TABLE OF CONTENTS - UNIT CHARACTERISTICS OF FUNCTIONS TABLE OF CONTENTS - UNIT CHARACTERISTICS OF FUNCTIONS INTRODUCTION TO FUNCTIONS RELATIONS AND FUNCTIONS EXAMPLES OF FUNCTIONS 4 VIEWING RELATIONS AND FUNCTIONS FROM A VARIETY OF DIFFERENT PERSPECTIVES 5 SET OF ORDERED PAIRS PERSPECTIVE 5 Eercise 5 MACHINE PERSPECTIVE 5 Eercise 6 MAPPING DIAGRAM PERSPECTIVE 6 Domain and Range 6 Eercise 7 Eercise 7 NUMERICAL PERSPECTIVE TABLES OF VALUES 8 GEOMETRIC PERSPECTIVE GRAPHS OF FUNCTIONS 8 Discrete Relations 8 Definition of Discrete (from wwwdictionarcom) 8 Continuous Relations 9 Summar Vertical Line Test 9 Eercise 9 ALGEBRAIC PERSPECTIVE EQUATIONS OF RELATIONS Eamples of Discrete and Continuous Relations DETERMINING DOMAIN AND RANGE GIVEN AN EQUATION OF A FUNCTION HOMEWORK PHYSICAL PERSPECTIVE APPLYING FUNCTIONS TO A PHYSICAL SITUATION 4 Eercise 4 HOMEWORK 4 TRANSFORMATIONS OF THE BASE FUNCTIONS 5 THE BASE FUNCTIONS 5 HOMEWORK 6 TRANSFORMATION # - TRANSLATIONS OF FUNCTIONS 7 WHAT IS A TRANSLATION? 7 UNDERSTANDING TRANSLATIONS OF FUNCTIONS 7 ANALYSIS AND CONCLUSIONS 9 EXTREMELY IMPORTANT QUESTIONS AND EXERCISES 9 HOMEWORK 4 TRANSFORMATION # - REFLECTIONS OF FUNCTIONS 5 WHAT IS A REFLECTION? 5 EXAMPLES OF REFLECTIONS 5 IMPORTANT QUESTIONS 6 INVESTIGATION 6 CONCLUSIONS 7 IMPORTANT TERMINOLOGY 7 HOMEWORK 7 COMBINATIONS OF TRANSLATIONS AND REFLECTIONS 8 OVERVIEW 8 Deepening our Understanding A Machine View of g( ) = f( ( + )) 9 SUMMARY 9 IMPORTANT EXERCISES 9 TRANSFORMATION #: STRETCHES AND COMPRESSIONS OF FUNCTIONS TERMINOLOGY EXAMPLES Copright, Nick E Nolfi MCRUO Unit Characteristics of Functions COF-
INVESTIGATION SUMMARY EXTREMELY IMPORTANT QUESTION HOMEWORK PUTTING IT ALL TOGETHER - COMBINATIONS OF TRANSLATIONS AND STRETCHES SUMMARY EXPLANATION EXAMPLE 4 Solution 4 EXAMPLE 5 Solution 5 HOMEWORK 5 INVERSES OF FUNCTIONS 6 INTRODUCTION THE NOTION OF AN INVERSE 6 A CLASSIC EXAMPLE OF A FUNCTION AND ITS INVERSE 6 UNDERSTANDING THE INVERSE OF A FUNCTION FROM A VARIETY OF PERSPECTIVES 6 Eample 7 Eample 7 Observations 8 SUMMARY 8 IMPORTANT QUESTION: HOW ARE THE DOMAIN AND RANGE OF A FUNCTION RELATED TO THE DOMAIN AND RANGE OF ITS INVERSE? 8 IMPORTANT EXERCISE 9 GEOMETRIC VIEW OF THE INVERSE OF A FUNCTION 4 EXAMPLE 4 Solution 4 Alternative Method for Finding f ( ) 4 EXTREMELY IMPORTANT FOLLOW-UP QUESTIONS 4 HOMEWORK 4 APPLICATIONS OF QUADRATIC FUNCTIONS AND TRANSFORMATIONS 4 INTRODUCTION PREREQUISITE KNOWLEDGE 4 PROBLEM SOLVING ACTIVITY 4 SUMMARY OF PROBLEM SOLVING STRATEGIES USED IN THIS SECTION 47 APPENDIX - ONTARIO MINISTRY OF EDUCATION GUIDELINES - UNIT 48 Copright, Nick E Nolfi MCRUO Unit Characteristics of Functions COF-
INTRODUCTION TO FUNCTIONS Relations and Functions Equations called formulas epress mathematical relationships eg + = 6 The equation The equation The equation + = 6 epresses a relationship between and + = 6 can describe the set of points P(, ) ling on a circle of radius 4 with centre (, ) + = 6 can also describe ever right triangle with a hpotenuse of length 4 B solving + = 6 for, we obtain two answers, = 6 and = 6 As we can see from the graph and the accompaning table of values, for an given value of, where 4< < 4, there are two possible values of + = 6 P(, ) A(, 7 ) 4 A(, 7 ) Since for certain values of, there are two possible values of, + = 6 is NOT a function An equation that epresses a relationship between two unknowns is called a RELATION If for each possible value of there is one and onl one corresponding value of, a relation is called a FUNCTION A vertical line can intersect the graph of a function at one and onl one point You have become accustomed to the convention of writing for the independent variable and for the dependent variable To indicate that a relation is a function, we use the notation f( ) instead of to denote the value of the dependent variable This is called function notation This notation also helps to remind us that the value of depends on the value of Consider the function f defined b the equation f( ) = We read this as f of equals squared and it means that the value of f( ) (ie the value of the dependent variable) is obtained b squaring the value of the independent variable Note that f is the name of the function and f( ) is the -value Note also that f of can be seen as a short form for f is a function of, which means that the function s value depends on If we write, for instance, f (), we mean the -value obtained when = The notation f () is read f at Traditionall, the letter f is used to denote functions because the word function begins with the letter f This does not disqualif other letters! Functions can equall well be given other names such as g, h and q Copright, Nick E Nolfi MCRUO Unit Characteristics of Functions COF-
