8.2 Graphing More Complicated Rational Functions

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1 1 Locker LESSON 8. Graphing More Complicated Rational Functions PAGE 33 Name Class Date 8. Graphing More Complicated Rational Functions Essential Question: What features of the graph of a rational function should you identify in order to sketch the graph? How do you identify those features? Resource Locker Common Core Math Standards The student is epected to: F-IF.7d(+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. Also A-SSE.1b, A-APR.6, F-IF., F-IF.8, F-BF.1b Mathematical Practices MP.8 Patterns Language Objective Students describe and eplain the key features of the graph of a complicated rational function. Eplore 1 Investigating Domains and Vertical Asympotes of More Complicated Rational Functions 1 You know that the rational function ƒ () = has the domain ǀ because the function is undefined at =. Its graph has the vertical asymptote = because as + ( approaches from the right), ƒ () +, and as - ( approaches from the left), ƒ () -. In this Eplore, you will investigate the domains and vertical asymptotes of other rational functions. Complete the table by identifying each function s domain based on the -values for which the function is undefined. Write the domain using an inequality, set notation, and interval notation. Then state the equations of what you think the vertical asymptotes of the function s graph are. Function Domain Possible Vertical Asymptotes ENGAGE Essential Question: What features of the graph of a rational function should you identify in order to sketch the graph? How do you identify those features? Possible answer: Identify vertical asymptotes and holes, based on the factors of the function s denominator, horizontal asymptotes, and slant asymptotes, based on an analysis of the function s end behavior; -intercepts based on the factors of the function s numerator, where the graph lies above or below the -ais, based on an analysis of the signs of the factors in both the numerator and denominator. PREVIEW: LESSON PERFORMANCE TASK View the Engage section online. Discuss the photo and how the radius of a baseball is a function of its volume. Then preview the Lesson Performance Task. f () = _ ( + 5) ( - 1) f () = _ + 1 f () = _ - ( + 1) ( - 1) f () = _ Inequality: < 1 or > 1 ǀ 1 Interval notation: (-, 1) (1, + ) Inequality: < 1 or > -1 ǀ -1 Interval notation: (-, -1) (-1, + ) Inequality: < -1 or -1 < < 1 or > 1 ǀ -1 and 1 Interval notation: (-, -1) (-1, 1) (1, + ) Inequality: < - or - < < 3 or > 3 ǀ - and 3 Interval notation: (-, -) (-, 3) (3, + ) = 1 = -1 = -1, = 1 = -, = 3 Module 8 5 Lesson Name Class Date 8. Graphing More Complicated Rational Functions Essential Question: What features of the graph of a rational function should you identify in order to sketch the graph? How do you identify those features? Eplore 1 Investigating Domains and Vertical Asympotes of More Complicated Rational Functions has the domain You know that the rational function ƒ () = ǀ because the function is undefined at =. Its graph has the vertical asymptote = because as + ( approaches from the right), ƒ () +, and as - ( approaches from the left), ƒ () -. In this Eplore, you will investigate the domains and vertical asymptotes of other rational functions. F-IF.7d(+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. Also A-SSE.1b, A-APR.6, F-IF. Complete the table by identifying each function s domain based on the -values for which the function is undefined. Write the domain using an inequality, set notation, and interval notation. Then state the equations of what you think the vertical asymptotes of the function s graph are. f () = Function Domain Inequality: < 1 or > 1 ǀ 1 Interval notation: (-, 1) (1, + ) _ ( + 5) ( - 1) f () = + 1 f () = - ( + 1) ( - 1) _ f () = Inequality: < 1 or > -1 ǀ -1 Interval notation: (-, -1) (-1, + ) Inequality: < -1 or -1 < < 1 or > 1 ǀ -1 and 1 Interval notation: (-, -1) (-1, 1) (1, + ) Inequality: < - or - < < 3 or > 3 ǀ - and 3 Interval notation: (-, -) (-, 3) (3, + ) Resource Possible Vertical Asymptotes = 1 = -1 = -1, = 1 = -, = 3 Module 8 5 Lesson HARDCOVER PAGES Turn to these pages to find this lesson in the hardcover student edition. 5 Lesson 8.

