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1 Advanced Functions Page 1 of Reciprocal of a Linear Function Concepts Rational functions take the form andq ( ) 0. The reciprocal of a linear function has the form P( ) f ( ), where P () and Q () are both polynomial functions Q( ) 1 f ( ) kc The restriction on the domain of a reciprocal linear function can be determined by finding the value of c that makes the denominator equal to zero, that is,. k The vertical asymptote of a reciprocal linear function has an equation of the form The horizontal asymptote of a reciprocal linear function has equation y 0. c. k If k 0, the left branch of a reciprocal linear function has a negative, decreasing slope, and the right branch has a negative, increasing slope. If k 0, the left branch of a reciprocal linear function has a positive, increasing slope, and the right branch has a positive, decreasing slope. Eample 1: Consider the function f ( ) a) State the domain b) Describe the behaviour of the function near the vertical asymptote. c) Describe the end behaviour d) Determine the intercepts e) Determine the horizontal asymptote f) State the range h) Sketch a graph of the function () f f () f () f () RHHS Mathematics Department

2 Advanced Functions Page of Reciprocal of a Linear Function Solutions Eample : Consider the function a) State the domain 1 f ( ) 5 b) Describe the behaviour of the function near the vertical asymptote. c) Describe the end behaviour d) Determine the intercepts e) Determine the horizontal asymptote f) State the range h) Sketch a graph of the function 0. 4 f () 0. 4 f () f () f () Solutions RHHS Mathematics Department Homework p.15 #1-,5-10,1

3 Advanced Functions Page 1 of Reciprocal of a Quadratic Function Concepts Rational functions can be analysed using key features: asymptotes, intercepts, slope (positive or negative, increasing or decreasing), domain, range, and positive and negative intervals. Reciprocals of quadratic functions with two zeros have three parts, with the middle one reaching a maimum or minimum point. This point is equidistant from the two vertical asymptotes. The behaviour near asymptotes is similar to that of reciprocals of linear functions. All of the behaviours listed above can be predicted by analysing the roots of the quadratic relation in the denominator. Think about the discriminants of quadratics b 4ac,, 0 Eample 1: (Reciprocal of Quadratic with Two Vertical Asymptotes) Consider the function f ( ). 4 a) State the domain. b) Find the equations of the asymptotes. Describe the behaviour of the function near the asymptotes. c) Determine the - and y- intercepts. d) Sketch a graph of the function. e) State the range. d) f () RHHS Mathematics Department

4 Advanced Functions Page of Reciprocal of a Quadratic Function Eample : (Reciprocal of Quadratic with Two Vertical Asymptotes) Describe the increasing and decreasing intervals in the branches of the function graph the function. f () f ( ) 1. Then, 6 Eample : (Reciprocal of Quadratic with No Vertical Asymptote) 1 Analyse the key features of the function f ( ) and sketch its graph. 4 Eample 4: (Reciprocal of Quadratic with one Vertical Asymptote) Analyse the key features of the function 1 f ( ) and sketch its graph. f () RHHS Mathematics Department Homework: P.16 #-4, 5ace,6,7,8adgh,9-11

5 MHF 4U Page 1 of 5 Investigating Asymptotes (Vertical & Horizontal) Graph each of the following rational functions on the TI-8 calculator and complete the following chart. 1 y 4 y 6 y 1 4 y Domain : Vertical Asymptote (s) : Horizontal Asymptote (s) : Domain : Vertical Asymptote (s) : Horizontal Asymptote (s) : y 4 y 6 y y Domain : Vertical Asymptote (s) : Domain : Vertical Asymptote (s) : Horizontal Asymptote (s) : Horizontal Asymptote (s) : For rational functions h f, where the degree of h() is n and the degree of g () is m, g 1. Vertical asymptotes (may) occur where.. If n m, then the equation of the horizontal asymptote is.