Eamples of Functions Equation of Function written without Function Notation Equation of Function written with Function Notation Graph Eamples = f( ) = f (4) = 4 = (4, ) lies on the graph of f f (5) = 5 = 5 (5, 5) lies on the graph of f f () = 44 (, ) lies on the graph of f f ( ) =, which is undefined in the set of real numbers = g ( ) = g () = = (, ) lies on the graph of g g () = = 4 (, 4) lies on the graph of g g (5) = 5 = 5 (5, 5) lies on the graph of g g( ) = ( ) = 9 (, 9) lies on the graph of g = 5+ 6 h ( ) = 5+ 6 h () = () 5() + 6 = (, ) lies on the graph of h h () = () 5() + 6 = 4 (, 4) lies on the graph of h h (5) = 5 (5) 5(5) + 6 = 56 (5, 56) lies on the graph of h h( ) = ( ) ( ) 5( ) + 6 = 8 (, 8) lies on the graph of h 4 = + 4 p ( ) = + 4 p () = () + = (, ) lies on the graph of p 4 p () = () + = 9 (, 9) lies on the graph of p 4 p () = () + = (, ) lies on the graph of h 4 ( ) ( ) ( ) 9 p = + = (, 9) lies on the graph of p Copright, Nick E Nolfi MCRUO Unit Characteristics of Functions COF-4
VIEWING RELATIONS AND FUNCTIONS FROM A VARIETY OF DIFFERENT PERSPECTIVES Set of Ordered Pairs Perspective Ordered Pair Two numbers written in the form (, ) form an ordered pair As the name implies, order is important Set A set is a group or collection of numbers, variables, geometric figures or just about anthing else Sets are written using set braces {} For eample, {,, } is the set containing the elements, and Note that order does not matter in a set The sets {a, b, c} and {c, a, b} are the same set Repetition does not matter either, so {a, b} and {a, a, b, b, b} are the same set The main idea of a set is to group objects that have common properties For eample, the smbol represents the set of rational numbers, the set of all fractions, including negative fractions and zero Alternativel, we can think of a rational number as a ratio of two integers That is, all rational numbers share the common propert that the can be written in the form a, where a and b are both integers and b b Formall, this is written a = : a, b, b Note that the smbol denotes the set of integers and the smbol means is an b element of Relation A relation is an set of ordered pairs For eample, the set {(,),(,),(,),(,),(,4),(,5), } is a relation In this relation, the number is associated with ever non-negative integer This relation can be epressed more precisel as follows: {(, ) : =,, } This is read as, the set of all ordered pairs (, ) such that is equal to and is an non-negative integer Function A function is a relation in which for ever -value, there is one and onl one corresponding -value The relation given above is not a function because for the -value, there are an infinite number of corresponding -values However, the relation {(,),(,),(,),(,),(4,),(5,), } = {(, ) :,, = } is a function because there is onl one possible -value for each -value Formall, this idea is epressed as follows: Suppose that R represents a relation, that is, a set of ordered pairs Suppose further that a = b whenever ( ac, ) R and (,) bc R(ie a must equal b if ( ac, ) and (,) bc are both in R) Then R is called a function Eercise Which of the following relations are functions? Eplain (,),(,),(,),(,) (a) { } (b) {(,),(,),(,),(5,)} (c) {(, ):, = } (d) {(, ) :,, + = 69} Machine Perspective Sets of ordered pairs allow us to give ver precise mathematical definitions of relations and functions However, being a quite abstract concept, the notion of a set ma be somewhat difficult to understand You can also think of a function as a machine that accepts an input and then produces a single output There is one and onl one output for ever input Function Machine Input operation Output Copright, Nick E Nolfi MCRUO Unit Characteristics of Functions COF-5
