Unit 5 Radical Functions & Combinatorics

1 Unit 5 Radical Functions & Combinatorics General Outcome: Develop algebraic and graphical reasoning through the study of relations. Develop algebraic and numeric reasoning that involves combinatorics. Specific Outcomes: 5.1 Graph and analyze radical functions (limited to functions involving one radical). 5.2 Apply the fundamental counting principle to solve problems. 5.3 Determine the number of permutations of n elements taken r at a time to solve problems. 5.4 Determine the number of combinations on n different elements taken r at a time to solve problems. 5.5 Expand powers of a binomial in a variety of ways, including using the binomial theorem (restricted to exponents that are natural numbers). Topics: Graph of y= x (Outcome 5.1) Page 2 Square Root of a Function (Outcome 5.1) Page 9 Fundamental Counting Principle (Outcome 5.2) Page 14 Permutations (Outcome 5.3) Page 21 Combinations (Outcome 5.4) Page 35 The Binomial Theorem (Outcome 5.5) Page 41

2 Graph of y Unit 5 Radical Functions & Combinatorics = x: Characteristics: Ex) Use your knowledge of the graph of y= x and transformations to sketch the graph of each of the following. a) y= x+ 5+ 3 b) f ( x) = 2x 6

3 c) f ( x) = x d) y= 3 x+ 8 e) y 2 x 6 = + f) y + 4= + 5 x 2

4 Ex) Determine the domain and range for each of the following radical functions. a) y= 5 x 6 + 3 b) 2y= 9 x 2 Ex) Determine the equation of the radical shown below. Consider a vertical stretch: Consider a horizontal stretch:

5 Graph of y = x Assignment: 1) Describe how the graph of y= x can be transformed to create the graph given by the function below. a) y= 7 x 9 b) 1 4+ y= x+ 6 3 2) Write the equation of the radical function that results by applying each set of transformations to the graph of y = x. a) vertical stretch about the x-axis by a factor of 4, then translated 6 units left. b) reflection in the y-axis, then translated 4 units right and 5 units down.

6 3) Without using technology, match each function with its graph. i) y= x 2 ii) y= x + 2 iii) y= x+ 2 iv) y= ( x 2) a) b) c) d) 4) Sketch the graph of each function given below by using transformations on y = x. a) f ( x) = x 3 b) r( x) = 3 x + 1

7 c) y 1= 4( x 2) d) 1 m( x) = x + 4 2 5) Consider the function 1 f ( x) = 5x. 4 a) Represent f( x ) in the form f ( x) = a x and describe the transformation that could be applied to the graph of y = x to create the graph of f( x ). b) Represent f( x ) in the form f ( x) = bx and describe the transformation that could be applied to the graph of y = x to create the graph of f( x ).

8 6) The function 4 y= 3x is translated 9 units up and then reflected in the x- axis. Without graphing, determine the domain and range of the transformed function. 7) For each graph, determine the equation of the radical function. a) b) c) d) 8) Write an equation of a radical function with the given domain and range. a) x x 6, x R b) x x 4, x R y y 1, y R y y 3, y R

9 Square Root of a Function: Complete the following table. 1 2 f ( x) = x + 5 g( x) = ( x + 1) 3 3 x f( x ) f( x ) x g( x ) gx ( ) 9 6 6 4 5 3 4 1 1 0 2 2 4 3 Sketch the graph of f( x ) Sketch the graph of gx ( )

Ex) If the following points exist on the graph of y f ( x) determine the corresponding points on the graph of y = f ( x). ( 4, 9 ) ( 5, 16) ( 8, 0) ( 4, 9) 10 =, Ex) Given the graph of y = f ( x), sketch the graph of y = f ( x). a) b) 2 Ex) If f ( x) = x 144, determine the domain and range of y = f ( x) and y = f ( x).

11 Square Root of a Function Assignment: 1) Given the following points on the graph of y = f ( x), determine the corresponding point on the graph of y = f ( x). If the point does not exist state so. a) ( 4, 12 ) b) ( 2, 0.4) c) ( 10, 2) d) ( 0.09, 1 ) e) ( 5, 0) f) ( m, n ) 2) Determine the domains and ranges of each pair of functions. a) y= 2x+ 6 y= 2x+ 6 b) y= x+ 9 y= x+ 9 c) y 2 = x 9 y 2 = x 9 d) y = 2 x 2 y= 2 x 2

12 3) Match each graph of y = f ( x) to the corresponding graph of y = f ( x). i) ii) iii) iv) a) b) c) d)