6 MHF 4U Page of 5 Investigating Asymptotes (Vertical & Horizontal) Graph each of the following rational functions on the TI-8 calculator and complete the following chart. 1 y 1 y 6 y 4 1 y Domain : Vertical Asymptote (s) : Horizontal Asymptote (s) : Domain : Vertical Asymptote (s) : Horizontal Asymptote (s) : y 4 y 6 y 1 1 y Domain : Vertical Asymptote (s) : Domain : Vertical Asymptote (s) : Horizontal Asymptote (s) : For rational functions Horizontal Asymptote (s) : h f, where the degree of h() is n and the degree of g () is m, g. If n m, then the equation of the horizontal asymptote can be found by.

7 MHF 4U Page of 5 Investigating Asymptotes (Vertical & Horizontal) Graph each of the following rational functions on the TI-8 calculator and complete the following chart. y y 6 y 1 y Domain : Vertical Asymptote (s) : Horizontal Asymptote (s) : Domain : Vertical Asymptote (s) : Horizontal Asymptote (s) : y y 6 y y Domain : Vertical Asymptote (s) : Domain : Vertical Asymptote (s) : Horizontal Asymptote (s) : For rational functions Horizontal Asymptote (s) : h f, where the degree of h() is n and the degree of g () is m, g 4. If n m, then there is a horizontal asymptote (true or false). 5. If n m, and n m 1, then there is asymptote. This type of asymptote is a line that the graph approaches that is neither vertical nor horizontal.

8 MHF 4U Page 4 of 5 Investigating Asymptotes (Vertical & Horizontal) For rational functions h f, where the degree of h() is n and the degree of g () is m, g 1. Vertical asymptotes (may) occur where the function is undefined, that is, where g ( ) 0.. If n m, then there is a horizontal asymptote at y 0. n coefficien t of. If n m, then there is a horizontal asymptote at y. m coefficien t of 4. If n m, then there is no horizontal asymptote. 5. If n m, and n m 1, then there is an oblique asymptote. An oblique asymptote is a line that the graph approaches that is neither vertical nor horizontal. 1. For each of the following, state the domain, and the equations of any vertical or horizontal asymptotes. Also state if an oblique asymptote eists. a) y b) 1 y c) 1 y 1 d) y e) y f) 4 y 5 6 g) y h) y i) y a) Simplify the rational epression. Don t forget to state restrictions. 6 b) Eplain the difference between the graph of y and that of 6 1 y.. For each case, create a function that has a graph with the given features. a) a vertical asymptote at 1 and a horizontal asymptote at y 0. b) Two vertical asymptotes at 1 and and a horizontal asymptote at y 1.

9 MHF 4U Page 5 of 5 Investigating Asymptotes (Vertical & Horizontal) Answers 1a) D: 1; VA: 1 ; HA: y 1 b) D: 1; VA: 1 ; OA: y 1 c) D: 1 ; VA: 1 ; HA: y ; VA: none ; HA: y 0 e) D: ; VA: ; HA: y 0 d) D: R ; VA: 1,, ; HA: y 0 g) D:, ; VA:, ; HA: y 0 f) D: 1, h) D: ; VA: ; OA: y 1 i) D: 1, 4 ; VA: 1, 1, 4 ; HA: y 0 ) y 6 1 y y 6 ( )( ), VA at Discontinue at 1 y VA at ) Answers may vary a) b y, a b R a 1, b) a b c y, a, b, c R a 1 ( )

10 MHF 4U Page 1 of Investigating Behaviour near the Asymptotes In order to sketch the graphs of rational functions, we need to know how the graphs behave near the asymptotes. We can do this by selecting appropriate values for and substituting these into the function. That is, create a simple table of values. Eample 1: Create a sketch for f ( ). 1 Step 1: State the vertical and horizontal asymptotes if they eist. Step : To determine the behaviour of the function near a vertical asymptote, select values that are very close to the asymptote on either side. To determine the behaviour of the function near a horizontal asymptote, select very small values (approaching ) and very large values (approaching ) f ( ) 1 This information can be summarized by stating that: As approaches As approaches, f approaches the horizontal asymptote, y =, from. f approaches the horizontal asymptote, y =, from., As approaches the vertical asymptote, = 1, from the left, As approaches the vertical asymptote, = 1, from the right, Step : Determine the and y intercepts. f approaches. f approaches.