For eample, consider the function machine that adds 4 to the input to produce the output add 4 + 4 Eercise Consider the function machine that takes an input and outputs square the input What is the output if the input is: 4, 4? Is it possible for a specific input to have more than one output? Eplain What is the input if the output is 9? Is it possible for a specific output to have more than one input in this case? Eplain The following ordered pairs show the inputs and outputs of a function machine What is the operation? (a) (, 5), (, ), (, 6) operation: (b) (, ), (, 8), (, 7), (, 5), (, ) operation: Mapping Diagram Perspective In addition to function machines, functions can be represented using mapping diagrams, tables of values, graphs and equations You ma not know it but ou alread have a great deal of eperience with functions from previous math courses A mapping diagram uses arrows to map each element of the input to its corresponding output value Remember that a function has onl one output for each input Can a function have more than one input for a particular output? Eplain Domain and Range The set of all possible input values of a function is referred to as the domain That is, if D represents the domain of a D= :(, ) f function f, then { } The set of all possible output values of a function is referred to as the range That is, if R represents the domain of a R= :(, ) f function f, then { } Copright, Nick E Nolfi MCRUO Unit Characteristics of Functions COF-6
Eercise In the table, the relations are shown using a mapping diagram The domain (or inputs) on the left have arrows pointing to the range (or output) on the right Eplain whether the relation is a function or not and justif our answer The first one is done for ou Relation as a Mapping Diagram Domain Range Is it a function? Wh or wh not? Set of ordered pairs 6 5 Since ever element of the domain has onl one corresponding element in the range, this relation is a function This function has a one-to-one mapping {(, ), (,), (, 6), (, 5), (, )} 6 5 6 5 6 5 A one-to-one mapping is eplained above Which relation above has a man-to-one mapping? a one-to-man mapping? a man-to-man mapping? Eercise Use the Internet to find definitions of the following terms: one-to-one mapping, one-to-man mapping, man-to-man mapping, surjection, injection, bijection Copright, Nick E Nolfi MCRUO Unit Characteristics of Functions COF-7
Numerical Perspective Tables of Values Consider the following relations epressed in table form (a) Which relations are functions? Justif our answers (b) Draw a mapping diagram for each relation (c) Write the set of ordered pairs for each relation Relation as a Table of Values 9 9 5 5 5 5 5 Is it a function? Wh or wh not? Mapping Diagram Set of Ordered Pairs Geometric Perspective Graphs of Functions Discrete Relations A discrete relation either has a finite number of ordered pairs OR the ordered pairs can be numbered using integers Graphs of discrete relations consist of either a finite or an infinite number of disconnected points, much like a connectthe-dots picture before the dots are connected Eamine the following graphs of discrete relations and then complete the table DO NOT CONNECT THE DOTS! Relation in Graphical Form Is it a function? Wh or wh not? Table of Values Mapping Diagram Set of Ordered Pairs Definition of Discrete (from wwwdictionarcom) apart or detached from others; separate; distinct: si discrete parts consisting of or characterized b distinct or individual parts; discontinuous Mathematics a (of a topolog or topological space) having the propert that ever subset is an open set b defined onl for an isolated set of points: a discrete variable c using onl arithmetic and algebra; not involving calculus: discrete methods Copright, Nick E Nolfi MCRUO Unit Characteristics of Functions COF-8
Continuous Relations Unlike the graphs on the previous page, the following graphs do not represent discrete relations The are called continuous relations because their graphs do not consist of disconnected points (You need to stud calculus to learn a more precise definition of continuit For now, it suffices to think of continuous relations as those whose graphs are unbroken ) Eamine the graphs and decide whether the represent functions Summar Vertical Line Test When ou look at a graph, how can ou decide whether it represents a function? Eercise Draw two graphs of functions, one that is discrete and one that is continuous Draw two graphs of relations that are NOT functions, one that is discrete and one that is continuous Copright, Nick E Nolfi MCRUO Unit Characteristics of Functions COF-9