2 B C Using a graphing calculator, graph each of the functions from Step A, and check to see if vertical asymptotes occur where you epect. Are there any unepected results? The graphs of the first three functions have the epected vertical asymptotes. The graph of the fourth function, however, has a vertical asymptote at = - but not at = Eamine the behavior of ƒ () = _ near = First, complete the tables. PAGE 3 EXPLORE Investigating Domains and Vertical Asymptotes of More Complicated Rational Functions approaches 1 from the right f () = + 3 _ Net, summarize the results. As 1 +, ƒ () +. As 1 -, ƒ () -. approaches 1 from the left f () = + 3 _ The behaviour of ƒ () = _ + 3 near = 1 shows that the graph of ƒ () has/does not have a - 1 vertical asymptote at = 1. AVOID COMMON ERRORS Remind students that if the numerator and denominator have the same zero, the result is a point discontinuity, which will not be visible on a graphing calculator. The -value at which the point discontinuity eists must be ecluded from the domain, and the corresponding y-value must be ecluded from the range. D ( + 5) ( - 1) Eamine the behavior of ƒ () = near = -1. ( + 1) First, complete the tables. approaches -1 from the right ( + 5) ( - 1) f () = Net, summarize the results. As - 1 +, ƒ () -. As - 1 -, ƒ () +. approaches -1 from the left ( + 5) ( - 1) f () = ( + 5) ( - 1) The behavior of ƒ () = near = -1 shows that the graph of ƒ () ( + 1) has/does not have a vertical asymptote at = -1. Module 8 6 Lesson PROFESSIONAL DEVELOPMENT Learning Progressions Previously, students learned to graph, find asymptotes, and describe domain, range, and end behavior of rational functions of the form a + b f () =. They used long division to write linear functions divided by linear c + d functions as a quotient plus a remainder, learned the basics about domain, range, and end behavior, and transformed the parent function f () = 1 to graph rational functions. Students apply these skills to more comple rational functions such a quadratic divided by linear functions. Graphing More Complicated Rational Functions 6

3 PAGE 35 E Eamine the behavior of ƒ () = - _ ( + 1) ( - 1) near = -1 and = 1. First, complete the tables. Round results to the nearest tenth. approaches -1 from the right f () = _ - ( + 1) ( - 1) approaches -1 from the left f () = - ( + 1) ( - 1) approaches 1 from the right f () = - ( + 1) ( - 1) approaches 1 from the left f () = - ( + 1) ( - 1) Net, summarize the results. As - 1 +, ƒ () +. As - 1 -, ƒ () -. As 1 +, ƒ () -. As 1 -, ƒ () +. - The behavior of ƒ () = near = -1 shows that the graph of ƒ () has/does ( + 1) ( - 1) - not have a vertical asymptote at = -1. The behavior of f () = ( + 1) ( - 1) near = 1 shows that the graph of ƒ () has/does not have a vertical asymptote at = 1. F Eamine the behavior of ƒ () = near = - and = First, complete the tables. Round results to the nearest ten thousandth if necessary. approaches - from the right f () = _ approaches - from the left f () = _ Module 8 7 Lesson COLLABORATIVE LEARNING Peer-to-Peer Activity Have students work in pairs. Give one student in each pair a rational function of ( + a) ( + b) the form f () =, where a, b, c, and d are integers. Have that student ( + c) ( + d) describe the function solely in terms of its zeros, asymptotes, and holes, if any. The partner should attempt to write a rational function that fits this description. Pairs can then discuss their results. Point out that more than one rational function can have the same zeros, asymptotes, and holes. 7 Lesson 8.

4 approaches 3 from the right f () = _ Net, summarize the results As - +, ƒ () -. As - -, ƒ () +. approaches 3 from the left f () = _ As 3 +, ƒ () 1.8. As 3 -, ƒ () 1.8. PAGE 36 The behavior of ƒ () = near = - shows that the graph of ƒ () has/does not have a vertical asymptote at = -. The behavior of ƒ () = near = 3 shows that the graph of ƒ () has/does not have a vertical asymptote at = 3. Reflect 1. Rewrite ƒ () = so that its numerator and denominator are factored. How does this form of the function eplain the behavior of the function near = 3? ( + ) ( - 3) ( + 3) ( - 3) With the function written in the form f () =, you can see that the numerator and denominator have a common factor of - 3. In effect, the function behaves just like the function g () = + 3 ecept that f () is not defined at = 3. However, f () approaches + g (3) = 1.8 as approaches 3.. Discussion When you graph ƒ () = _ on a graphing calculator, you can t tell that the function is undefined for = 3. How does using the calculator s table feature help? What do you think the graph should look like to make it clear that the function is undefined at = 3? The table feature will display an error message for the function s value at = 3. Because the graph has a hole at = 3, there should be an open circle at the point (3, 1.8). Module 8 8 Lesson DIFFERENTIATE INSTRUCTION Critical Thinking Show how to find the oblique asymptote for a quadratic/linear function by synthetic division rather than long division. Graphing More Complicated Rational Functions 8

