Using a Table of Values to Sketch the Graph of a Polynomial Function

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1 A point where the graph changes from decreasing to increasing is called a local minimum point. The -value of this point is less than those of neighbouring points. An inspection of the graphs of polnomial functions in this lesson and in Lesson 1.3 illustrates that the graph of a polnomial function of degree n can have at most n -intercepts and at most (n 1) local maimum or minimum points. For eample, the graph of a cubic function can have at most 3 -intercepts and at most local maimum or local minimum points. To sketch the graph of a polnomial function, use a table of values and knowledge of the end behaviour of its graph. f() 0 local maimum points local minimum point Eample 1 Using a Table of Values to Sketch the Graph of a Polnomial Function Remember that a sketch does not have to be accurate. Sketch the graph of each polnomial function. a) f() = b) g() = SOLUTION a) The equation represents an odd-degree polnomial function. Since the leading coefficient is negative, as : - q, the graph rises and as : q, the graph falls. The constant term is 4, so the -intercept is 4. Use a table of values to create the graph. Check Your Understanding 1. Sketch the graph of each polnomial function. a) f() 3 3 b) g() f() f() b) The equation represents an even-degree polnomial function. Since the leading coefficient is positive, the graph opens up. The constant term is 0, so the -intercept is 0. Use a table of values to create the graph. P DO NOT COPY. 1.4 Relating Polnomial Functions and Equations 41

2 4 g() g() Eample Using Intercepts to Sketch the Graph of a Polnomial Function Check Your Understanding. Sketch the graph of the polnomial function: f() 3 6 Sketch the graph of the polnomial function: f() = SOLUTION Factor the polnomial. Use the factor theorem. List the factors of the constant term, 6: 1, 1,,, 3, 3, 6, 6 Use mental math. When 1, f(1) 0 So, 1 is a factor of Divide to determine the other factor So, = ( - 1)( ) Factor the cubic polnomial. Use the factor theorem. Let P() Use mental math. When 1, P(1) Z 0 When 1, P( 1) Z 0 So, neither 1 nor 1 is a factor. Tr : P() () 3 () 13() So, is a factor of Divide to determine the other factor = ( - 1)( - )( + 5-3) 4 Chapter 1: Polnomial Epressions and Functions DO NOT COPY. P

3 Factor the trinomial: 5 3 ( 3)( 1) So, f() ( 1)( )( 3)( 1) Determine the zeros of f(). Let f() 0. 0 ( 1)( )( 3)( 1) Solve the equation. So, 1 0 or 0 or 3 0 or The zeros are: 1,, 3, So, the -intercepts of the graph are: 1,, 3, and Plot points at the intercepts. The equation has degree 4, so it is an even-degree polnomial function. The leading coefficient is positive, so the graph opens up. The constant term is 6, so the -intercept is 6. Draw a smooth curve through the points, beginning at the top left and ending at the top right f() THINK FURTHER Suppose the graph of a polnomial function is smmetrical about the -ais. What do ou know about the function? In Eample, the equation 0 = ( - 1)( - )( + 3)( - 1) is the factored form of a polnomial equation. This Eample illustrates that when the equation of a polnomial function is factorable, the -intercepts of its graph can be determined b factoring. The -intercepts are the zeros of the polnomial function because the are the values of when the function is 0. The zeros of the function are the roots of the related polnomial equation. P DO NOT COPY. 1.4 Relating Polnomial Functions and Equations 43

4 A polnomial equation ma have a repeated root. Here are two eamples: 1 0 can be written as ( 1) 0. The equation has root: 1 Theeponentofthefactor( 1) is, so 1 is a root with multiplicit. The related function has a zero of multiplicit can be written as ( 1) 3 0. The equation has root: 1 Theeponentofthefactor( 1) is 3, so 1 is a root with multiplicit 3. The related function has a zero of multiplicit 3. The behaviour of the graph at a zero depends on its multiplicit. Here are the graphs of f() ( 1) and g() ( 1) 3. This graph has a zero of multiplicit. This graph has a zero of multiplicit f() ( 1) 0 4 g() ( 1) 3 Both graphs have -intercept 1. The graph of f() ( 1) touches the -ais at 1, but does not cross the ais at this point. The graph of g() ( 1) 3 crosses the -ais at 1. This difference in behaviour is related to the multiplicit of the zero. In general, when a zero has even multiplicit, the graph touches the -ais at the related -intercept, but does not cross it; we sa that the graph just touches the -ais. When a zero has odd multiplicit, the graph crosses the -ais at the related -intercept. 44 Chapter 1: Polnomial Epressions and Functions DO NOT COPY. P