13 4) Using each graph of y = f ( x), sketch the graph of y = f ( x). a) b) c) 5) If ( 24, 12) is a point on the graph of the function y = f ( x), identify one point on the graph of each of the following functions. a) y = 4 f ( x + 3) b) y = f (4 x) + 12 c) y f ( x ) = 2 ( 2) 4 + 6

14 Fundamental Counting Principle: If a task is made up of stages, the total number of ways a task can be completed is given by: m n p... Where m = # of possibilities for stage 1 n = # of possibilities for stage 2 p = # of possibilities for stage 3 and so on Ex) When purchasing a specific model of car the following choices must be made: Color Transmission Interior Sterio Options -white -automatic -cloth -CD -seat warmers -black -standard -leather -MP3 and sun roof -red -satellite -no seat warmers -yellow radio or sun roof How many different cars can be created based on the above choices?

15 Ex) How many postal codes are possible in Canada? Ex) How many even 4 digit numbers can be created with the numbers 2, 3, 4, 5, 7, 8, and 9 a) if digits can be repeated? b) if no digits can be repeated? Ex) If 24 people are lining up to buy concert tickets, how many ways can they be arranged?

16 Factorial Notation: Ex) 5! read as 5 Factorial Means: 5! = 5 4 3 2 1! in calculator can be found: Math PRB 4:! Ex) Find the following: 4! 7! 4! 103! 100!

17 Fundamental Counting Principle Assignment: 1) A football team has the following kit: Jersey: Red, or Black Pants: White, Red, or Black Socks: Red, or White If the team plays in a different uniform each week, for how many weeks can it play before it has to repeat a previous uniform? 2) The score at the end of the second period of a hockey game is: Flames 6 Oilers 3 How many different possibilities are there for the score at the end of the first period? 3) With the new renovations completed at Prestwick High School, there will be seven entrances. In how many different ways can a student coming for Math tutorials; a) enter the school and exit through a different entrance? b) enter and exit through any entrance? c) enter and exit through the same entrance?

18 4) Find the number of four letter creations that can be formed from the letters of the word PRODUCE if each letter can only be used once and: a) it does not matter which letters are used. b) only consonants can be used. c) the creation must begin and end with a consonant. d) it must begin with a vowel. e) it must contain the letter P. f) it must begin with D and end with a vowel. 5) How many even four digit numerals have no repeated digits?

19 6) A vehicle license plate consists of 3 letters followed by 3 digits. How many different license plates are possible if: a) there are no restrictions on the letters or digits used? b) no letters may be repeated? c) the first digit cannot be zero and no digit can be repeated? 7) a) How many different three-digit numeral can be formed from the digits 1, 5, and 8 if the digits cannot be repeated? b) How many different three-digit numerals can be formed using the digits 1, 3, 5, 7, and 9 if the digits may be repeated? c) How many four-digit numerals can be formed from the digits 0, 2, and 3 fi the digits may be repeated? (Note: 0223 is classified as a 3-digit numeral 223.) d) How many different non-zero numerals are possible using some or all of the numerals 0, 1, 2, and 3 if the digits cannot be repeated?

20 8) How many different sums of money can be made from two pennies, four nickels, two quarters, and five dollar coins? 9) Mr. and Mrs. McDonald want a family picture taken with their children, Hamish, Flora, and James. In how many different ways can all five line up in a straight line for the picture if; a) there are no restrictions? b) the parents must be at either end of the line? c) baby James must be in the middle? d) the children alternate with the adults? 10) Ocean going ships have used coloured flags hung vertically for signalling. By changing the order of the coloured flags, the ships can send out different signals. If ships carry six different coloured flags, one flag of each colour, how many different signals are possible if; a) all six flags are used? b) four flags are used? c) at least two flags are used?

21 Permutations: A permutation refers to the number of ways a specific number of objects can be arranged in which the order is important (each object has a specific location or title assigned to it). Case I -There are n distinct objects and all are used. # of Permutations = n! Ex) How many ways can 6 people be lined up? Ex) How many ways can a 4 person committee be designated the roles of president, vice president, secretary, and treasurer?

