In this section, rational functions were discussed. f (x) = n(x) ____ d(x) • Domain: All real numbers except the x-values that make the denominator equal to zero, d(x) = 0. • Vertical Asymptotes: Vertical lines, x = a, where d(a) = a, after all common factors have been divided out. Vertical asymptotes steer the graph are never touched. • Horizontal Asymptotes: Horizontal lines, y = b, that steer the graph as x → ± ∞. 1. If degree of the numerator < degree of the denominator, then y = 0 is a horizontal asymptote. 2. If degree of the numerator = degree of the denominator, then y = c is a horizontal asymptote where c is the ratio of the leading coefficients of the numerator and denominator, respectively. 3. If degree of the numerator > degree of the denominator, then there is no horizontal asymptote. • Slant Asymptotes: Slant lines, y = mx + b, that steer the graph as x → ± ∞. 1. If degree of the numerator − degree of the denominator = 1, then there is a slant asymptote. 2. Divide the numerator by the denominator. The quotient corresponds to the equation of the line (slant asymptote). Procedure for Graphing Rational Functions 1. Find the domain of the function. 2. Find the intercept(s). • y-intercept • x-intercepts (if any) 3. Find any holes. • If x − a is a common factor of the numerator and denominator, then x = a corresponds to a hole in the graph of the rational function if the multiplicity of a in the numerator is greater than or equal to the multiplicity of a in the denominator.

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