5 Eample 3 Using the Multiplicit of a Zero to Graph a Polnomial Function Sketch the graph of each polnomial function. a) f() = ( - 1) ( + 3) b) g() = -( + ) 3 ( - 1) SOLUTION a) f() = ( - 1) ( + 3) f() ( 1) ( 3) To determine the zeros, solve f () 0. 0 = ( - 1) ( + 3) The roots of the equation are 1 and 3. So, the zeros of the function are 1 and 3. 9 The zero 1 has multiplicit. The zero 3 has multiplicit So, the graph just touches the -ais at 1 and at 3. The equation has degree 4, so it is an even-degree polnomial function. The leading coefficient is positive, so the graph opens up. The -intercept is: ( 1) (3) 9 Plot points at the intercepts, then draw a smooth curve that rises to the left and rises to the right. Check Your Understanding 3. Sketch the graph of each polnomial function. a) f() ( 1) 4 ( ) b) g() ( 1) 3 ( 3) b) g() = -( + ) 3 ( - 1) To determine the zeros, solve g() 0. g() ( ) 3 ( 1) and 1. So, the zeros of the function are and 1. The zero has multiplicit 3. The zero 1 has multiplicit. So, the graph crosses the -ais at, and just touches the -ais at 1. The equation has degree 5, so it is an odddegree polnomial function. 0 = -( + ) 3 ( - 1) The roots of the equation are The leading coefficient is negative, so as : - q, the graph rises and as : q, the graph falls. The -intercept is: () 3 ( 1) 8 Plot points at the intercepts, then draw a smooth curve that rises to the left and falls to the right. P DO NOT COPY. 1.4 Relating Polnomial Functions and Equations 45

6 Discuss the Ideas 1. Suppose a is a factor of a polnomial. What else do ou know about the corresponding polnomial equation and the graph of the corresponding polnomial function?. How does the multiplicit of a zero of a polnomial function affect its graph? Eercises A 3. Which functions are polnomial functions? Justif our choices. a) f() = - b) g() = c) h() = d) k() = e) p() = Chapter 1: Polnomial Epressions and Functions DO NOT COPY. P

7 4. Which graphs are graphs of polnomial functions? Justif our answers. a) b) 4 4 f() g() 0 4 c) d) h() k() Complete the table below. The first row has been done for ou. Odd or Even Leading -intercept Equation Degree Degree Tpe coefficient of its graph f() 3 1 even quadratic 3 1 a) g() b) h() c) k() P DO NOT COPY. 1.4 Relating Polnomial Functions and Equations 47

8 B 6. Use a table of values to sketch the graph of each polnomial function. a) f() = b) g() = Chapter 1: Polnomial Epressions and Functions DO NOT COPY. P

9 7. Use intercepts to sketch the graph of each polnomial function. a) f() = b) h() = P DO NOT COPY. 1.4 Relating Polnomial Functions and Equations 49

10 8. Identif the graph that corresponds to each function. Justif our choices. a) f() = b) g() = c) h() = d) k() = i) Graph A ii) Graph B iii) Graph C iv) Graph D Determine the zeros of each polnomial function. State the multiplicit of each zero. How does the graph of each function behave at the related -intercepts? Use graphing technolog to check. a) f() = ( + 3) 3 b) g() = ( - ) ( + 3) 50 Chapter 1: Polnomial Epressions and Functions DO NOT COPY. P

11 c) h() = ( - 1) 4 ( + 1) d) j() = ( - 4) 3 ( + 1) 10. Sketch the graph of this polnomial function. h() = ( + 1) ( - 1)( + ) 11. a) Write an equation in standard form for each polnomial function described below. i) a cubic function with zeros 3, 3, and 0 ii) a quartic function with zeros and 1 of multiplicit 1, and a zero of multiplicit P DO NOT COPY. 1.4 Relating Polnomial Functions and Equations 51

12 b) Is there more than one possible equation for each function in part a? Eplain. 1. Sketch a possible graph of each polnomial function. a) cubic function; leading coefficient is positive; zero of 4 has multiplicit 3 b) quintic function; leading coefficient is positive; zero of 3 has multiplicit ; zero of has multiplicit ; zero of 4 has multiplicit 1 c) quartic function; leading coefficient is negative; zero of 4 has multiplicit 3; zero of 3 has multiplicit A cubic function has zeros, 3, and 1. The -intercept of its graph is 18. Sketch the graph, then determine an equation of the function. 5 Chapter 1: Polnomial Epressions and Functions DO NOT COPY. P