11 MHF 4U Page of Investigating Behaviour near the Asymptotes Now you try! Create sketches for f ( ) and 5 6 f ( ) in your notebooks. 6 8 Practice Questions: 1. State the equations of the vertical and horizontal asymptotes of the curves shown. a) b). Under what condition does a rational function have a vertical asymptote?. Under what condition does a rational function have a horizontal asymptote? 4. For each of the following rational functions i) Determine if there are any vertical and / or horizontal asymptotes ii) Determine the behaviour of the function near the asymptotes. (Use a table of values) iii) Determine the and y intercepts. iv) Create a sketch of the graph using the above information. a) y b) 5 y c) y 7 d) 4 y e) 7 1 y f) 4 y 1 g) 5 y h) y i) 1 y 6

13 Rational Functions Rational functions take the form f P Q(), where P() and Q() are both polynomial functions and Q( ) 0. Rational functions can be analyzed using key features: asymptotes, intercepts, slope (positive or negative, increasing or decreasing), domain, range, and positive and negative intervals. Linear over Linear 017 E. Choi MHF4U - All Rights Reserved

Asymptote A line that a curve approaches more and more closely but does not intersect. The line may be a horizontal asymptote, a vertical asymptote, or a linear oblique asymptote.

14 Asymptote A line that a curve approaches more and more closely but does not intersect. The line may be a horizontal asymptote, a vertical asymptote, or a linear oblique asymptote. Vertical Asymptote Vertical asymptotes are vertical lines which correspond to the zeroes of the denominator of a rational function. Horizontal Asymptote horizontal asymptotes are horizontal lines that the graph of the function approaches as tends to + or. The graph of a function with a horizontal (y = 0), vertical ( = 0), and oblique asymptote (purple line, given by y=). Oblique (Slant) Asymptote An oblique asymptote has a slope that is non-zero but finite, such that the graph of the function approaches it as tends to + or. Reciprocal of Linear & Quadratic Functions 017 E. Choi MHF4U - All Rights Reserved

15 Linear over Linear A rational function of the form features: f a b c d has the following key The equation of the vertical asymptote can be found by setting the denominator equal to zero and solving for, provided the numerator does not have the same zero. The equation of the horizontal asymptote can be found by dividing each term in both the numerator and the denominator by and investigating the behaviour of the function as. The coefficient b acts to stretch the curve but has no effect on the asymptotes, domain, or range. The coefficient d shifts the vertical asymptote. The two branches of the graph of the function are equidistant from the point of intersection of the vertical and horizontal asymptotes. Horizontal asymptotes at a y c Linear over Linear 017 E. Choi MHF4U - All Rights Reserved

16 Key features of a Function (MHF) - Domain & Range - Asymptotes (V.A., H.A. or O.A.) - Behaviours near the asymptotes - Intercepts - Hole(s) - Cross Over Linear over Linear 017 E. Choi MHF4U - All Rights Reserved

18 Eample 1: Linear over Linear Determine the key features of Domain: {, R} VA: HA: y 1 -int: 0 y-int: y 0 f ( ) and use them to sketch the function. VA: f () 0 0 HA: y 1 f ( ) 1 1 above below Range: { y y 1, y R} Linear over Linear 017 E. Choi MHF4U - All Rights Reserved

19 Eample : Linear over Linear g( ) 5 {.5, R a) Compare the graphs of f ( ), and g( ). 1 f ( ) 5 Domain: {.5, R} VA:. 5 HA: y int: 1 y-int: y 1/ f () Above Below Domain: } VA:. 5 HA: y int: 5 y-int: y f () Above Below 1 f ( ) 5 5 g( ) 5 Range: { y y 0.5, y R} Range: { y y 0.5, y R} Linear over Linear 017 E. Choi MHF4U - All Rights Reserved