5 EXPLAIN 1 Sketching the Graphs of More Complicated Rational Functions AVOID COMMON ERRORS Students may try to graph without factoring and will miss holes and discontinuities. Remind students to factor quadratic epressions when possible. If they determine that the numerator and denominator share a common linear factor, it should be crossed out and the function simplified. There is a hole in the graph at the -value where the shared linear factor equals zero. PAGE 37 Eplain 1 Sketching the Graphs of More Complicated Rational Functions As you have seen, there can be breaks in the graph of a rational function. These breaks are called discontinuities, and there are two kinds: 1. When a rational function has a factor in the denominator that is not also in the numerator, an infinite discontinuity occurs at the value of for which the factor equals 0. On the graph of the function, an infinite discontinuity appears as a vertical asymptote.. When a rational function has a factor in the denominator that is also in the numerator, a point discontinuity occurs at the value of for which the factor equals 0. On the graph of the function, a point discontinuity appears as a hole. The graph of a rational function can also have a horizontal asymptote. It is determined by the degrees and leading coefficients of the function s numerator and denominator. If the numerator is a polynomial p() in standard form with leading coefficient a and the denominator is a polynomial q() in standard form with leading coefficient b, then an eamination of the function s end behavior gives the following results. Relationship between Degree of p () and Degree of q () Equation of Horizontal Asymptote (if one eists) Degree of p () < degree of q () y = 0 Degree of p () = degree of q () Degree of p () > degree of q () y = a_ b There is no horizontal asymptote. The function instead increases or decreases without bound as increases or decreases without bound. In particular, when the degree of the numerator is 1 more than the degree of the denominator, the function s graph approaches a slanted line, called a slant asymptote, as increases or decreases without bound. You can sketch the graph of a rational function by identifying where vertical asymptotes, holes, and horizontal asymptotes occur. Using the factors of the numerator and denominator, you can also establish intervals on the -ais where either an -intercept or a discontinuity occurs and then check the signs of the factors on those intervals to determine whether the graph lies above or below the -ais. Module 8 9 Lesson 9 Lesson 8. LANGUAGE SUPPORT Vocabulary Development Have students work in pairs. They should complete a chart like the following, using the different types of rational functions identified in this lesson and the previous lesson. Rational Function Eample Vertical Asymptotes Graph f() = constant/linear f() = linear/linear f() =quadratic/linear f() = quadratic/quadratic

6 Eample 1 Sketch the graph of the given rational function. (If the degree of the numerator is 1 more than the degree of the denominator, find the graph s slant asymptote by dividing the numerator by the denominator.) Also state the function s domain and range using inequalities, set notation, and interval notation. (If your sketch indicates that the function has maimum or minimum values, use a graphing calculator to find those values to the nearest hundredth when determining the range.) ƒ () = + 1 _ - Identify vertical asymptotes and holes. The function is undefined when - = 0, or =. Since - does not appear in the numerator, there is a vertical asymptote rather than a hole at =. QUESTIONING STRATEGIES How do you find the vertical asymptotes of a rational equation? Simplify the equation by eliminating any factors that are in both the numerator and denominator. There is a linear asymptote at -values for which the denominator equals zero in the simplified equation. Identify horizontal asymptotes and slant asymptotes. The numerator and denominator have the same degree and the leading coefficient of each is 1, so there is a horizontal asymptote at y = 1 1 = 1. Identify -intercepts. An -intercept occurs when + 1 = 0, or = -1. Check the sign of the function on the intervals < -1, -1 < <, and >. Interval Sign of + 1 Sign of - _ + 1 Sign of f () = - < < < > Sketch the graph using all this information. Then state the domain and range y Inequality: < or > Set notation: ǀ Interval notation: (-, ) (, + ) Range: Inequality: y < 1 or y > 1 Set notation: yǀy 1 Interval notation: (-, 1) (1, + ) PAGE 38 Module 8 50 Lesson Graphing More Complicated Rational Functions 50

7 B ƒ () = + - _ + 3 Factor the function s numerator. ƒ () = _ + - ( = - 1) ( + ) Identify vertical asymptotes and holes. The function is undefined when + 3 = 0, or = -3. Since + 3 does not appear in the numerator, there is a vertical asymptote rather than a hole at = -3. Identify horizontal asymptotes and slant asymptotes. Because the degree of the numerator is 1 more than the degree of the denominator, there is no horizontal asymptote, but there is a slant asymptote. Divide the numerator by the denominator to identify the slant asymptote So, the line y = - Identify -intercepts. is the slant asymptote. There are two -intercepts: when - 1 = 0, or = 1, and when + = 0, or = -. Check the sign of the function on the intervals <-3, -3 < < -, - < < 1, and > 1. PAGE 39 Interval Sign of + 3 Sign of + Sign of - 1 Sign of f () = < < < - - < < 1 > 1 Sketch the graph using all this information. Then state the domain and range. y Inequality: < -3 or > -3 ǀ -3 Interval notation: (-, -3 ) ( -3, + ) ( - 1) ( + ) Module 8 51 Lesson 51 Lesson 8.