13 C 14. Investigate pairs of graphs of even-degree polnomial functions of the form shown below for different values of the variables a, b, c, and d H. Describe one graph as a transformation image of the other graph. What conclusions can ou make? h() ( a)( b)( c)( d) and k() ( a)( b)( c)( d) 15. Investigate pairs of graphs of odd-degree polnomial functions of the form shown below for different values of the variables a, b, c, d, and e H. Describe one graph as a transformation image of the other graph. What conclusions can ou make? h() ( a)( b)( c)( d)( e) and k() ( a)( b)( c)( d)( e) P DO NOT COPY. 1.4 Relating Polnomial Functions and Equations 53

14 16. Each of the functions f() = and g() = has one zero of multiplicit and one different zero. Use onl this information to determine the values of b for which the function h() = b has each number of zeros. Eplain our strateg. a) 3 different zeros b) 1 zero of multiplicit 1 and no other zeros Multiple-Choice Questions 1. The graph of a polnomial function rises to the left and falls to the right. Which statement describes the function? A. The function has an odd degree and its leading coefficient is positive. B. The function has an even degree and its leading coefficient is positive. C. The function has an odd degree and its leading coefficient is negative. D. The function has an even degree and its leading coefficient is negative. 54 Chapter 1: Polnomial Epressions and Functions DO NOT COPY. P

15 . The graph of a polnomial function of degree 4 is shown. Which statements are true? I. The function has an even degree. II. The function has a zero of multiplicit 3. III. The -intercept is negative. IV. The function is a quartic function. A. I, II, IV B. I, III, IV C. I, II D. I, III 0 f() Stud Note Generalize the rules for graphing polnomial functions of odd or even degree. ANSWERS Eercises 3. a) no b) es c) es d) no e) no 4. a) no b) es c) no d) es 5. a) 5; odd; quintic; 1; 0 b) 3; odd; cubic; 3; 7 c) 4; even; quartic; 1; 5 8. a) B b) D c) A d) C 9. a) 3, multiplicit 3 b), multiplicit ; 3, multiplicit c) 1, multiplicit 4; 0.5, multiplicit 1 d) 4, multiplicit 3; 1, multiplicit 11. b) es 13. = a) -54 < b < 54 b) b > 54 or b < -54 Multiple Choice 1. C. B P DO NOT COPY. 1.4 Relating Polnomial Functions and Equations 55

16 Additional Workspace 56 Chapter 1: Polnomial Epressions and Functions DO NOT COPY. P

17 1.5 Modelling and Solving Problems with Polnomial Functions FOCUS Analze the graph of a polnomial function to solve a problem. Get Started This graph represents the path of a ball kicked into the air. What was the maimum height of the ball? How far had the ball travelled horizontall when it reached this height? Vertical distance (m) f() Horizontal distance (m) What total horizontal distance did the ball travel? Construct Understanding Use graphing technolog. A certain airline s regulations state that the sum of the length, width, and depth of a piece of carr-on luggage must not eceed 100 cm. Several models of carr-on luggage have length 10 cm greater than their depth. Write a polnomial function to represent the volume of this luggage. Graph the function. To the nearest cubic centimetre, what is the maimum possible volume of this luggage? What are its dimensions to the nearest tenth of a centimetre? What assumptions did ou make? P DO NOT COPY. 1.5 Modelling and Solving Problems with Polnomial Functions 57

18 THINK FURTHER In the problem above, suppose the sum of the length, width, and depth was halved. Would the maimum volume also be halved? Eplain. Polnomial functions and their graphs can be used to solve real-world problems. Eample 1 Using the Graph of a Cubic Function to Solve a Problem Check Your Understanding 1. A piece of cardboard 30 cm long and 5 cm wide is used to make a bo with no lid. Equal squares of side length centimetres are cut from the corners and the sides are folded up. a) Write a polnomial function to represent the volume, V, of the bo in terms of. A piece of sheet metal 5 cm long and 18 cm wide is used to make a bo with no lid. Equal squares of side length centimetres are cut from the corners and the sides are folded up. a) Write a polnomial function to represent the volume, V,of the bo in terms of. b) Graph the function. What is the domain? c) To the nearest cubic centimetre, what is the maimum volume of the bo? What size of square should be cut out to create a bo with this volume? To the nearest tenth of a centimetre, what are the dimensions of the bo? 58 Chapter 1: Polnomial Epressions and Functions DO NOT COPY. P

