Slide 1 / 11 Slide 2 / 11 Algebra I Functions 2015-11-02 www.njctl.org Slide / 11 Slide / 11 Table of Contents click on the topic to go to that section Relations and Functions Domain and Range Relations and Functions Evaluating Functions Eplicit and Recursive Functions Multiple Representations of Functions Return to Table of Contents Slide 5 / 11 Vocabular A relation is an set of ordered pairs. Slide 6 / 11 Determine if each of the relations below is a function and provide an eplanation to support our : A function is a relation where each value in the domain has eactl ONE value in the range. The -value does NOT repeat in a function.
-1 2 5 9 Slide 7 / 11 Determine if each of the relations below is a function and provide an eplanation to support our : -2-5 X Y 1 2 5-5 9 7 Slide / 11 Determine if each of the relations below is a function and provide an eplanation to support our : X Y - -1 5 0-1 9 11 X Y 1 2 2 2 5 2 2 7 - Slide 9 / 11 Graphs On a graph, a function does not have a point in the same vertical location as another point. The Vertical Line Testcan determine if a graph represents a function. Place a ruler or imaginar vertical line on the graph and move it from left to right. If the vertical line intersects onl one point at a time on the ENTIRE graph, then it represents a function. If the vertical line intersects more than one point at ANY time on the graph, then it is NOT a function. Slide 10 / 11 Equations An equation is a function onl if when a number is substituted in for, there is onl 1 output -value. Function = + Not a Function = 5 Function 10 6 2 Not a Function 10 6 2 = 5-10 - -6 - -2 0-2 2 6 10-10 - -6 - -2 0-2 2 6 10 - - -6-6 - - -10-10 Slide 11 / 11 Determine if each of the relations below is a function and provide an eplanation to support our : Slide 12 / 11 Determine if each of the relations below is a function and provide an eplanation to support our :
Slide 1 / 11 1 Is the following relation a function? Slide 1 / 11 2 Is the following relation a function? Yes No {(,1), (2,-1), (1,1)} -1 0-2 -1 0 Yes No Slide 15 / 11 Is the following relation a function? Slide 16 / 11 Is the following relation a function? X Y -2 0 2-1 -1 2 Yes No Yes No -2 0 Slide 17 / 11 Slide 1 / 11 5 Is the following relation a function? 6 Is the following relation a function? Yes No Yes No
Slide 19 / 11 Slide 20 / 11 7 Is the following relation a function? Yes Domain and Range No Return to Table of Contents Slide 21 / 11 Vocabular The domain of a function/relation is the set of all possible input values (-values). Relation Domain {1,, 6} {2,, 5} Slide 22 / 11 State the domain for each eample below and tell whether the relation is a function. 2-2 -5 7 - s {1,, 1} 1 2 5 9 Slide 2 / 11 State the domain for each eample below and tell whether the relation is a function. Slide 2 / 11 State the domain for each eample below and tell whether the relation is a function. X Y X Y X Y 1 1 2-2 5-5 2 2-1 5 0 s 9-1 9 7 5 2 11 s
Slide 25 / 11 Slide 26 / 11 State the domain for each eample below: s Slide 27 / 11 Slide 2 / 11 9 What is the domain of the following: (Choose all that appl) -1 0-2 -1 0 10 What is the domain of the following: (Choose all that appl) X Y -2 0 2-1 -1 2 A - G H < 0 I > 0 J All Real Numbers A - -2 0 G H < 0 I > 0 J All Real Numbers Slide 29 / 11 Slide 0 / 11 11 What is the domain of the following: 12 What is the domain of the following: (Choose all that appl) (Choose all that appl) A - G H < 0 I > 0 J All Real Numbers A - G H < 0 I > 0 J All Real Numbers