8 The sketch indicates that the function has a maimum value and a minimum value. Using 3:minimum from the CALC menu on a graphing calculator gives -1 as the minimum value. Using :maimum from the CALC menu on a graphing calculator gives -5 as the maimum value. Range: Inequality: y < -5 or y > -1 yǀy < or y > Interval notation: (-, ) (, + ) EXPLAIN Modeling with More Complicated Rational Functions Your Turn Sketch the graph of the given rational function. (If the degree of the numerator is 1 more than the degree of the denominator, find the graph s slant asymptote by dividing the numerator by the denominator.) Also state the function s domain and range using inequalities, set notation, and interval notation. (If your sketch indicates that the function has maimum or minimum values, use a graphing calculator to find those values to the nearest hundredth when determining the range.) ƒ () = y f () = = + 1 ( - 1) ( + ) There are vertical asymptotes at = - and = 1. There is a horizontal asymptote at y = 0. There is an -intercept at = -1. Check the sign of the function on the intervals < -, - < < -1, -1 < < 1, and > 1. Inequality: < - or - < < 1 or > Range: Inequality: - < y < + AVOID COMMON ERRORS Students may not realize when to look for a slant asymptote. Remind students to find the slant asymptote of quadratic/linear rational functions. ǀ - and 1 Interval notation: (-, -) (-, 1) (1 + ) yǀ- < y < + Interval notation: (-, + ) Eplain Modeling with More Complicated Rational Functions When two real-world variable quantities are compared using a ratio or rate, the comparison is a rational function. You can solve problems about the ratio or rate by graphing the rational function. Eample Write a rational function to model the situation, or use the given rational function. State a reasonable domain and range for the function using set notation. Then use a graphing calculator to graph the function and answer the question. 3 A baseball team has won 3 games out of 56 games played, for a winning percentage of How 56 many consecutive games must the team win to raise its winning percentage to 0.600? Let w be the number of consecutive games to be won. Then the total number of games won is the function T won (w) = 3 + w, and the total number of games played is the function T played (w) = 56 + w. Module 8 5 Lesson Graphing More Complicated Rational Functions 5

9 QUESTIONING STRATEGIES What are three ways to write the domain and range of a function? set notation, inequalities, interval notation PAGE 350 Image Credits: AVAVA/ Shutterstock The rational function that gives the team s winning percentage p (as a decimal) is T won (w) p (w) = T played (w) = 3 + w 56 + w. The domain of the rational function is wǀw 0 and w is a whole number. Note that you do not need to eclude -56 from the domain, because only nonnegative whole-number values of w make sense in this situation. Since the function models what happens to the team s winning percentage from consecutive wins (no losses), the values of p(w) start at and approach 1 as w increases without bound. So, the range is pǀ0.571 p < Graph y = on a graphing calculator using a viewing window that 56 + shows 0 to 10 on the -ais and 0.5 to 0.7 on the y-ais. Also graph the line y = 0.6. To find where the graphs intersect, select 5: intersect from the CALC menu. So, the team s winning percentage (as a decimal) will be if the team wins consecutive games. Two friends decide spend an afternoon canoeing on a river. They travel miles upstream and 6 miles downstream. In still water, they know that their average paddling speed is 5 miles per hour. If their canoe trip takes hours, what is the average speed of the river s current? To answer the question, use the rational function t ( c ) = 5 - c c = 50 - c (5 - c) (5 + c) where c is the average speed of the current (in miles per hour) and t is the time (in hours) spent canoeing miles against the current at a rate of 5 - c miles per hour and 6 miles with the current at a rate of 5 + c miles per hour. In order for the friends to travel upstream, the speed of the current must be less than their average paddling speed, so the domain of the function is cǀ 0 c < 5. If the friends canoed in still water (c = 0), the trip would take a total of = hours. 6 As c approaches 5 from the left, the value of 5 + c approaches 6 = 0.6 hour, but the value 10 of increases without bound 5 - c. So, the range of the function is tǀ t. Graph y = _ 50 - on a graphing calculator using a viewing window that shows 0 (5 - ) (5 + ) to 5 on the -ais and to 5 on the y-ais. Also graph the line y =. To find where the graphs intersect, select 5:intersect from the CALC menu. The calculator shows that the average speed of the current is about 3.8 miles per hour. Module 8 53 Lesson 53 Lesson 8.