19 SOLUTION a) Sketch a diagram. The bo has height centimetres, width (18 ) centimetres, and length (5 ) centimetres. So, a polnomial function that represents the volume, V, of the bo is: V() (18 )(5 ) 18 cm (18 ) cm (5 ) cm 5 cm cm cm (18 ) cm b) Graph the function. What is the domain? c) To the nearest cubic centimetre, what is the maimum volume of the bo? What size of square should be cut out to create a bo with this volume? To the nearest tenth of a centimetre, what are the dimensions of the bo? (5 ) cm b) Use a graphing calculator. Enter the equation: (18 )(5 ) The dimensions of the bo are positive. The sheet metal has width 18 cm. So, the side length of a square cut from each corner must be less than 18 cm, or 9 cm. So, the domain is: 0 9 Use these window settings: c) To determine the maimum volume, press: r, then use the arrow kes to move the cursor to determine the approimate coordinates of the local maimum point. So, the maimum volume of the bo is approimatel 693 cm 3. This occurs when each square that is cut out has a side length of approimatel 3.4 cm. The approimate dimensions of the bo are: Height: 3.4 cm Width: 18 cm (3.4 cm) 11. cm Length: 5 cm (3.4 cm) 18. cm So, for maimum volume, the dimensions of the bo are approimatel 18. cm b 11. cm b 3.4 cm. P DO NOT COPY. 1.5 Modelling and Solving Problems with Polnomial Functions 59

20 Eample Using the Graph of a Quartic Function to Solve a Problem Check Your Underanding. Clara and 3 friends were born on March 11. Lesle is 5 ears ounger than Clara. Mike is ears ounger than Clara. Thomas is 3 ears older than Clara. On March 11, 011, the product of their ages was greater than the sum of their ages. How old was Clara and each friend on that da? Leo and 3 friends each have birthdas on December 13. Sanda is 3 ears ounger than Leo. Leo is 4 ears ounger than Vince. Hunter is 1 ear older than Leo. On December 13, 010, the product of their ages was greater than the sum of their ages. How old was Leo and each friend on that da? SOLUTION Let Leo s age in ears be. Then, Sanda s age in ears is 3, Vince s age in ears is 4, and Hunter s age in ears is 1. The sum of their ages is: ( 3) ( 4) ( 1) 4 Sum of ages product of ages So, = ( - 3)( + 4)( + 1) 0 = ( - 3)( + 4)( + 1) Enter the equation ( - 3)( + 4)( + 1) into a graphing calculator. Graph the function using these window settings: Since age cannot be negative, the positive -intercept of the graph represents Leo s age. To determine the positive -intercept, press: rá, then use the arrow kes. The -intercept is 15. So, Leo was 15 ears old on that da. The ages of the other friends, in ears, were: Sanda: Vince: Hunter: On December 13, 010, Leo was 15, Sanda was 1, Vince was 19, and Hunter was Chapter 1: Polnomial Epressions and Functions DO NOT COPY. P

21 Discuss the Ideas 1. Wh might the domain and range of a polnomial function differ from the domain and range of a function that models a situation using the same polnomial?. When ou use a graphing calculator to graph a polnomial function, how do ou determine the window settings? Eercises Use technolog to graph the functions. A 3. A bo has length ( ) units, width ( 5) units, and height ( 3) units. Write a polnomial function to represent its volume, V, in terms of. P DO NOT COPY. 1.5 Modelling and Solving Problems with Polnomial Functions 61

22 4. a) Write a polnomial function to represent the volume, V, of this square prism in terms of. cm ( 6) cm cm b) Graph the function. Sketch the graph. Use the graph to determine the dimensions of the prism if its volume is 81 cm 3. B 5. A piece of cardboard 36 cm long and 8 cm wide is used to make an open bo. Equal squares of side length centimetres are cut from the corners and the sides are folded up. a) Write epressions to represent the length, width, and height of the bo in terms of. b) Write a polnomial function to represent the volume of the bo in terms of. c) Graph the function. Sketch the graph. What is the domain? 6 Chapter 1: Polnomial Epressions and Functions DO NOT COPY. P