Slide 1 / 11 Slide 2 / 11 1 What is the domain of the following: 1 What is the domain of the following: (Choose all that appl) (Choose all that appl) A - G H < 0 I > 0 J All Real Numbers A - G H < 0 I > 0 J All Real Numbers Slide / 11 Vocabular The range of a function/relation is the set of all possible output values (-values). Relation Domain {1,, 6} {2,, 5} Range {-1,, 7} {} 2-2 -5 7 - Slide / 11 State the range for each eample below: s {1, } {-2, 2, 6} 1 2 5 9 Slide 5 / 11 State the range for each eample below: Slide 6 / 11 State the range for each eample below: X Y 1 2 5-5 9 7 X Y 1 2 2 2 5 2 X Y - -1 5 0-1 9 11 s s
Slide 7 / 11 State the range for each eample below: Slide / 11 15 What is the range of the following: (Choose all that appl) {(,1), (2,-1), (1,1)} s A - G H < 0 I > 0 J All Real Numbers Slide 9 / 11 Slide 0 / 11 16 What is the range of the following: (Choose all that appl) -1 0-2 -1 0 17 What is the range of the following: (Choose all that appl) X Y -2 0 2-1 -1 2 A - G H < 0 I > 0 J All Real Numbers -2 0 A - G H < 0 I > 0 J All Real Numbers Slide 1 / 11 Slide 2 / 11 1 What is the range of the following: 19 What is the range of the following: (Choose all that appl) (Choose all that appl) A - A - G G H < 0 H < 0 I > 0 I > 0 J All Real Numbers J All Real Numbers
Slide / 11 Slide / 11 20 What is the range of the following: 21 What is the range of the following: (Choose all that appl) (Choose all that appl) A - A - G G H < 0 H < 0 I > 0 I > 0 J All Real Numbers J All Real Numbers Slide 5 / 11 Slide 6 / 11 Function Notation Equations for relations have been in the form of = + 2 Evaluating Functions When a relation is a function, it can also be written in function notation: f() = + 2 f() = + 2 is still a line with a slope of and a -intercept of 2. Return to Table of Contents When a relation is a function, = can be substituted with the notation of f() = Function Notation is read: "f of " Slide 7 / 11 Function Notation Slide / 11 Evaluating a Function So wh the new notation? 1) It lets the reader know the relation is a function. 2) A second function can be added, such as g() = and the reader will be able to distinguish between the different functions. To Evaluate in = Form: Find the value of = 2 + 1 when = = 2 + 1 = 2() + 1 = 7 When is = 7 To Evaluate in Function Notation Given f() = 2 +1 find f() f() = 2() + 1 f() = 7 "f of is 7" ) The notation makes evaluating a value of easier to read. Similar methods are used to solve but function notation makes asking and ing questions more concise.
Slide 9 / 11 Slide 50 / 11 Slide 51 / 11 Slide 52 / 11 22 Given and Find the value of. Slide 5 / 11 Slide 5 / 11 2 Given and Find the value of.
26 Given and Slide 55 / 11 Find the value of. 27 Given and Find the value of. Slide 56 / 11 Slide 57 / 11 Slide 5 / 11 Vocabular Arithmetic Sequence - is a sequence of numbers with a constant common difference. Eplicit and Recursive Functions a 1 - is the first term of the sequence. d - is the common difference s: (5, 7, 9, 11, 1,...) a 1 = 5 and d = 2 (2, 17, 11, 5, -1,...) a 1 = 2 and d = -6 Return to Table of Contents Eplicit and Recursive Arithmetic Sequences Video (Youtube Video) Eplicit and Recursive Arithmetic Sequences Video (Khan Academ Video) Slide 59 / 11 Eplicit Function Form - a(n) = a 1 + d(n-1) Slide 60 / 11 Write the eplicit form for the following sequences: 1) 9, 16, 2, 0,... Recursive Function Form - is written in two parts 1. The first part is the first term a 1 2. The second part is a(n) = a n-1 + d (previous term plus the common difference) Note: In the recursive formula the previous term is used to produce the net term. 2) 1,, 6,, 11,... ) 6, 57, 6, 5,...