10 Your Turn Write a rational function to model the situation, or use the given rational function. State a reasonable domain and range for the function using set notation. Then use a graphing calculator to graph the function and answer the question.. A saline solution is a miture of salt and water. A p% saline solution contains p% salt and (100 - p) % water by mass. A chemist has 300 grams of a % saline solution that needs to be strengthened to a 6% solution by adding salt. How much salt should the chemist add? Let s be the amount (in grams) of salt added to the saline solution. The mass of the salt in the solution before any more salt is added is 0.0 (300) = 1 grams, so the mass m(in grams) of the salt in the solution after more salt is added is m salt (s) = 1 + s. The mass m(in grams) of the solution after more salt is added is m solution (s) = s. So, after more salt is added, the percent p (as a decimal) of the mass of solution that is salt is the rational M function p (s) = salt (s) M solution (s) = 1 + s s. The domain of the rational function is sǀ s 0. The percent of salt in the solution starts at 0.0 and approaches 1 as s increases without bound, so the range is pǀ0.0 p < 1. Graphing y = 1 + and y = 0.06 on a graphing calculator with a window that shows to 0.07 on the -ais and 0 to 10 on the y-ais and then locating the point where the graphs intersect, you find that the amount of salt that the chemist should add is about 6. grams. Elaborate 5. How can you show that the vertical line = c, where c is a constant, is an asymptote for the graph of a rational function? Eamine the behavior of the function as approaches c from both the left and the right. If the values of the function increase or decrease without bound, the line is an asymptote. 6. How can you determine the end behavior of a rational function? Divide the function s numerator by the denominator. The quotient gives the function s end behavior. For instance, if the quotient is a constant c, then the values of the function approach c as increases or decreases without bound (that is, the horizontal line y = c is an asymptote for the graph). 7. Essential Question Check-In How do you identify any vertical asymptotes and holes that the graph of a rational function has? Factor the function s numerator and denominator. If a factor of the denominator is not also a factor of the numerator, then a vertical asymptote occurs at the -value for which the factor equals 0. If a factor of the denominator is also a factor of the numerator, then a hole occurs at the -value for which the factor equals 0. PAGE 351 ELABORATE INTEGRATE MATHEMATICAL PRACTICES Focus on Patterns MP.8 The rules for finding horizontal asymptotes covered in this lesson can be etended to rational functions other than linear/linear, linear/quadratic, quadratic/linear, and quadratic/quadratic functions. When the degree of the numerator is smaller than the degree of the denominator, the horizontal asymptote is y = 0. When the degree of the numerator is larger than the degree of the denominator, there is no horizontal asymptote. When the degrees of the numerator and denominator are equal, the horizontal a asymptote is y = 1 a. SUMMARIZE THE LESSON How do you graph rational functions? Factor if possible, cancel factors common to numerator and denominator, find horizontal, oblique, and vertical asymptotes, plug in -values to find points, and remember to graph a hole (or point discontinuity) where a cancelled factor equals zero. Module 8 5 Lesson Graphing More Complicated Rational Functions 5

11 EVALUATE ASSIGNMENT GUIDE Concepts and Skills Eplore Investigating Domains and Vertical Asymptotes of More Complicated Rational Functions Eplain 1 Sketching the Graphs of More Complicated Rational Functions Eplain Modeling With More Complicated Rational Functions INTEGRATE MATHEMATICAL PRACTICES Focus on Math Connections Practice Eercises 1 Eercises 7 1 Eercises MP.1 A function in the form f () = a - h + k can be graphed by transforming the parent function f () = 1. It can also be rewritten as a linear/linear rational function by finding a common denominator and rewriting the function as a single fraction. Evaluate: Homework and Practice State the domain using an inequality, set notation, and interval notation. For any -value ecluded from the domain, state whether the graph has a vertical asymptote or a hole at that -value. Use a graphing calculator to check your answer ƒ () = _. ƒ () = Inequality: < -1 or > -1 ǀ -1 Set notation: Interval notation: (-, -1) (-1, + ) vertical asymptote at = 3 remainder Divide the numerator by the denominator to write the function in the form ƒ () = quotient + divisor and determine the function s end behavior. Then, using a graphing calculator to eamine the function s graph, state the range using an inequality, set notation, and interval notation ƒ () = _. ƒ () = - ( - ) ( + 3) 7 So, f () = 3 +. As increases or So, f () = 0 +. As increases or - ( - ) ( + 3) decreases without bound, f () approaches 3. decreases without bound, Range: f () approaches 0. Inequality: y < 3 or y > 3 Range: yǀy 3 Inequality: - < y < - Interval notation: (-, 3) (3, + ) yǀ- < y < - Interval notation: (-, + ) 5. ƒ () = _ - 1 _ So, f () = - +. As increases without - 1 bound, f () increases without bound, and as decreases without bound, f () decreases without bound. Range (approimate): Inequality: y < or y > yǀy < or y > Interval notation: (-, -5.83) (-0.17, + ) 6. ƒ () = _ Online Homework Hints and Help Etra Practice Inequality: < 1 or 1 < < 3 or > 3 ǀ 1 and 3 Interval notation: (-, 1) (1, 3) (3, + ) a hole at = 1; a vertical asymptote at = 3 So, f () = As increases + - or decreases without bound, f () approaches. Range (approimate): Inequality: y < 0.51 or y > 3.9 yǀy < 0.51 or y > 3.9 Interval notation: (-, 0.51) (3.9, + ) Module 8 55 Lesson Eercise Depth of Knowledge (D.O.K.) Mathematical Practices 1 6 Skills/Concepts MP. Modeling 7 1 Skills/Concepts MP.5 Using Tools Skills/Concepts MP. Modeling 17 Skills/Concepts MP. Reasoning 18 3 Strategic Thinking MP.6 Precision 19 3 Strategic Thinking MP.3 Logic 55 Lesson 8.

