2.6 Graphs of Rational Functions

Transcription

1 8 CHAPTER Polynomial, Power, and Rational Functions.6 Graphs of Rational Functions What you ll learn about Rational Functions Transformations of the Reciprocal Function Limits and Asymptotes Analyzing Graphs of Rational Functions Exploring Relative Humidity... and why Rational functions are used in calculus and in scientific applications such as inverse proportions. Rational Functions Rational functions are ratios (or quotients) of polynomial functions. DEFINITION Rational Functions Let ƒ and g be polynomial functions with gx Z 0. Then the function given by is a rational function. rx = ƒx gx The domain of a rational function is the set of all real numbers except the zeros of its denominator. Every rational function is continuous on its domain. EXAMPLE Finding the Domain of a Rational Function Find the domain of ƒ and use limits to describe its behavior at value(s) of x not in its domain. ƒx = x - SOLUTION The domain of ƒ is all real numbers x Z. The graph in Figure.46 strongly suggests that ƒ has a vertical asymptote at x =. As x approaches from the left, the values of ƒ decrease without bound. As x approaches from the right, the values of ƒ increase without bound. Using the notation introduced in Section. on page 9 we write lim ƒx = - q x: - and lim ƒx = q. x: + The tables in Figure.47 support this visual evidence numerically. Now try Exercise. [ 4.7, 4.7] by [ 5, 5] X Y = /(X ) Y ERROR (a) X Y = /(X ) Y ERROR (b) FIGURE.46 The graph of ƒx = /x -. (Example ) FIGURE.47 Table of values for ƒx = /x - for values of x (a) to the right of, and (b) to the left of. (Example ) In Chapter we defined horizontal and vertical asymptotes of the graph of a function y = ƒx. The line y = b is a horizontal asymptote of the graph of ƒ if lim ƒx = b x: -q or lim ƒx = b. x: q

2 SECTION.6 Graphs of Rational Functions 9 The line x = a is a vertical asymptote of the graph of ƒ if lim ƒx = q or lim ƒx = q. x:a - x:a + We can see from Figure.46 that lim /x - = lim /x - = 0, so the line x: -q x: q y = 0 is a horizontal asymptote of the graph of ƒx = /(x - ). Because lim ƒx = - q and lim ƒx = q, the line x = is a vertical asymptote of x: - x: + ƒx = /x -. Transformations of the Reciprocal Function One of the simplest rational functions is the reciprocal function ƒx = x, which is one of the basic functions introduced in Chapter. It can be used to generate many other rational functions. Here is what we know about the reciprocal function. BASIC FUNCTION The Reciprocal Function [ 4.7, 4.7] by [ 3., 3.] FIGURE.48 The graph of ƒx = /x. ƒx = x Domain: - q, 0 0, q Range: - q, 0 0, q Continuity: All x Z 0 Decreasing on - q, 0 and 0, q Symmetric with respect to origin (an odd function) Unbounded No local extrema Horizontal asymptote: y = 0 Vertical asymptote: x = 0 End behavior: lim ƒx = lim ƒx = 0 x: -q x: q Comparing Graphs of Rational Functions EXPLORATION. Sketch the graph and find an equation for the function g whose graph is obtained from the reciprocal function ƒx = /x by a translation of units to the right.. Sketch the graph and find an equation for the function h whose graph is obtained from the reciprocal function ƒx = /x by a translation of 5 units to the right, followed by a reflection across the x-axis. 3. Sketch the graph and find an equation for the function k whose graph is obtained from the reciprocal function ƒx = /x by a translation of 4 units to the left, followed by a vertical stretch by a factor of 3, and finally a translation units down.