Slide 61 / 11 Write the recursive form for the following sequences: 1) 9, 16, 2, 0,... 2) -2, 1,, 7, 10,... Slide 62 / 11 Write the first five terms of the sequence given the eplicit formula. a(n) = 1 - (n - 1) ) 6, 57, 6, 5,... Slide 6 / 11 Slide 6 / 11 2 What is the eplicit formula of the sequence? Write the first five terms of the sequence given the recursive formula., 2, 51, 60,... a 1 = -10 A a(n) = 9 + (n-1) a(n) = a n-1 + a(n) - 9(n-1) C a(n) = + 9(n-1) D a(n) = 9 - (n-1) Slide 65 / 11 Slide 66 / 11 29 What is the recursive formula for the sequence? 11,, -5, -1,... 0 What is the sequence that corresponds to the formula? f(n) = 2 - (n-1) A a 1 = 11 a(n) = a n-1 + C a 1 = 11 a 1 = 11 a(n) = a n-1 + 7 D a 1 = 11 A 2, 5,, 51,... 2, 9, 6,,... C 2, 6, 50, 5,... a(n) = a n-1 + 9 a(n) = a n-1 - D 2, 9, 6,,...
Slide 67 / 11 1 What is the sequence that corresponds to the formula? f 1 = 77 f(n) = f n-1 + 1 A 77, 90, 10, 116, 129,... 77, 6, 9, 5,... Slide 6 / 11 Vocabular Geometric Sequence - is a sequence of numbers with a constant common ratio. a 1 - is the first term of the sequence. r - is the common ratio s: (, 12, 6, 10,...) a 1 = and r = (2, 1, 7,.5,...) a 1 = 2 and r = 1/2 C 77, 91, 105, 119, 1,... D 77, 62, 7, 2,... Eplicit and Recursive Geometric Sequences (Youtube Video) Eplicit and Recursive Geometric Sequences (Khan Academ Video) Slide 69 / 11 Slide 70 / 11 Eplicit Formula - a 1(r) (n-1) Recursive Formula - is written in two parts 1. The first part is the first term a 1 2. The second part is a n = r a n-1 for n > 1 (common ratio times the previous term) Note: In the recursive formula the previous term is used to produce the net term. Write an eplicit formula for the following sequences: 1) 7, 5, 175, 75,... 2) 20, 5, 5/, 5/16,... ) 2.5, 5, 10, 20,... Slide 71 / 11 Write an recursive formula for the following sequences: 1) 7, 5, 175, 75,... Slide 72 / 11 Write the first five terms of the sequence given the eplicit formula. a(n) = 27(1/) n-1 2) 20, 5, 5/, 5/16,... ) 2.5, 5, 10, 20,...
Slide 7 / 11 Slide 7 / 11 2 What is the eplicit formula of the sequence? Write the first five terms of the sequence given the recursive formula. a 1 = -2 a(n) = (5)a n-1 for n>1 A f(n) = 2(2) n-1 2, 12, 6,,... f(n) = 2(.5) n-1 C f(n) = 2-2(n-1) D f(n) = 2(2) n-1 Slide 75 / 11 Slide 76 / 11 What is the recursive formula for the sequence? What is the sequence that corresponds to the formula?, 1, 10, 6,... f 1 = -16 A f(n) = (-1/)f n-1 for n > 1 A -16, -, -1, 1/ C D -16,, -1, 1/ 6, -, 1, -1/ D 16, -, 1, -1/ Slide 77 / 11 Slide 7 / 11 5 What is the sequence that corresponds to the formula? f(n) = 2() n-1 A 2,, 2, 12,...,, 16, 2 C 2, 1/2, 1/, 1/16 Multiple Representations of Functions D 2,, 2, 11 Return to Table of Contents
Slide 79 / 11 Multiple Representations An function can be written as a table, graph, verbal model, or equation. We can find the rate of change and the -intercept from an of these representations. Slide 0 / 11 Multiple Representations Look at the given table. We can use an two values to determine slope. We can find where = 0 to determine the -intercept. Remember, to find slope we can use the formula: Slope = 2-1 2-1 To find the -intercept (initial value) we look to where = 0-2 -1 0 1 2 f() -5-2 1 7 Slope = -intercept = 1 2-1 7 - = 2-1 2-1 = = 1 Slide 1 / 11 Multiple Representations Slide 2 / 11 Multiple Representations Sometimes a table will not show the -coordinate of zero. In that case ou need to figure it out. There are a