23 d) What is the maimum volume of the bo? What is the side length of the square that should be cut out to create a bo with this volume? Give our answers to the nearest tenth. 6. The volume, in cubic centimetres, of an epandable gift bo can be represented b the polnomial function V() = The height of the bo in centimetres is 40. Assume the length is greater than the width. a) Determine binomial epressions for the length and width of the bo in terms of. b) Graph the function. Sketch the graph. What do the -intercepts represent? c) To the nearest cubic centimetre, what is the maimum volume of the gift bo? P DO NOT COPY. 1.5 Modelling and Solving Problems with Polnomial Functions 63

24 7. Fred and Ted are twins. The were born 3 ears after their older sister, Bethan. This ear, the product of their three ages is 576 greater than the sum of their ages. How old are the twins? 8. Ann, Stan, and Fran are triplets. The were born 4 ears before their sister, Kim. This ear, the product of their four ages is greater than the sum of their ages. How old is Kim? 9. A carton of juice has dimensions 6.4 cm b 3.8 cm b 10.9 cm. The manufacturer wants to design a bo with double the capacit b increasing each dimension b centimetres. To the nearest tenth of a centimetre, what are the dimensions of the larger carton? 64 Chapter 1: Polnomial Epressions and Functions DO NOT COPY. P

25 10. A package sent b a courier has the shape of a square prism. The sum of the length of the prism and the perimeter of its base is 100 cm. a) Write a polnomial function to represent the volume V of the package in terms of. cm cm cm b) Graph the function. Sketch the graph. c) To the nearest tenth of a centimetre, what are the dimensions of the package for which its volume is maimized? P DO NOT COPY. 1.5 Modelling and Solving Problems with Polnomial Functions 65

26 C 11. An open bo with locking tabs is to be made from a square piece of cardboard with side length 8 cm. This is done b cutting equal squares of side length centimetres from the corners and folding along the dotted lines as shown. cm a) Write a polnomial cm function to represent the volume, V, of the bo in terms of. 8 cm 8 cm b) Graph the function. Sketch the graph. State the domain. c) To the nearest centimetre, what is the value of for the bo with maimum volume? 66 Chapter 1: Polnomial Epressions and Functions DO NOT COPY. P

27 1. A manufacturer designs a clindrical can with no top. The surface area of the can is 300 cm. The can has base radius r centimetres. a) Write a polnomial function to model the capacit, C cubic centimetres, of the can as a function of r. b) Graph the function. Sketch the graph. To the nearest tenth of a centimetre, what are the radius and height of the can when it has a maimum capacit? Multiple-Choice Questions 1. A bo has the shape of a rectangular prism. The height of the bo is centimetres. The length of the bo is 4 centimetres. The sum of its length, width, and height is 18 cm. What is a polnomial function that represents the volume, V, of the bo? A. V() = (4)(18-5) B. V() = (4)(18-4) C. V() = (4)(18 - ) D. V() = (18-4)(18-5) P DO NOT COPY. 1.5 Modelling and Solving Problems with Polnomial Functions 67

28 . Here is the graph of = 50 - p 3. Suppose is the volume of a closed clindrical can with radius centimetres and surface area 500 cm. Which statements are correct? I. The maimum volume is approimatel 858 cm 3. II. The minimum volume is approimatel 858 cm 3. III. The volume is 0 when is approimatel 9 cm. IV. The can has no maimum volume. A. I and II B. I and III C. I and IV D. II and IV Stud Note Which characteristics of the graph of a polnomial function ma be used to solve a problem, involving volume, that is modelled b a polnomial function? ANSWERS Check Your Understanding 1. a) V() (5 )(30 ) b) 0 < < 1.5 c) 151 cm 3 ; 4.5 cm; 1.0 cm b 16.0 cm b 4.5 cm. In ears: Clara: 17; Lesle: 1; Mike: 15; Thomas: 0 Eercises 3. V() = ( + )( - 5)( + 3) 4. a) V() = ( + 6) b) 3 cm b 3 cm b 9 cm 5. a) height: centimetres; length: (36 ) centimetres; width: (8 ) centimetres b) V() = (36 - )(8 - ) c) 0 < < 14 d) approimatel 34.9 cm 3 ; approimatel 5. cm 6. a) length: ( 5) centimetres; width: centimetres c) approimatel cm ears old 8. 1 ears old cm b 5.4 cm b 1.5 cm 10. a) V() = 4 (5 - ) c) approimatel 16.7 cm b 16.7 cm b 33.3 cm 11. a) V() = 8(7 - )(14 - ) 300r - pr3 b) 0 < < 7 c) 3 cm 1. a) C(r) = b) radius height 5.6 cm Multiple Choice 1. A. B 68 Chapter 1: Polnomial Epressions and Functions DO NOT COPY. P