3 0 CHAPTER Polynomial, Power, and Rational Functions The graph of any nonzero rational function of the form gx = ax + b cx + d, c Z 0 can be obtained through transformations of the graph of the reciprocal function. If the degree of the numerator is greater than or equal to the degree of the denominator, we can use polynomial division to rewrite the rational function. EXAMPLE Transforming the Reciprocal Function Describe how the graph of the given function can be obtained by transforming the graph of the reciprocal function ƒx = /x. Identify the horizontal and vertical asymptotes and use limits to describe the corresponding behavior. Sketch the graph of the function. (a) gx = (b) hx = 3x - 7 x - SOLUTION (a) gx = = a b = ƒ The graph of g is the graph of the reciprocal function shifted left 3 units and then stretched vertically by a factor of. So the lines x = -3 and y = 0 are vertical and horizontal asymptotes, respectively. Using limits we have lim gx = lim gx = 0, lim gx = q, and lim gx = - x: q x: - q q. The graph is shown in Figure.49a. x: -3 + x: -3 - (b) We begin with polynomial division: y (a) 4 x 3 x - 3x - 7 3x So, hx = 3x - 7 x - = 3 - = -ƒx x - Thus the graph of h is the graph of the reciprocal function translated units to the right, followed by a reflection across the x-axis, and then translated 3 units up. (Note that the reflection must be executed before the vertical translation.) So the lines x = and y = 3 are vertical and horizontal asymptotes, respectively. Using limits we have lim hx = lim hx = 3, lim gx = - q, and lim gx = q. x: q x: -q x: + x: - The graph is shown in Figure.49b. Now try Exercise y (b) FIGURE.49 The graphs of (a) gx = / and (b) hx = 3x - 7/x -, with asymptotes shown in red. x Limits and Asymptotes In Example we found asymptotes by translating the known asymptotes of the reciprocal function. In Example 3, we use graphing and algebra to find an asymptote. EXAMPLE 3 Finding Asymptotes Find the horizontal and vertical asymptotes of ƒx = x + /x +. Use limits to describe the corresponding behavior of ƒ. SOLUTION Solve Graphically The graph of ƒ in Figure.50 suggests that lim ƒx = lim ƒx = x: q x: -q and that there are no vertical asymptotes. The horizontal asymptote is y =.

4 SECTION.6 Graphs of Rational Functions [ 5, 5] by [, 3] FIGURE.50 The graph of ƒx = x + /x + with its horizontal asymptote y =. Solve Algebraically Because the denominator x + 7 0, the domain of ƒ is all real numbers. So there are no vertical asymptotes. Using polynomial long division, we find that ƒx = x + x + = + x +. When the value of ƒ x ƒ is large, the denominator x + is a large positive number, and /x + is a small positive number, getting closer to zero as ƒ x ƒ increases. Therefore, lim ƒx = lim ƒx =, x: q x: -q so y = is indeed a horizontal asymptote. Now try Exercise 9. Example 3 shows the connection between the end behavior of a rational function and its horizontal asymptote. We now generalize this relationship and summarize other features of the graph of a rational function: Graph of a Rational Function The graph of y = ƒx/gx = a n x n + Á /b m x m + Á has the following characteristics:. End behavior asymptote: If n 6 m, the end behavior asymptote is the horizontal asymptote y = 0. If n = m, the end behavior asymptote is the horizontal asymptote y = a n /b m. If n 7 m, the end behavior asymptote is the quotient polynomial function y = qx, where ƒx = gxqx + rx. There is no horizontal asymptote.. Vertical asymptotes: These occur at the real zeros of the denominator, provided that the zeros are not also zeros of the numerator of equal or greater multiplicity. 3. x-intercepts: These occur at the real zeros of the numerator, which are not also zeros of the denominator. 4. y-intercept: This is the value of ƒ0, if defined. It is a good idea to find all of the asymptotes and intercepts when graphing a rational function. If the end behavior asymptote of a rational function is a slant line, we call it a slant asymptote, as illustrated in Example 4. EXAMPLE 4 Graphing a Rational Function Find the asymptotes and intercepts of the function ƒx = x 3 /x - 9 and graph the function. SOLUTION The degree of the numerator is greater than the degree of the denominator, so the end behavior asymptote is the quotient polynomial. Using polynomial long division, we obtain ƒx = x 3 x - 9 = x + 9x x - 9. So the quotient polynomial is qx = x, a slant asymptote. Factoring the denominator, x - 9 = x - 3, (continued)