12 Sketch the graph of the given rational function. (If the degree of the numerator is 1 more than the degree of the denominator, find the graph s slant asymptote by dividing the numerator by the denominator.) Also state the function s domain and range using inequalities, set notation, and interval notation. (If your sketch indicates that the function has maimum or minimum values, use a graphing calculator to find those values to the nearest hundredth when determining the range.) 7. ƒ () = _ There is a vertical asymptote at = -1. There is a horizontal asymptote at y = 1. There is an -intercept at = 1. Check the sign of the function on the intervals < -1, -1 < < 1, and > 1. Inequality: < -1 or > -1 Range: Inequality: y < 1 or y > y AVOID COMMON ERRORS Students may assume that when quadratic/linear functions cannot be simplified by cancelling a factor common to the numerator and denominator, that no asymptote eists. Remind students to find the oblique asymptote by dividing the numerator by the denominator. The quotient is the oblique asymptote. ǀ -1 Interval notation: (-, -1) (-1, + ) yǀy 1 Interval notation: (-, 1) (1, + ) 8. ƒ () = - 1 ( - ) ( + 3) y There is a vertical asymptote at = -3 and =. There is a horizontal asymptote at y = 0. There is an -intercept at = 1. Check the sign of the function on the intervals < -3, -3 < < 1, 1 < <, and > Inequality: < -3 or -3 < < or > ǀ -3 and Interval notation: (-, -3) (-3, ) (, + ) Range: Inequality: - < y < + yǀ- < y < + Interval notation: (-, + ) Module 8 56 Lesson Graphing More Complicated Rational Functions 56

13 9. ƒ () = ( + 1) ( - 1) + There is a vertical asymptote at = -. ( + 1) ( - 1) f () = = _ The line y = - is a slant asymptote. There are -intercepts at = -1 and = 1. Check the sign of the function on the intervals < -, - < < -1, -1 < < 1, and > 1. y Range (approimate): Inequality: < - or > - Inequality: y < -0.5 or y > -7.6 ǀ - Interval notation: (-, -) (-, + ) yǀy < -0.5 or y > -7.6 Interval notation: (-, -0.5) (-7.6, + ) PAGE 35-3( - ) 10. ƒ () = ( - ) ( + ) There is a vertical asymptote at = - There is a hole at =. There is a horizontal asymptote at y = -3. There is an -intercept at = 0. Check the sign of the function on the intervals < -, - < < 0, 0 < <, and >. Inequality: < - or - < < or > ǀ - and Interval notation: (-, -) (-, ) (, + ) y Range: Inequality: y < -3 or y > -3 Interval notation: (-, -3) (-3, + ) Module 8 57 Lesson 57 Lesson 8.