20 Eample : Linear over Linear b) Eplain the effects of the coefficient b in y a b c d The constant b in the a b y c d has an effect that is similar to both vertical and a horizontal stretch factor but does not affect the asymptotes, domain or range. Linear over Linear 017 E. Choi MHF4U - All Rights Reserved

24 Holes Recall if the degree of the numerator is smaller than the degree of the denominator in a rational function, then there is a horizontal asymptote at y = 0 It is possible that the function crosses over a horizontal or oblique asymptote at a certain value of. To determine if the graph crosses over the asymptote (H.A. or O.A.) or not, Let = H.A. or O.A. and solve the equation: If LS RS the graph does not cross over the H.A. or O.S., else, the graph crosses over the asymptote at where a is the intersection point of f() & H.A. or O.A A hole can be eisted in a rational function if ( a, f ( a)) will be a hole. f m n( k a ) ka, then Investigating Cross-Over & Holes 017 E. Choi MHF4U - All Rights Reserved

25 Key features of a Function (MHF) - Domain & Range - Asymptotes (V.A., H.A. or O.A.) - Behaviours near the asymptotes - Intercepts - Hole(s) - Cross Over Investigating Cross-Over & Holes 017 E. Choi MHF4U - All Rights Reserved

26 Eample 1: Investigating Hole For the following function, state the domain, and the equations of any vertical or horizontal asymptotes. Determine the behaviors near the asymptotes, check if there is a cross-over or a hole in the graph. f f 1 Domain: { 0, R} 1 Hole at: 0 0, VA: None HA: y 0 Draw y y 1 the reciprocal then do 1, 1, 1 Notice all reciprocal values will be positive and Range: { y 0 y, y R} Investigating Cross-Over & Holes HA: y f () Above above 017 E. Choi MHF4U - All Rights Reserved

27 Eample : Investigating Crossover & Hole For the following function, state the domain, and the equations of any vertical or horizontal asymptotes. Determine the behaviors near the asymptotes, check if there is a cross-over or a hole in the graph. f 4 ( )( ) ( 1)( ) 1 Domain: {, 1, R} 4 Hole at:, VA: 1 HA: y 0 Cross-over: Let f() = HA f () crosses over at -int: y-int: y Investigating Cross-Over & Holes f () HA: y 0 VA: 1 VA: E. Choi MHF4U - All Rights Reserved

30 Advanced Functions Page 1 of 4 Sketching Rational Functions Eample 1: Testing End Behaviours near HORIZONTAL ASYMPTOTES when it is not ZERO! Sketch the functions 9 a) f Testing values b) f 9 Testing values RHHS Mathematics Department

31 Advanced Functions Page of 4 Sketching Rational Functions Eample Sketch y State the domain Check if there are any holes in the graph. Asymptotes Check the behaviours near the asymptotes Cross-over Intercepts Sketch Determine the intervals of f 0 & f 0 (Optional) Testing values y RHHS Mathematics Department

32 Advanced Functions Page of 4 Sketching Rational Functions h Steps to Graphing Rational Functions f : g 1. Simplify the function to determine if there are any holes in the graph.. Determine if there are any vertical or horizontal asymptotes.. Determine the behaviour of f() near the vertical and horizontal asymptotes by creating a table of values 4. Determine if the graph crosses over its horizontal asymptote. 5. Determine the equation of the oblique asymptote, if it eists. If it eists, determine the behaviour of the f() near the asymptote, and if the f() crosses over the oblique asymptote. 6. Determine the and y intercepts. Notice the added step for the oblique asymptote. Eercise 1. Create sketches for each of the following by following the algorithm above. Also, when you are finished the sketch, determine (if possible, you may need to approimate) the domain and range, and the intervals of increasing and decreasing. a) y b) y 1 c) 6 y d) 4 y e) 6 y f) y 6 g) 5 y h) 0 5 y 6. The model for the concentration y of a drug in the bloodstream, hours after it is taken orally, is 7 f ( ). a) What is the domain of f() in this contet? b) What is the range of f()? (optional) c) Using the graph, describe what happens to the concentration of the drug over 4 hours. Does the model seem reasonable? RHHS Mathematics Department