5 CHAPTER Polynomial, Power, and Rational Functions shows that the zeros of the denominator are x = 3 and x = -3. Consequently, x = 3 and x = -3 are the vertical asymptotes of ƒ. The only zero of the numerator is 0, so ƒ0 = 0, and thus we see that the point (0, 0) is the only x-intercept and the y-intercept of the graph of ƒ. The graph of ƒ in Figure.5a passes through (0, 0) and suggests the vertical asymptotes x = 3 and x = -3 and the slant asymptote y = qx = x. Figure.5b shows the graph of ƒ with its asymptotes overlaid. Now try Exercise 9. [ 9.4, 9.4] by [ 5, 5] (a) Analyzing Graphs of Rational Functions Because the degree of the numerator of the rational function in Example 5 is less than the degree of the denominator, we know that the graph of the function has y = 0 as a horizontal asymptote. [ 9.4, 9.4] by [ 5, 5] (b) FIGURE.5 The graph of ƒx = x 3 /x - 9 (a) by itself and (b) with its asymptotes. (Example 4) [ 4.7, 4.7] by [ 4, 4] FIGURE.5 The graph of ƒx = /x - x - 6. (Example 5) EXAMPLE 5 Analyzing the Graph of a Rational Function Find the intercepts and asymptotes, use limits to describe the behavior at the vertical asymptotes, and analyze and draw the graph of the rational function ƒx = x - x - 6. SOLUTION The numerator is zero when x =, so the x-intercept is. Because ƒ0 = /6, the y-intercept is /6. The denominator factors as x - x - 6 = x - 3x +, so there are vertical asymptotes at x = - and x = 3. From the comment preceding this example we know that the horizontal asymptote is y = 0. Figure.5 supports this information and allows us to conclude that lim ƒx = - q, lim ƒx = q, lim ƒx = - q, and lim ƒx = q. x: - - x: - + x:3 - x:3 + Domain: - q, - -, 3 3, q Range: All reals Continuity: All x Z -, 3 Decreasing on - q, -, -, 3, and 3, q Not symmetric Unbounded No local extrema Horizontal asymptotes: y = 0 Vertical asymptotes: x = -, x = 3 End behavior: lim ƒx = lim ƒx = 0 Now try Exercise 39. x: - q x: q The degrees of the numerator and denominator of the rational function in Example 6 are equal. Thus, we know that the graph of the function has y =, the quotient of the leading coefficients, as its end behavior asymptote. EXAMPLE 6 Analyzing the Graph of a Rational Function Find the intercepts, analyze, and draw the graph of the rational function SOLUTION The numerator factors as ƒx = x - x - 4. x - = x - = x +,

6 SECTION.6 Graphs of Rational Functions 3 so the x-intercepts are - and. The y-intercept is ƒ0 = /. The denominator factors as x - 4 = x + x -, so the vertical asymptotes are x = - and x =. From the comment preceding this example we know that y = is the horizontal asymptote. Figure.53 supports this information and allows us to conclude that [ 4.7, 4.7] by [ 8, 8] FIGURE.53 The graph of ƒx = x - /x - 4. It can be shown that ƒ takes on no value between /, the y-intercept, and, the horizontal asymptote. (Example 6) lim ƒx = q, lim ƒx = - q, lim ƒx = - q, lim ƒx = q. x: - - x: - + x: - x: + Domain: - q, - -,, q Range: - q, /4, q Continuity: All x Z -, Increasing on - q, - and -, 04; decreasing on [0, ) and, q Symmetric with respect to the y-axis (an even function) Unbounded Local maximum of / at x = 0 Horizontal asymptotes: y = Vertical asymptotes: x = -, x = End behavior: lim ƒx = lim ƒx = Now try Exercise 4. x: -q x: q In Examples 7 and 8 we will investigate the rational function ƒx = x 3-3x + 3x +. The degree of the numerator of ƒ exceeds the degree of the denominator by. Thus, there is no horizontal asymptote. We will see that the end behavior asymptote is a polynomial of degree. [ 4.7, 4.7] by [ 8, 8] FIGURE.54 The graph of ƒx = x 3-3x + 3x + / as a solid black line and its end behavior asymptote y = x - x + as a dashed blue line. (Examples 7 and 8) EXAMPLE 7 Finding an End Behavior Asymptote Find the end behavior asymptote of ƒx = x 3-3x + 3x + and graph it together with ƒ in two windows: (a) one showing the details around the vertical asymptote of ƒ, (b) one showing a graph of ƒ that resembles its end behavior asymptote. SOLUTION The graph of ƒ has a vertical asymptote at x =. Divide x 3-3x + 3x + by to show that ƒx = x 3-3x + 3x + = x - x + +. [ 40, 40] by [ 500, 500] FIGURE.55 The graphs of ƒx = x 3-3x + 3x + / and its end behavior asymptote y = x - x + appear to be identical. (Example 7) The end behavior asymptote of ƒ is y = x - x +. (a) The graph of ƒ in Figure.54 shows the details around the vertical asymptote. We have also overlaid the graph of its end behavior asymptote as a dashed line. (b) If we draw the graph of ƒx = x 3-3x + 3x + / and its end behavior asymptote y = x - x + in a large enough viewing window, the two graphs will appear to be identical (Figure.55). Now try Exercise 47.