14 ƒ() = ( + )( - ) f() = = There is a vertical asymptote at = The line y = + 3 is a slant asymptote y There are -intercepts at = and = -. Check the sign of the function on the intervals - - < -, - < < 1, 1 < <, and >. 0 - Range: Inequality: - < y < + Inequality: < 1 or > 1 Set notation: ǀ 1 Set notation: yǀ- < y < + Interval notation: (-, 1) (1, + ) Interval notation: (-, + ) - 1. ƒ() = + + y ( - ) - = _ f() = + + ( + ) 6 There is a vertical asymptote at = There is a horizontal asymptote at y =. There are -intercepts at = 0 and =. Check the sign of the function on the intervals < -, - < < 0, 0 < <, and >. Range (approimate): Inequality: y -0.5 Inequality: < - or > - Set notation: ǀ - Interval notation: (-, -) (-, + ) Module 8 IN3_MNLESE389885_UM08L.indd Set notation: yǀy -0.5 Interval notation: [-0.5, + ) Lesson 8/0/17 9:6 AM Graphing More Complicated Rational Functions 58

15 AVOID COMMON ERRORS Students may try to graph first and find asymptotes afterward. Remind students to always start by identifying vertical and horizontal asymptotes, before graphing a rational function. The asymptotes are guidelines, or a framework, for the graph. Write a rational function to model the situation, or use the given rational function. State a reasonable domain and range for the function using set notation. Then use a graphing calculator to graph the function and answer the question. 13. A basketball team has won 16 games out of 3 games played, for a winning percentage (epressed as a decimal) of How many consecutive games must the team 3 win to raise its winning percentage to 0.750? Let w be the number of consecutive games to be won. Then the total number of games won is the function T won (w) = 16 + w, and the total number of games played is the function T played (w) = 3 + w. Image Credits: Lindsay Hebberd/Corbis The rational function that gives the team s winning percentage p (as a decimal) is p (w) = T won (w) T played (w) = 16 + w. The domain of the rational 3 + w function is wǀw 0 and w is a whole number. Since the values of p(w) start at and approach 1 as w increases without bound, the range is pǀ0.696 p < 1. Graphing y = 16 + and y = 0.75 on a graphing calculator with a window 3 + that shows 0 to 10 on the -ais and 0.6 to 0.8 on the y-ais and then locating the point where the graphs intersect, you find that the number of consecutive wins needed to bring the team s winning percentage to is So far this season, a baseball player has had 8 hits in 9 times at bat, for a batting average of How many consecutive hits must the player get to raise his 9 batting average to 0.300? Let h be the number of consecutive hits. Then the total number of hits is the function T hits (h) = 8 + h, and the total number of times at bat is the function T at bats (h) = 9 + h. The rational function that gives the player s batting average a (as a decimal) is a (h) = T hits (h) T at bats (h) = 8 + h 9 + h. The domain of the rational function is hǀh 0 and h is a whole number. Since the values of a(h) start at 0.86 and approach 1 as h increases without bound, the range is aǀ0.86 a < 1. Graphing y = 8 + and y = 0.3 on a graphing calculator with a 9 + window that shows 0 to 10 on the -ais and 0. to 0. on the y-ais and then locating the point where the graphs intersect, you find that the number of consecutive hits needed to bring the player s batting average to is 6. Module 8 59 Lesson 59 Lesson 8.

16 15. A kayaker traveled 5 miles upstream and then 8 miles downstream on a river. The average speed of the current was 3 miles per hour. If the kayaker was paddling for 5 hours, what was the kayaker s average paddling speed? To answer the question, use the rational function s - 9 t(s) = + = where s is the kayaker s s-3 s+3 (s - 3)(s + 3) average paddling speed (in miles per hour) and t is the time (in hours) spent kayaking 5 miles against the current at a rate of s - 3 miles per hour and 8 miles with the current at a rate of s + 3 miles per hour. In order for the kayaker to travel upstream, the paddling speed must eceed the speed of the current, so the domain of the function is sǀs > 3. As s approaches 3 from the right, t(s) increases without bound, and as s increases without bound, t(s) approaches 0. So, the range of the function is tǀt > Graphing y = and y = 5 on a graphing calculator with a ( - 3)( + 3) window that shows 3 to 10 on the -ais and 0 to 6 on the y-ais and then locating the point where the graphs intersect, you find that the kayaker s paddling speed was about.3 miles per hour. 16. In aviation, air speed refers to a plane s speed in still air. A small plane maintains a certain air speed when flying to and from a city that is 00 miles away. On the trip out, the plane flies against a wind, which has an average speed of 0 miles per hour. On the return trip, the plane flies with the wind. If the total flight time for the round trip is 3.5 hours, what is the plane s average air speed? To answer this question, use s the rational function t(s) = + =, where s is the air speed s - 0 s + 0 (s - 0)(s + 0) (in miles per hour) and t is the total flight time (in hours) for the round trip. PAGE 353 Image Credits: David Madison/Corbis In order for the plane to make progress when flying against the wind, the plane s air speed must be greater than the speed of the wind, so the domain of the function is sǀs > 0. As s approaches 0 from the right, t(s) increases without bound, and as s increases without bound, t(s) approaches 0. So, the range of the function is tǀt > and y = 3.5 on a graphing calculator with Graphing y = ( - 0)( + 0) a window that shows 0 to 500 on the -ais and 0 to 5 on the y-ais and then locating the point where the graphs intersect, you find that the plane s average air speed is about 35 miles per hour. Module 8 IN3_MNLESE389885_UM08L.indd Lesson 8/0/17 1:37 PM Graphing More Complicated Rational Functions 60