29 Additional Workspace P DO NOT COPY. 1.5 Modelling and Solving Problems with Polnomial Functions 69

30 STUDY GUIDE Concept Summar Big Ideas Some polnomials can be factored b using long division. The zeros of a polnomial function or the -intercepts of its graph can be determined b solving the corresponding polnomial equation. Appling the Big Ideas This means that: Polnomials can be divided b other polnomials. When a polnomial P() is divided b a binomial of the form a, a H, the remainder is P(a). A binomial is a factor of a polnomial when the division results in a remainder of 0. When a polnomial can be factored, the zeros of the related polnomial function or the -intercepts of its graph can be determined b equating each factor to 0. The behaviour of the graph of a polnomial function at its -intercepts depends on the multiplicit of each zero of the function: When a zero has even multiplicit, the graph just touches the -ais at the related -intercept. When a zero has odd multiplicit, the graph crosses the -ais at the related -intercept. Chapter Stud Notes What do ou need to remember when ou divide a polnomial b a binomial? What characteristics of the graph of a polnomial function can be determined from its equation? 70 Chapter 1: Polnomial Epressions and Functions DO NOT COPY. P

31 Skills Summar Skill Divide a polnomial b a binomial of the form a, a H. (1.1) Use the remainder and factor theorems, and the factor propert to factor a polnomial. (1.) Sketch the graph of a polnomial function. (1.4) Description To divide using snthetic division, remove the variables and record onl the coefficients. Include a 0 for an term that is not included in the polnomial. If a is a factor of a polnomial, then a is a factor of the constant term in the polnomial. The binomial a is a factor of P() if P(a) 0. In the equation of a polnomial function, the leading coefficient determines the end behaviour of the graph and the constant term is the -intercept of the graph. The zeros of the function are the -intercepts of the graph. The multiplicit of a zero determines the behaviour of the graph at the related -intercept. Eample Divide b So, ( ) ( ) 4 8 3R3 Determine a binomial factor of P() The factors of 3 are: 1, 1, 3, 3 Tr 1: P(1) 8(1) 3 6(1) 11(1) 3 0 So, 1 is a factor. Sketch a graph of f() ( ) ( ). -intercepts:, The zero has multiplicit ; the graph just touches the -ais at. The zero has multiplicit 1; the graph crosses the -ais at. The equation has degree 3, so it is an odddegree polnomial function. The leading coefficient is positive, so as : q, the graph falls and as : q, the graph rises. The constant term is 8, so the -intercept is 8. f() ( ) ( ) 0 8 Solve a problem b analzing the graph of a polnomial function. (1.5) Write a polnomial function to model the situation, then graph the function and analze the graph to solve the problem. A rectangular prism has width metres, length ( ) metres, and height ( 5) metres. What are the dimensions of a prism with volume 10 m 3? A polnomial function that models the volume is: ( )( 5) Graph the function and 10. Determine the -coordinate of the point of intersection: (3, 10) The prism has dimensions 3 m b 5 m b 8 m. P DO NOT COPY. Stud Guide 71

32 REVIEW Use long division to divide b 1. Write the division statement.. Use snthetic division to divide. Write the division statement. a) ( ), ( 1) b) ( ), (1 ) 7 Chapter 1: Polnomial Epressions and Functions DO NOT COPY. P

33 1. 3. Determine the remainder when is divided b each binomial. Which binomials are factors of the polnomial? How do ou know? a) b) 3 4. For each polnomial, determine one factor of the form a, a H. a) b) P DO NOT COPY. Review 73

34 5. Factor: a) For each polnomial function below, predict the end behaviour of the graph. Justif our prediction, then use graphing technolog to check. i) g() = ii) h() = iii) k() = Chapter 1: Polnomial Epressions and Functions DO NOT COPY. P

35 b) What do ou notice about the graphs and equations of the functions in part a, ii and iii? 7. Match each function to its graph. Justif our choices. a) = b) = c) = d) = i) Graph A ii) Graph B iii) Graph C iv) Graph D P DO NOT COPY. Review 75

36 Sketch the graph of each polnomial function. a) g() = b) f() = ( - 3)( - 1)( + ) 76 Chapter 1: Polnomial Epressions and Functions DO NOT COPY. P