7 4 CHAPTER Polynomial, Power, and Rational Functions EXAMPLE 8 Analyzing the Graph of a Rational Function Find the intercepts, analyze, and draw the graph of the rational function ƒx = x 3-3x + 3x +. SOLUTION ƒ has only one x-intercept and we can use the graph of ƒ in Figure.54 to show that it is about The y-intercept is ƒ0 = -. The vertical asymptote is x = as we have seen. We know that the graph of ƒ does not have a horizontal asymptote, and from Example 7 we know that the end behavior asymptote is y = x - x +. We can also use Figure.54 to show that ƒ has a local minimum of 3 at x =. Figure.54 supports this information and allows us to conclude that lim ƒx = - q and lim ƒx = q. x: - x: + Domain: All x Z Range: All reals Continuity: All x Z Decreasing on - q, and, 4; increasing on 3, q Not symmetric Unbounded Local minimum of 3 at x = No horizontal asymptotes; end behavior asymptote: y = x - x + Vertical asymptote: x = End behavior: lim ƒx = lim ƒx = q Now try Exercise 55. x: -q x: q Exploring Relative Humidity The phrase relative humidity is familiar from everyday weather reports. Relative humidity is the ratio of constant vapor pressure to saturated vapor pressure. So, relative humidity is inversely proportional to saturated vapor pressure. Chapter Opener Problem (from page 57) Problem: Determine the relative humidity values that correspond to the saturated vapor pressures of, 4, 36, 48, and 60 millibars, at a constant vapor pressure of millibars. (In practice, saturated vapor pressure increases as the temperature increases.) Solution: Relative humidity (RH) is found by dividing constant vapor pressure (CVP) by saturated vapor pressure (SVP). So, for example, for SVP = 4 millibars and CVP = millibars, RH = /4 = / = 0.5 = 50%. See the table below, which is based on CVP = millibars with increasing temperature. SVP (millibars) RH (%)

8 SECTION.6 Graphs of Rational Functions 5 QUICK REVIEW.6 (For help, go to Section.4.) Exercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. In Exercises 6, use factoring to find the real zeros of the function.. ƒx = x + 5x - 3. ƒx = 3x - x gx = x gx = x - 5. hx = x 3-6. hx = x + In Exercises 7 0, find the quotient and remainder when ƒx is divided by dx. 7. ƒx = x +, dx = x ƒx = 4, dx = 9. ƒx = 3x - 5, dx = x 0. ƒx = 5, dx = x SECTION.6 EXERCISES In Exercises 4, find the domain of the function ƒ. Use limits to describe the behavior of ƒ at value(s) of x not in its domain.. ƒx =. ƒx = ƒx = 4. ƒx = x - 4 x - In Exercises 5 0, describe how the graph of the given function can be obtained by transforming the graph of the reciprocal function gx = /x. Identify the horizontal and vertical asymptotes and use limits to describe the corresponding behavior. Sketch the graph of the function. 5. ƒx = 6. ƒx = - x - 3 x ƒx = 8. ƒx = 3x - 9. ƒx = 5 - x 0. ƒx = 4-3x x + 4 x - 5 In Exercises 4, evaluate the limit based on the graph of ƒ shown. [ 5.8, 3] by [ 3, 3]. x:3 -. lim x: x: q 4. lim x: -q ƒx In Exercises 5 8, evaluate the limit based on the graph of ƒ shown. 5. lim ƒx 6. x: lim 8. x: -q ƒx [ 9.8, 9] by [ 5, 5] lim ƒx x: lim x: q ƒx In Exercises 9, find the horizontal and vertical asymptotes of ƒx. Use limits to describe the corresponding behavior. 3x 9. ƒx = x - 0. ƒx = x + 3 x +. ƒx = x +. ƒx = x - 3 x - x x + 3x In Exercises 3 30, find the asymptotes and intercepts of the function, and graph the function. x - x + 3. gx = 4. gx = x - x - 3 x + x hx = 6. hx = x 3 - x x 3-4x gx = -3x + x + ƒx = x + x - x - x gx = x - 3x - 7 ƒx = x - x +