33 Advanced Functions Page 4 of 4 Sketching Rational Functions Answers 1a) b) c) d) VA =, 8 = -1 = = - HA y = 0 y = y = 1 y = 0 OA / / / / Cross over / / / = 0 -int / / 1.5 y-int 1/ domain, 8 1 Hole / / = - / e) f) g) h) VA / = - = 5 =, HA / / / / OA y = + y 1 y 5 5 y 1 Cross over / / / -6/11 -int - - & 0 & -/ y-int - -/5 0 domain R 5, hole = / / / a) 0 b) 0 y.5 c) RHHS Mathematics Department

34 MHF 4U Page 1 of Investigating Oblique Asymptotes h Recall that for rational functions f, where the degree of h() is n and the degree of g () is g m, if n m, and n m 1, then there is an oblique asymptote. An oblique asymptote is a line that the graph approaches that is neither vertical nor horizontal. We will now investigate how to determine the equation of the oblique asymptote and the behaviour of the graph near the asymptote. Eample 1: Sketching Rational Functions 6 Graph f. Testing Values Difference function f () Behaviour Testing Values Difference function f () Behaviour

35 MHF 4U Page of Investigating Oblique Asymptotes Eample Sketch y 4 6 Testing Values Difference function f () Behaviour Testing Values Difference function f ()

36 MHF 4U Page of Investigating Oblique Asymptotes Eercise 1. Determine the equation of the oblique asymptote for each of the following: a) c) y b) y 7 y d) y Create sketches for each of the following by determining all possible asymptotes and the behaviour of the graph near the asymptote. Also determine the and y intercepts and the domain. a) 1 y b) 1 4 y 4 c) 9 14 y d) 4 17 y e) 1 y f) y 9 Answers 1a) y 4 b) y c) y d) y 5 ) a) VA HA OA Cross over -int y-int Domain -1 / y 1 / b) 4 / y 4 /, c) d) 1, / y /, / y 7 / / 1, 17/ e) 0, - / y 1/4 1 / 0, f) -/ / y / /

37 Advanced Functions Page 1 of 4 Word Problems with Rational Functions Eample 1: Problem Solving with Rational equations David s rowing speed is 10 km/h at still water, if it took him 0 hours to row 5 km down a river and 5 km back. Find the speed of the current. Recall: Rate of Change (R) In general, Rate of change of F with respect to time t at t = a is: (1) () F F( t) F( a) Find the slopeof secant : ms (in simplified form) t t a Let t a 0.001, find m and m, ( m m ) () Then m s1 m t m s s1 R m s t s1 s can be determined. Eample : Rate of change to the slope of a tangent a) Estimate the slope of the tangent to the graph of f ( ) at the point where = -5. b) Determine the equation of the tangent line and sketch the function and the tangent line. RHHS Mathematics Department

38 Advanced Functions Page of 4 Word Problems with Rational Functions Eample : Rate of change to population The frog population in a newly created conservation area can be predicted over time by the model 500t p( t) 50, where p represents the population size and t is the time in years since the opening of the 5 t conservation area. Estimate the instantaneous rate of change in population after years. Eercise 1) On the 4-km go-cart course at Sportsworld, Arshia drives 0.4 km/h faster than Sarah, but she has engine trouble part way around the course and has to stop to get the go-cart fied. This stop costs Arshia one-half hour, and so she arrives 15 min after Sarah at the end of the course. How fast did each girl drive and how long did each girl take to finish the course? (1 decimal place) ) Rowing at 8 km/h, in still water, Rima and Bhanu take 16 h to row 9 km down and a river and 9 km back. Find the speed of the current. ) A river flows at km/h, and John takes 6 hours to row 16 km up the river and 16 km back. How fast did he row? 4) Jaime bought a case of concert T-shirts for $450. She kept two for herself and sold the rest for $560, making a profit of $10 on each shirt. How many shirts were in the case. RHHS Mathematics Department