37 A piece of cardboard 6 cm long and 0 cm wide is used to make a gift bo that has a top. The diagram shows the net for the bo. The shaded parts are discarded. 13 cm The squares cut from each corner have side length centimetres. What is the maimum volume of the bo? What is the side length of the 0 cm square that should be cut out to create a bo with this volume? Give our answers to the nearest tenth. 6 cm ANSWERS = ( - 1)( ) + 5. a) = ( - 1)( ) + 3 b) = ( + 1)( ) a) 0 b) 0 4. a) or 3 b) or = ( - 4)( - 3)( + ) 7. a) B b) A c) D d) C 9. approimatel cm 3 ; approimatel 3.7 cm P DO NOT COPY. Review 77

38 PRACTICE TEST 1. Multiple Choice Which statement is true? A. When is divided b, the remainder is 3. B. The binomial 1 is a factor of C. When is divided b 3, the remainder is 5. D. The binomial is a factor of Multiple Choice Which statement about the graph of a quartic function is false? A. The graph ma open up. B. The graph ma have a zero of multiplicit 3. C. The graph ma fall to the left and rise to the right. D. The graph ma have a zero of multiplicit. 3. Divide b. Write the division statement. 4. Does the polnomial have a factor of 3? How do ou know? 5. Factor: Chapter 1: Polnomial Epressions and Functions DO NOT COPY. P

39 6. Sketch the graph of this polnomial function. g() = P DO NOT COPY. Practice Test 79

40 7. Canada Post defines a small packet as one for which the sum of its length, width, and height is less than or equal to 90 cm. A compan produces several different small packets, each with length 15 cm longer than its height. a) Write a polnomial function to represent possible volumes of one of these packets in terms of its height. Assume the sum of the dimensions is maimized. b) Graph the function. c) To the nearest cubic centimetre, what is the maimum possible volume of one of these packets? What are its dimensions to the nearest tenth of a centimetre? ANSWERS 1. D. C = ( + )( ) 4. no 5. ( + 1)( - 3)( - 5)( - 1) 7. a) V() = ( + 15)(75 - ) c) cm 3 ; 38.1 cm b 8.8 cm b 3.1 cm 80 Chapter 1: Polnomial Epressions and Functions DO NOT COPY. P

41 Radical and Rational Functions BUILDING ON graphing polnomial functions solving radical and rational equations algebraicall simplifing rational epressions BIG IDEAS The graph of a function, f(), can be used to graph the corresponding radical function, = f(). For the graph of a rational function, = f(), the non-permissible values g() of correspond to vertical asmptotes or holes. The roots of a rational equation or radical equation are the -intercepts of the graph of a corresponding function. LEADING TO curve sketching using calculus NEW VOCABULARY radical function invariant point rational function oblique (slant) asmptote P DO NOT COPY.

42 .1 Properties of Radical Functions FOCUS Graph and analze radical functions, and solve radical equations. Get Started What are the domain and range of each function? 3 3( ) Construct Understanding Complete the table of values for and. Sketch a graph of each function, then state its domain and range. When is the graph of above the graph of? When is it below? Suppose ou are given the graph of. How could ou use it to graph? Chapter : Radical and Rational Functions DO NOT COPY. P

43 A radical function has the form f(),where f() is a function. The square root of a number is onl defined for non-negative numbers, so the domain of f() is the set of values of for which f() 0. Here is the graph of. The domain of is 0 and the range is 0. Since both 0 and 0, the graph starts at the origin and etends in onl one direction. Here are the graphs of two linear functions and their related radical functions. 4 and + 4 and ( 3, 1) ( 4, 0) The points ( 4, 0) and ( 3, 1) lie on both graphs. Between these points, the graph of + 4 lies above the graph of 4. Since 4 0 then 4 So, the domain of + 4 is 4; and the range is 0. The graph of + 4 opens to the right (1, 1) (, 0) 4 The points (1, 1) and (, 0) lie on both graphs. Between these points, the graph of - + lies above the graph of. Since 0 then So, the domain of - + is ; and the range is 0. The graph of + opens to the left. P DO NOT COPY..1 Properties of Radical Functions 83

