1 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
2 c) f ( x) x d) y 3 x 8 e) y 2 x 6 f) y 4 5 x 2
3 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: Now Try Page 72 #2, 3, 4, 5, 6, 9, 10, 11,
4 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 gx ( ) 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 5, 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). Now Try Page 86 #2, 3, 5, 7, 8, 13, 16
6 Fundamental Counting Principle: If a task is made up of stages, the total number of ways a task can be completed is given by: mn 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?
7 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?
8 Factorial Notation: Ex) 5! read as 5 Factorial Means: 5! 5432 1! in calculator can be found: Math PRB 4:! Ex) Find the following: 4! 7! 4! 103! 100! Now Try Worksheet
9 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?
10 Case II -There are n distinct objects, but only r of these are used. # of Permutations n P 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
11 Ex) Express 68 67 6665 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)! 42 ( n 1)!
12 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? Now Try Worksheet
13 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. Now Try Worksheet
14 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 n C 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?
15 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 Now Try Worksheet
16 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)
17 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 k1 n k x y) n is: * This formula will give individual terms of an expansion. Ex) Find the 4 th 7 term when ( a 3 b) is expanded.
18 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. Now Try Worksheet