39 Advanced Functions Page of 4 Word Problems with Rational Functions 5) Stuart agrees to a house-painting job for $900. He takes 4 days longer than epected, and he has earned $18.75 less per day than epected. In how many days did he epect to complete the house? 6) A grade 11 class, on a field trip to Montreal, had lunch in a restaurant. The bill came to $9.5. Four students had birthdays that day, and it was agreed that these four should not have to pay for lunch. The other students had to pay $1 more than if all the students had paid. How many students had lunch? 7) Eplain the steps for adding and subtracting rational epressions. Use an eample and eplain each step fully as 1 1 8) Find A and B such that 1 9) Find A, B and C such that 1 1 A A 1 B. B C. 10) Determine the slope of the line that is tangent to the graph of each function at the given point. Then find the equation of the tangent line. a) 5 f ( ), where b) 6 f ( ), where 7 5 c) 6 f ( ), where d) 5 5 f ( ), where 4 6 s s 10s 11) The electrical current in a circuit varies with time according to c, where the current, c, 4 s 8s is in amperes, and time s is in seconds. Find the average rate of change from 0.75 seconds to 1.5 seconds, and find the instantaneous rate of change at 1.5 seconds. Identify any vertical asymptotes. 1) As you get farther from Earth s surface, gravity has less effect on you. For this reason, you actually 6400(55) weigh less at high altitudes. A person who weighs 55 kg can use the function, W ( h) to find h their weight, W in kgs, at a specific height, h in feet above sea level, above the Earth s surface. Find the average rate of change from heights of 750 ft to 100 ft above sea level, and find the instantaneous rate of change at 100 ft above sea level. t 8t 5 1) The average speed of a certain particle in meters per second is given by the equation S ( t). t Find the average rate of change as time changes from 0.5 seconds to.5 seconds, and find the instantaneous rate of change at.5 seconds. RHHS Mathematics Department

40 Advanced Functions Page 4 of 4 Word Problems with Rational Functions 14) After you eat something that contains sugar, the ph or acid level in your mouth changes. This can be modeled by the function 0.4m L ( m) 6.5, where L is the ph level and m is the number of minutes m 6 that have elapsed since eating. Find the average rate of change from 1.5 minutes to minutes, and find the instantaneous rate of change at minutes. 15) Eplain why the line y is an asymptote for the graph of y ) Eplain why the line y is an asymptote for the graph of y. Answers 1) Arshia: 8.4 km/h (5.5 h) Sarah: 8.0 km/h (5.5 h) ) 5 km/h ) 6 km/h 4) 18 t-shirts 5) 1 days 6) students 7) i)factor the numerators and denominators for both epressions. ii) Epress the sum or difference of the epressions with a common denominator. iii) Simplify. Eplanation 1 of steps may vary. 8) A =, B = -7 9) A= B = C = 9 9 Slopes 10a. m = 11 b. m = 4 15, 49 Equation of tangent line y y 4 4 c. m = - 46 y d. m = 5 4 Average Rate of Change y Instantaneous Rate of Change amp/sec amp/sec kg/ft kg/ft m/sec. 06 m/sec ph/min 0. 7 ph/min & 16. has an oblique asymptote because the degree of the numerator is eactly one greater that that of the denominator. The equation of that asymptote is found through synthetic division (the quotient). RHHS Mathematics Department

MHF 4U Unit 2 Rational Functions Outline

MHF 4U Unit 2 Rational Functions Outline MHF 4U Unit Rational Functions Outline Day 1 Lesson Title Specific Epectations Rational Functions and Their Essential Characteristics C.1,.,.3 (Lesson Included) Rational Functions and Their Essential Characteristics