44 THINK FURTHER Wh does the graph of f() lie above the graph of f() between the invariant points? The graphs at the bottom of page 83 illustrate characteristics that are common to all graphs of f() and f(). If 0 and 1 are in the range of f(), then points with these -coordinates lie on both graphs; these are invariant points. Where the graph of f() lies between the graph of 1 and the -ais, the graph of f() lies above the graph of f(). Where the graph of f() lies above the graph of 1, the graph of f() lies below the graph of f(). Where the graph of f() lies below the -ais, the graph of f() does not eist. These characteristics can be used to sketch a graph of f() when the graph of f() is given. Eample 1 Sketching the Graph of f() Given the Graph of a Linear Function f() Check Your Understanding 1. For each graph of f() below: Sketch the graph of f(). State the domain and range of f(). a) 8 b) f() f() 4 6 For each graph of f()below: Sketch the graph of f(). State the domain and range of f(). a) f() b) SOLUTION a) From the graph: f() f() 0 for all values of 3, f() so the domain of f() is Mark the points where 0 or 1; the graph of f() lies above the graph of f() between these points. Identif the coordinates of another point that lies above the -ais on the graph of f() : f() 4 f() f() Join the points with a smooth curve for the graph of f(). The range of f() is Chapter : Radical and Rational Functions DO NOT COPY. P

45 b) From the graph: f() 0 for all values of, so the domain of f() is. Mark the points where 0or 1. Identif the coordinates of other points that lie above the -ais on the graph of f() : f() f() f() = 3 0 f() 4 Join the points with a smooth curve for the graph of f(). The range of f() is 0. In Eample 1b, the functions f() and f() have different domains because the square root of a negative number is not defined; and, if <, then f() is negative and f() is not defined. The ranges are different because f() is the principal square root of f(), which is alwas 0 or positive. Here are tables of values for and graphs of 1 and THINK FURTHER Describe the graph of a linear function, f(), for which the graph of f() is a horizontal line, and f() > f() for all real values of. The domain of 1 is H, and the range is 1. The domain of + 1 is H, and the range is 1. The onl invariant point is (0, 1). P DO NOT COPY..1 Properties of Radical Functions 85

46 THINK FURTHER Consider the quadratic function 1. Would the domain and range of this function be the same as the domain and range of 1? Eample Sketching the Graph of f() Given the Graph of a Quadratic Function f() Check Your Understanding. For the graph of each quadratic function f() below: Sketch the graph of f(). State the domain and range of f(). a) 4 f() For the graph of each quadratic function below: Sketch the graph of = = f() f(). State the domain and range of = f(). a) f() b) 6 f() b) f() 4 SOLUTION a) The graph of lies below the -ais for and for, so is undefined for these values of. The domain of = = = f() < - > 4 f() f() is 4. Mark points A and B where, and points C and D where. The graph of = = 0 = 1 f() lies above the graph of = f() between A and C, and between B and D. Identif the coordinates of another point with -coordinate in the domain of = f(). f() 1 9 f() 9 = 3 Join the points with a smooth curve for the graph of = f(). f() 8 4 f() C A 0 B D The range of = f() is Chapter : Radical and Rational Functions DO NOT COPY. P

47 b) The graph of lies below the -ais for 0, so = = f() -4 < < is undefined for these values of. The domain of = f() f() is -4 or 0. Mark points A and B where and points C and D where. The graph of = = 0 = 1 f() lies above the graph of = f() between A and C, and between B and D. Identif the coordinates of other points with -coordinates in the domain of = f(). f() f() Join the points with two smooth curves for the graph of = f(). 6 4 f() 6 C A 4 D f() 0 B The range of = f() is 0. To sketch the graph of a radical function, it ma not alwas be necessar to determine the coordinates of additional points beond those with -coordinates 0 and 1. THINK FURTHER The graph of a radical function = g() is the point (, 0). What is an eplicit equation for a possible quadratic function g()? P DO NOT COPY..1 Properties of Radical Functions 87

48 Eample 3 Sketching the Graph of f() Given the Graph of a Cubic Function f() Check Your Understanding 3. For the graph of the cubic function f(): Sketch the graph of f(). State the domain and range of f(). 1 ( 0.5, 0.15) f() 1 For the graph of the cubic function : Sketch the graph of = = f() f(). State the domain and range of = f(). f() 6 4 ( 0.5, 0.35) 0 SOLUTION The graph of f() lies below the -ais for 1 and for 0 1, so the graph of = < < < f() is undefined for these values of. The domain of = f() is -1 0 or 1. Mark the three points where = 0 and the point where =. Identif the coordinates of other points on the graph of = 1 f(). f() f() Join the points with two smooth curves for the graph of = f(). 6 f() 4 0 f() The range of = f() is 0. The graph of a radical function can be used to solve a related radical equation. The zeros of the graph are the roots of the equation. 88 Chapter : Radical and Rational Functions DO NOT COPY. P