22 Case II -There are n distinct objects, but only r of these are used. # of Permutations = P = n r n! ( n r)! Ex) In a class of 20 people how many ways can a president, a vice president and a secretary be selected? Ex) A picture is to be taken of 3 people. The 3 people are to be arranged as shown below. If there are 15 people to select from, how many different pictures are possible? Person # 1 Person # 2 Person # 3

23 Ex) Express 68 67 66 65 64 in the form n P r. Ex) Simplify the expression given by ( n + 1)!. ( n 2)! Ex) Solve the following for n. ( n + 3)! = ( n + 1)! 42

24 Case III -There are n objects, all are used but they are not all distinct. # of Permutations = n! abc!!! where a = # of a objects that are the same b = # of b objects that are the same c = # of c objects that are the same etc. Ex) How many ways can 3 green, 4 red, 2 blue, 1 orange, and 2 yellow chairs be lined up? Ex) How many ways can the letters in the word COMMITTEE be rearranged? Ex) How many ways can the letters in the word REARRANGE be rearranged?

25 Permutations with Restrictions: Ex) How many ways can the letters in the word ORANGE be arranged if the vowels must be together? Ex) How many ways can Lindsey, Mike, Alexie, Alyssa, and Dillon sit together in 1 row of the bleachers if: a) Lindsey and Dillon must sit together. b) Lindsey and Dillon cannot sit together.

26 Permutations Assignment: 1) Evaluate the following. a) 10! b) 8! 4! c) 15! 10!5! d) 25! 7! 21! 11! 2) Simplify each expression. a) n! n b) ( n 3)! ( n 2)! c) ( n + 1)! ( n 1)! d) (3 n)! (3n 2)!

27 3) Express the following as a quotient of factorials. a) 9 8 7 6 b) 20 19 18 c) ( n + 2)( n + 1) n 4) Solve the following. a) ( n + 1)! = 6 b) ( n+ 1)! = 6( n 1)! n! c) ( n + 2)! = 12 d) n! ( n + 1)! = 20( n 1) ( n 2)!

28 5) How many arrangements are there of the letters: a) DOG b) DUCK c) SANDWICH d) CANMORE 6) How many five-digit numbers can be made from the digits 2, 3, 4, 7, and 9 if no digit can be repeated? 7) If n P r is the number of ways that n objects can be arranged r at a time, explain why 7 P 0 = 1. 8) Use a permutation formula ( n P r ) to determine how many arrangements there are of a) two letters from the word b) three letters from the word GOLDEN CHAPTERS

29 9) How many numbers can be made from the digits 2, 3, 4, and 5 if no digit can be repeated? (Hint: consider 4 cases four-digit numbers, three-digit numbers, two-digit numbers, one-digit numbers.) 10) Solve each equation, where n is an integer. n! = b) np4 = 8( n 1 P3 ) 84 a) n 2Pn 4 11) In a ten-team basketball league, each team plays every other team twice, once at home and once away. How many games are scheduled?

30 12) How many arrangements could be made of the word: a) FATHER if F is first? b) UNCLE if C is first and L is last? c) DAUGHTER if UG is last? d) MOTHER if the vowels are first and last? 13) How many arrangements of the following words can be made if all the vowels must be kept together? a) FATHER b) DAUGHTER c) UNCLE d) EQUATION

31 14) Find the number of different arrangements of the letters of in the word ANSWER under each condition: a) without restrictions b) that begin with an S c) that begin with a vowel and d) that have the three letters A, N, and end with a consonant S adjacent and in the order ANS e) that have the three letters A, N, and S adjacent but not necessarily in that order 15) How many even four-digit numbers can be made from the digits 0, 2, 3, 4, 5, or 7 if no digit can be repeated?

32 16) Ann, Brian, Colin, Diane, and Eric go to watch a movie and sit in 5 adjacent seats. In how many ways can this be done under each condition? a) without restrictions? b) if Brian sits next to Diane? c) if Ann refuses to sit next to Eric? 17) In how many ways can four adults and five children be arranged in a single line under each condition? a) without restrictions b) if children and adults are alternated c) if the adults are all together and d) if the adults are all together the children are all together

33 18) How many different arrangements can be made using all of the letters of each word? a) COCHRANE b) WINNIPEG c) RED DEER d) MILLARVILLE 19) Find the number of arrangements of the letters of the word TATTOO under each condition: a) begins with a T b) begins with two T s c) begins with three T s d) begins with one T and the next e) begins with exactly two T s letter is not a T

34 20) How many permutation are there of the letters of the word MONOTONOUS under each condition? a) without restrictions b) if each arrangement begins with a T c) if each arrangement begins d) if the four O s are to be together with a O 21) Naval signals are made by arranging colored flags in a vertical line and the flags are then read from top to bottom. How many signals using six flags can be made if you have? a) 3 red, 1 green, and 2 blue flags? b) 2 red, 2 green, and 2 blue flags? c) unlimited supplies of red, green and blue flags? 22) Determine the number of different arrangements that can be made using all of the letters of the word SASKATOON.