9 6 CHAPTER Polynomial, Power, and Rational Functions In Exercises 3 36, match the rational function with its graph. Identify the viewing window and the scale used on each axis. 3. ƒx = 3. ƒx = - x ƒx = ƒx = ƒx = ƒx = x In Exercises 5 56, find the intercepts, analyze, and graph the given rational function. 4x + x 5. ƒx = 3x - x gx = x - 4x + 5 x - 4x kx = x 3 - hx = x 3 - x - x ƒx = x 3 - x gx = x 3 - x - x + 5 x - (a) (c) (b) (d) In Exercises 57 6, find the intercepts, vertical asymptotes, and end behavior asymptote, and graph the function together with its end behavior asymptote kx = x 5 + x - x + hx = x 4 + x + x gx = x 5 + ƒx = x 5 - x kx = 3x 3 + x - 4 hx = x 3-3x + x 3 - x 3 + Standardized Test Questions 63. True or False A rational function must have a vertical asymptote. Justify your answer. 64. True or False ƒx = x - x is a rational function. x + 4 Justify your answer. (e) In Exercises 37 44, find the intercepts and asymptotes, use limits to describe the behavior at the vertical asymptotes, and analyze and draw the graph of the given rational function. 37. ƒx = 38. gx = x - x - 3 x + 4 x hx = 40. kx = x - x gx = x - x - ƒx = x + x - x - 9 x - x kx = x - x - hx = x + x - 3 x + x - 3 In Exercises 45 50, find the end behavior asymptote of the given rational function ƒ and graph it together with ƒ in two windows: (a) One showing the details around the vertical asymptote(s) of ƒ. (b) One showing a graph of ƒ that resembles its end behavior asymptote ƒx = x + x - 3 ƒx = x - x ƒx = x 3 - x x + (f) ƒx = x ƒx = x 5 + ƒx = x 4 - x + x - x + In Exercises 65 68, you may use a graphing calculator to solve the problem Multiple Choice Let ƒx = What values of x x + 3x. have to be excluded from the domain of ƒ? (A) Only 0 (B) Only 3 (C) Only -3 (D) Only 0, 3 (E) Only 0, Multiple Choice Let gx = Which of the transformations of ƒx =. produce the graph of g? x (A) Translate the graph of ƒ left 3 units. (B) Translate the graph of ƒ right 3 units. (C) Translate the graph of ƒ down 3 units. (D) Translate the graph of ƒ up 3 units. (E) Vertically stretch the graph of ƒ by a factor of. 67. Multiple Choice Let ƒx = Which of the x + 5. following statements is true about the graph of ƒ? (A) There is no vertical asymptote. (B) There is a horizontal asymptote but no vertical asymptote. (C) There is a slant asymptote but no vertical asymptote. (D) There is a vertical asymptote and a slant asymptote. (E) There is a vertical and horizontal asymptote. x

10 SECTION.6 Graphs of Rational Functions Multiple Choice What is the degree of the end behavior asymptote of ƒx = x 8 + x 4 +? (A) 0 (B) (C) (D) 3 (E) 4 Explorations 69. Group Activity Work in groups of two. Compare the functions ƒx = x - 9 and gx =. x - 3 (a) Are the domains equal? (b) Does ƒ have a vertical asymptote? Explain. (c) Explain why the graphs appear to be identical. (d) Are the functions identical? 70. Group Activity Explain why the functions are identical or not. Include the graphs and a comparison of the functions asymptotes, intercepts, and domain. (a) ƒx = x + x - and gx = x + (b) ƒx = x - and gx = x + x - (c) ƒx = and gx = x 3 - x - x + (d) ƒx = and gx = x + x - x + 7. Boyle s Law This ideal gas law states that the volume of an enclosed gas at a fixed temperature varies inversely as the pressure. (a) Writing to Learn Explain why Boyle s Law yields both a rational function model and a power function model. (b) Which power functions are also rational functions? (c) If the pressure of a.59-l sample of nitrogen gas at a temperature of 9 K is atm, what would the volume be at a pressure of 0.53 atm if the temperature does not change? 7. Light Intensity Aileen and Malachy gathered the data in Table.9 using a 75-watt lightbulb and a Calculator-Based Laboratory (CBL ) with a light-intensity probe. (a) Draw a scatter plot of the data in Table.9. (b) Find an equation for the data assuming it has the form ƒx = k/x for some constant k. Explain your method for choosing k. (c) Superimpose the regression curve on the scatter plot. (d) Use the regression model to predict the light intensity at distances of. m and 4.4 m. Table.9 Light-Intensity Data for a 75-W Lightbulb Distance (m) Intensity ( W/m ) Extending the Ideas In Exercises 73 76, graph the function. Express the function as a piecewise-defined function without absolute value, and use the result to confirm the graph s asymptotes and intercepts algebraically. ƒ ƒ ƒ ƒ 73. hx = x - 3 x ƒx = 5-3x x hx = 3x + 5 ƒ x ƒ + 3 ƒx = - x ƒ x ƒ Describe how the graph of a nonzero rational function ƒx = ax + b cx + d, c Z 0 can be obtained from the graph of y = /x. (Hint: Use long division.) 78. Writing to Learn Let ƒx = + //x and gx = x 3 + x - x/x 3 - x. Does f = g? Support your answer by making a comparative analysis of all of the features of ƒ and g, including asymptotes, intercepts, and domain.