few was to do it. f() = - 1 2 5 6 7 f() -1 2 5 11 1 17 There are two was to find the -intercept here. f() = - 0 1 2 5 6 7 f() - -1 2 5 11 1 17 - + + + We can simpl continue the table so that we can find when = 0. We see that the 's are moving at intervals of +. In order to work backwards to where = 0 we need to subtract from. - - (-1) = - so when = 0, = -. Note: This technique works best when the table is close to = 0 Slide / 11 Multiple Representations Slide / 11 6 What is the slope of the following table? f() = - 1 2 5 6 7 f() -1 2 5 11 1 17 Another technique would be to substitute for and solve for using the equation. f() = - f(0) = (0) - f(0) = - f() -2-1 -1 1 0 1 5 2 7 9
Slide 5 / 11 Slide 6 / 11 7 What is the -intercept of the following table? What is the slope of this table? f() -2-1 -1 1 0 1 5 2 7 9 f() 10 11 5 12 6 1 7 1 Slide 7 / 11 Slide / 11 9 What is the -intercept of this table? f() = - 6 f() 10 11 5 12 6 1 7 1 Carla puts awa a certain amount of mone per week. She started out with a certain amount. How could we figure out what she started out with and what she puts in per week? week 0 1 2 5 6 amount in account 7 2 90 9 106 11 122 Slide 9 / 11 Slide 90 / 11 If we look at the slope or rate of change we can figure out how much she puts in each week. So what is the slope of this table? How much did Carla start out with? # of weeks 5 6 7 9 10 # of weeks account balance 5 6 7 9 10 7 2 90 9 106 11 122 account balance 7 2 90 9 106 11 122
Slide 91 / 11 Tr One Rob is training for a marathon. He is increasing his run each week. ased on the table: Slide 92 / 11 0 Evan is buing pizza from a store that sells each pie for a given price and each topping for set price as well. Use the table to identif the cost of each topping. How much more will he run each week? How man miles was he running before training? How man weeks until he runs a full marathon 26.2 miles? # of toppings 1 2 5 Cost 15.25 16.50 17.75 19.00 20.25 # of weeks training 2 5 6 7 Miles 7 10 1 16 19 22 Slide 9 / 11 1 Evan is buing pizza from a store that sells each pie for a given price and each topping for set price as well. How much is a pie without an toppings. Slide 9 / 11 2 Sandra is going to a buffet. The meal is a fied price but she has to pa for each soda she drinks. What is the initial value? e prepared to eplain how it relates to the scenario. 5 0 25 # of toppings 1 2 5 Cost 15.25 16.50 17.75 19.00 20.25 Cost 20 15 10 5 0 1 2 5 6 Number of drinks Slide 95 / 11 Sandra is going to a buffet. The meal is a fied price but she has to pa for each soda she drinks. What is the slope? Use the points (0, 15) and (6, 25). e prepared to eplain how it relates to the scenario. Cost 5 0 25 20 15 Slide 96 / 11 What does the circled coordinate mean? A C D This tree grows feet ever ear This tree was planted when it was feet There are trees planted The tree was planted when it was ears old. Height 2 2 2 20 16 12 10 5 0 1 2 5 6 Number of drinks 1 2 5 6 7 Years since planting