35 Combinations: A combination refers to the number of ways a specific number of objects can be grouped together in which the order does not matter. # of Combinations = C = n r n! ( n r)! r! Ex) How many ways can a group of 3 be formed from a class of 25 students? Ex) How many different tickets could you create for the LOTTO 649 draw?

36 Ex) How many 5 card hands can be formed from a standard deck of cards (52 cards being used) that have: a) no restrictions b) only diamonds c) no clubs d) exactly 3 tens e) 3 of kind and f) a full house 2 other non matching cards

37 Combinations Assignment: 1) Pete s Perfect Pizza Company has 9 choices of toppings available, how many different 3 topping pizzas can be made? 2) A theatre company consisting of 6 players is to be chosen from 15 actors. How many selections are possible if the company must include Mrs. Jones? 3) Edinburgh High School has a twelve-member student council. A four member sub-committee is to be selected to organize dances. a) How many different sub-committees are possible? b) How many four member sub-committees are possible if the council president and vice-president must be members? 4) A basketball coach has five guards and seven forwards on his basketball team. a) In how many different ways can he select a starting team of two guards and three forwards? b) How many different starting teams are there if the star player, who plays guard, must be included?

38 5) Twelve face cards are removed from a deck of fifty-two cards. From the face cards, three card hands are dealt. Determine the number of distinct three card hands that are possible which include: a) no restrictions b) 3 Kings c) 1 Queen and 2 Kings d) only 1 Jack 6) Consider a standard deck of 52 cards. Determine the number of distinct six card hands that are possible which include: a) no restrictions b) only clubs c) 2 clubs and 4 diamonds d) no sevens e) 4 tens f) only 1 Jack and 4 Queens 7) The Athletic Council decides to form a sub-committee of 6 council members to look at a new sports program. There are a total of 15 Athletic Council members, 6 females and 9 males. How many different ways can the sub committee consist of at most one male?

39 8) A group of 4 journalists is to be chosen to cover a murder trial. There are 5 male and 7 female journalists available. How many possible groups can be formed: a) consisting of 2 men and 2 women? b) consisting of at least one woman? 9) Consider a standard deck of 52 cards. How many different four cards hands have; a) at least one black card? b) at least 2 Kings? c) two pairs? d) at most 2 clubs? 10) City Council decides to form a sub-committee of five aldermen to investigate transportation concerns. There are 4 males and 7 females. How many different ways can the sub-committee be formed consisting of at least one female member? 11) A basketball squad of 11 players is to be chosen from 17 available players. In how many ways can this be done if: a) Colin and Darryl must be b) Jeff and Brent cannot both be selected? selected?

40 12) The number of ways that a selection of 7 students can be chosen from a class of 28 is the same as the number of ways that n students can be chosen from the same class. What is the value of n? 13) How many people are there in a class in which there are 20 ways to select a committee of three people? 14) In each of the following solve for n. a) C = n 3 84 b) 11C n = 330 (two answers) c) nc7 = n + 1C8

41 The Binomial Theorem: Pascal s Triangle: 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1 1 8 28 56 70 56 28 8 1 5 Ex) Expand ( a+ b)

42 Ex) Expand the following using Pascal s Triangle. 6 a) ( x+ y) 5 b) (3x 2) The General Term of the expansion of ( t C x y n k k k + 1 = n k x + y) n is: * This formula will give individual terms of an expansion. Ex) Find the 4 th term when ( a 3 ) 7 + b is expanded.

43 Ex) Find the 11 th 14 term when (5x 1) is expanded. 10 4 Ex) One term in the expansion of ( x+ a) is Determine the value of a. 3281250x. Ex) Determine the value of the constant term when 2x + 3 x 2 6 is expanded.

44 Binomial Theorem Assignment: 1) Find the indicated term of each expansion. a) the fifth term of ( a ) 5 b b) the second term of 6 ( x 2) c) the third term of (3x 2 ) 9 2 + y d) the forth term of ( ) 7 a 2a e) the middle term of x 2 2 6 2) Expand and write the following in simplest form. a) (2 x+ y) 3 b) 4 1 x x

45 3) Find the indicated term of each expansion. a) the 3 x term in (1 2 x) 12 b) the 4 3 xy term in (3x 7 y) 7 4) One term in the expansion of if a 0. ( x ) 8 + a is 6 448x. Determine the value of a 5) If 8 4455x is a term in the expansion of ( x ) 11 + b, determine the value of b.