17 JOURNAL Have students outline the process of graphing more comple rational functions in their journals. Suggest that they incorporate several eamples. 17. Multiple Response Select the statements that apply to the rational function ƒ () = _ A. The function s domain is ǀ - and 3. B. The function s domain is ǀ - and -3. C. The function s range is yǀy 0. D. The function s range is yǀ- < y < +. E. The function s graph has vertical asymptotes at = - and = 3. F. The function s graph has a vertical asymptote at = -3 and a hole at =. G. The function s graph has a horizontal asymptote at y = 0. H. The function s graph has a horizontal asymptote at y = 1. H.O.T. Focus on Higher Order Thinking + a 18. Draw Conclusions For what value(s) of a does the graph of ƒ () = have a hole? Eplain. Then, for each value of a, state the domain and the range of ƒ() using interval notation. The graph of f() has a hole if the numerator is a factor of the denominator. Since = ( + 1) ( + 3), there are two values of a for which the graph of f() has a hole : a = 1 and a = 3. The domain of + 1 the function f() = is (-, -3) (-3, -1) (-1, + ) and the range is (-, 0) 1 (0, ) ( 1, + ). The domain of the function + 3 f() = is (-, -3) (-3, -1) (-1, + ), and the range is (-, - 1 ) 1 (-, 0 ) (0, + ). 19. Critique Reasoning A student claims that the functions ƒ () = and g () = + have different domains but identical ranges. Which part of the - 1 student s claim is correct, and which is false? Eplain. The domain of f() is ǀ -1_, while the domain of g() is ǀ -1_ and 1_, so the student s claim is correct with respect to the domains. The range of f() is yǀy -1, while the range of g() is yǀy -1 and y 0, so the student s claim is incorrect with respect to the ranges. Module 8 61 Lesson 61 Lesson 8.

18 Lesson Performance Task In professional baseball, the smallest allowable volume of a baseball is 9.06% of the largest allowable volume, and the range of allowable radii is 0.0 inch. a. Let r be the largest allowable radius (in inches) of a baseball. Write epressions, both in terms of r, for the largest allowable volume of the baseball and the smallest allowable volume of the baseball. (Use the formula for the volume of a sphere, V = 3 π r 3.) b. Write and simplify a function that gives the ratio R of the smallest allowable volume of a baseball to the largest allowable volume. c. Use a graphing calculator to graph the function from part b, and use the graph to find the smallest allowable radius and the largest allowable radius of a baseball. Round your answers to the nearest hundredth. a. Largest allowable volume: _ 3 π r 3 _ 3 Smallest allowable volume: π (r - 0.0) 3 _ 3 π (r - 0.0) 3 3 b. R (r) = (r - 0.0) = c. Graph y = _ 3 π r 3 3 ( - 0.0) 3 r 3 and y = on a graphing calculator, and find the -coordinate of the point where the graphs intersect. The graphs intersect at = 1.7 to the nearest hundredth, so the largest allowable radius is 1.7 inches. The smallest allowable radius is = 1.3 inches. PAGE 35 AVOID COMMON ERRORS Students may set the ratio of volumes to 9.06 instead of Have students eplain what a percentage is, and ask them what 9.06% means in this situation. Have students graph y = 9.06 and y = 0.906, and ask them which one gives a solution for this problem. INTEGRATE MATHEMATICAL PRACTICES Focus on Critical Thinking MP.3 Have students discuss the significance of the percentage 9.06% when solving this problem by graphing. Ask them to describe the behavior of the solution as the percentage varies. Ask them what will happen to the graph as the range of allowable radii gets smaller or larger. Module 8 6 Lesson EXTENSION ACTIVITY Have students discuss whether the solution graphs would be different if the baseball were a cube with a side of length. Have students determine the largest and smallest allowable lengths for the side of the cube when the smallest allowable volume is 9.06% of the largest allowable volume, and the range of allowable lengths is 0.0 inches. Have students compare the function for this situation to the one they der ived in the Performance Task. Scoring Rubric points: Student correctly solves the problem and eplains his/her reasoning. 1 point: Student shows good understanding of the problem but does not fully solve or eplain his/her reasoning. 0 points: Student does not demonstrate understanding of the problem. Graphing More Complicated Rational Functions 6