Slide 97 / 11 = m + b Slide 9 / 11 Tr one!! Mica is having a pool part. The cost to rent the pool is $25 and $7.00 per person attending the part. Notice that regardless of how man people come, Mica will have to pa $25. This is the initial value, the - intercept, the "b", also known as the constant. Also notice that it costs $7.00 per person. This amount will change as the number of guests changes. This will be the slope, the rate of change, or the "m". Raul is at the gas station. He is filling up his gas tank at $.5 per gallon and is also buing $12 worth of food from the convenience store. Write an equation to show this scenario. f() =.5 + 12 So the equation of this problem becomes: = 7 + 25 Slide 99 / 11 Slide 100 / 11 5 Sand charges $5 per necklace that she makes and charges a flat fee of $9 for shipping. Which equation would best fit this scenario? A f() = 5 + 9 f() = 9 + 5 C 5 = 9 D 9 = 5 6 How much does it cost to bu 5 necklaces? A C D $9 $5 $225 $2 Sand charges $5 per necklace that she makes and charges a flat fee of $9 for shipping. Slide 101 / 11 Slide 102 / 11 7 How man necklaces can a person bu with $77? Multiple Representations A C D 6 10 We have learned how to represent a function several was: Table/Ordered Pairs Graph Equation Verbal Description (Scenario) Sand charges $5 per necklace that she makes and charges a flat fee of $9 for shipping. Net we will compare two different models to each other. We will look at the relationship between the two models in terms of the rate of change and domain.
Slide 10 / 11 Two Different Representations Slide 10 / 11 Greater Rate of Change In order to compare the rate of change of two different tpes of representations of functions we simpl find the rate of change of each and compare them. The higher the absolute value of the rate of change, the bigger it is. For eample, if a graph has a slope of - and an equation has a slope of, the slope of the graph is steeper because the absolute value of - = and the absolute value of =. > so The graph has a bigger slope, or rate of change. Let's tr one! Which has a greater rate of change? A f() = -5 +6 Slope = -5 1 (, 5) (2, ) (1, 1) Slope = -1 2 2-1 = 2 1 = 2 absolute value of -5 = 5 and absolute value of 2 = 2 5>2 so A had a greater rate of change than. Slide 105 / 11 Greater Rate of Change Slide 106 / 11 Greater Rate of Change Let's tr to compare a table and a verbal model. A Chris and Shari are going to have a bowling part. It costs $10 to rent a lane and $2 per pair of shoes. 2 6 10 12 1 7 1 19 25 1 7 1 A Chris and Shari are going to have a bowling part. It costs $10 to rent a lane and $2 per pair of shoes. We can turn this into a function. 10 is a constant fee. 2 changes depending on the amount of people at the part. So the equation is f() = 2 + 10. The rate of change = 2 Which has the greater rate of change? Slide 107 / 11 Greater Rate of Change 2 6 10 12 1 7 1 19 25 1 7 1 Slide 10 / 11 Let's tr to compare a table and a verbal model. A Greater Rate of Change Chris and Shari are going to have a bowling part. It costs $10 to rent a lane and $2 per pair of shoes. To find the rate of change we can use the slope formula. 1-7 6-2 = 2 = The rate of change is. 2 6 10 12 1 7 1 19 25 1 7 1 Which has the greater rate of change? Rate of change of A = 2 Rate of change of = Therefore, has the greater rate of change.
Slide 109 / 11 Slide 110 / 11 Which has the greater rate of change? A {(1, ), (2, 6), (, ), (, 10), (5, 12)} 9 Which has the greater rate of change? A f() = 1/ + 5 The school store is selling book covers $1 for 2. Slide 111 / 11 Slide 112 / 11 50 Which has the greater rate of change? 51 Which has the greatest rate of change? A f() = - -9-6 - 0 6 9 f() - - -2-1 0 1 2 A {(1, ), (2, ), (, 5), (, 6), (5, 7)} Ran and Andrew jump down the stairs steps at a time. C f() = 1/ - 2 D Slide 11 / 11 52 Which has the greatest rate of change? A A cable compan charges $12 for ever 2 premium channel. f() = 5 + 6 C {(9, ), (6, 2), (, 1), (0, 0), (-, -1)} D