46 6) Determine the value of the constant term for each of the following expansions. a) x x 2 1 6 b) 2x + 1 3 x 8 7) Express each row of Pascal s Triangle using combinations. Leave each term in the form n C r. a) 1 4 6 4 1 b) 1 7 21 35 35 21 7 1

47 Answers Graph of y = x Assignment: 1. a) The graph of y= x is stretched vertically about the x-axis by a factor of 7 and then it is translated 9 units to the right. b) The graph of y= x is stretched vertically about the x-axis by a factor of 1, and then it is translated 6 units to the left and 4 units down. 3 2. a) y= 4 x+ 6 b) y= x+ 4 5 3. a) ii b) i c) iii d) iv 4. a) b) c) d) 5. a) b) 5 f ( x) = x The graph of y= x could be stretched about the x-axis 4 by a factor of 5 4. 5 f ( x) = x The graph of y= x could be stretched about the y-axis by 16 a factor of 16 5. 6. Domain: x x 0, x R Range: y y 13, y R 7. a) y= x+ 3+ 4 b) 1 y= 2 x+ 5 3 c) y= 2 x+ 5 1 d) y= 4 x+ 4 + 5 8. a) y = x 6 + 1 b) y= x+ 4 3

48 Square Root of a Function Assignment: 1. a) ( 4, 2 3 ) b) e) ( 5, 0) 2, f) (, ) 2. a) y 2x 6 y 10 5 m n, where n 0 c) does not exist d) ( 0.09, 1 ) = +, x x R, y y R = 2x+ 6, x x 3, x R, y y 0, y R = x+, x x R, y y R = x+ 9, x x 9, x R, y y 0, y R 2 = x 9, x x R, y y 9, y R 2 = x 9, x x 3, x 3, x R, y y 0, y R 2 = 2 x, x x R, y y 2, y R 2 2 x x 2 x 2, x R, y y 0, y R b) y 9 c) d) y y y y y =, 3. a) iii b) iv c) i d) ii 4. a) b) c) 5. a) ( 27, 4 3) b) ( 6, 12 2 3) c) ( 26, 6 4 3) Fundamental Counting Principle Assignment: 1. 12 2. 28 3. a) 42 b) 49 c) 7 4. a) 840 b) 24 c) 240 d) 360 e) 480 f) 60 5. 2296 6. a) 17 576 000 b) 15 600 000 c) 11 389 248 7. a) 6 b) 125 c) 54 d) 48 8. 269 9. a) 120 b) 12 c) 24 d) 12

49 10. a) 720 b) 360 c) 1950 Permutations Assignment: 1. a) 3628800 b) 1680 c) 3003 d) 2. a) ( n 1)! b) 1 n 2 115 3 c) ( n+ 1) n d) 3 n(3n 1) 3. a) 9! 20! ( n + 2)! b) c) 5! 17! ( n 1)! 4. a) 5 b) 2 c) 2 d) 4 5. a) 6 b) 24 c) 40320 d) 5040 6. 120 8. a) 30 b) 336 9. 64 10. a) 7 b) 8 11. 90 12. a) 120 b) 6 c) 720 d) 48 13. a) 240 b) 4320 c) 48 d) 2880 14. a) 720 b) 120 c) 192 d) 24 e) 144 15. 156 16. a) 120 b) 48 c) 72 17. a) 362880 b) 2880 c) 5760 d) 17280 18. a) 20160 b) 10080 c) 210 d) 831600 19. a) 30 b) 12 c) 3 d) 18 e) 9 20. a) 75600 b) 7560 c) 30240 d) 2520 21. a) 60 b) 90 c) 729 22. 45360 Combinations Assignment: 1. 84 2. 2002 3. a) 495 b) 45 4. a) 350 b) 140 5. a) 220 b) 4 c) 24 d) 112 6. a) 20 358 520 b) 1716 c) 55 770 d) 12 271 512 e) 1128 f) 176 7. 55 8. a) 210 b) 490

50 9. a) 255 775 b) 6961 c) 2808 d) 258 856 10. 462 11. a) 5005 b) 7371 12. 21 13. 6 14. a) 9 b) 4 or 7 c) 7 Binomial Theorem Assignment: 1. a) 2. a) 4 5ab b) 5 12x c) 3 2 2 3 8x + 12x y + 6xy + y b) 7 2 314928x y d) x 4x + 6 + 4 2 11 280a e) 4 1 2 x 4 x 3 20x 3. a) 3 1760x b) 4 3 972405x y 4. 4 5. 3 6. a) 15 b) 1792 7. a) 4C0 4C1 4C2 4C3 4C 4 b) 7C0 7C1 7C2 7C3 7C4 7C5 7C6 7C 7