Algebraic Perspective Equations of Relations The perspectives given above help us to understand the properties of relations and functions However, when it comes time to computing with relations and functions, then equations become an indispensable tool! For each of the following (the first is done for ou) (a) determine whether it is a function (b) state an equation that describes the relation (c) complete a table of values and a mapping diagram for each relation Note that the given relations are all continuous, which means that it is impossible to create a complete table of values or mapping diagram (Wh?) (d) state the domain and range and the tpe of mapping Relation in Graphical Form Is it a function? Eplain This is a function because an vertical line will intersect at onl a single point Equation f( ) = Table of Values f() 4 4 Mapping Diagram 9 4 Domain and Range (D & R) The domain is the set of all real numbers, that is, D = The range is the set of all real numbers greater than or equal to zero, that is, R= : { } Tpe of Mapping man-toone Copright, Nick E Nolfi MCRUO Unit Characteristics of Functions COF-
Eamples of Discrete and Continuous Relations (a) Discrete Function with a Finite Number of Ordered Pairs Roll Two Dice and Observe the Sum Consider rolling two dice and observing the sum Clearl, there are onl eleven possible outcomes, the whole values from to inclusive It s also clear that some outcomes are more likel than others If ou have plaed board games that involve dice rolling, ou surel have noticed that rolls like and occur infrequentl while 7 occurs much more often The reason for this is that there is onl one wa of obtaining, for eample, but there are si was of obtaining 7 B working out all the possibilities, we obtain the probabilit of each outcome as shown in the table at the right Probabilit Roll Two Dice and Observe Sum The main point to remember here is that that we have a discrete Roll function In this case, there are onl a finite number of ordered pairs in this function Furthermore, the are disconnected from each other, as the graph clearl shows This disconnectedness is a natural consequence of the phsical nature of rolling a pair of dice It is possible to roll a or a but it is not possible to roll 459654 or an number that is not a whole number between and inclusive! Possible Outcomes 4 5 6 7 8 9 Probabilit 778 6 5556 6 8 6 4 6 5 889 6 6 6667 6 5 889 6 4 6 8 6 5556 6 778 6 (b) Discrete Function with an Infinite Number of Points Probabilit of n Consecutive Heads in n Coin Tosses Once again, it makes no sense to connect the dots in this case The number of coin tosses must be a positive whole number It is difficult to imagine how one could make 75, 499 or 5 coin tosses! Once again then, we have a discrete function The difference in this case is that there are an infinite number of possibilities There is no limit to the number of coin tosses that one can make Moreover, there is no bound on the number of consecutive heads that can be obtained Although it is etremel unlikel to obtain a ver large number of consecutive heads, it is not impossible! Probabilit of obtaining n Consecutive Heads p(n) We can take this eample equation Let p(n) obtaining n consecutive Then, n n pn ( ) = = =, n n Note that n means that n is an element of the set of natural numbers, that is, n is a positive whole number one step further and write an represent the probabilit of heads when a coin is tossed n times Number of Tosses (n) Probabilit of Obtaining n Consecutive Heads Number of Coin Tosses (n) 5 5% = = 5 5% 4 = = 5 5% 8 = = 4 65 65% 6 = = 5 5 5% = = The number of ordered pairs is infinite in this Copright, Nick E Nolfi MCRUO Unit Characteristics of Functions case because (at least in principle) an whole COF- number of consecutive heads is possible
(c) Continuous Function Now consider the growth rate of humans At the right is a graph that shows an eample of growth velocit (in cm/ear) versus age (in ears) Notice that this function is quite different from the discrete eamples in (a) and (b) Unlike the roll of a pair of dice or the number of coin tosses, it is not possible to assign a whole number to a person s age or growth rate Both quantities var continuousl over a certain range of values Hence, the graph of a continuous function will not consist of a series of distinct, disconnected points All the points fuse together to form an unbroken curve Determining Domain and Range given an Equation of a Function Complete the following table The first three rows are done for ou Equation of Function Graph Domain and Range Eplanation f( ) = D = { : } R= R The domain consists of all real numbers because an real number can be squared Since the square of an real number must be non-negative, the range consists of all real numbers greater than or equal to zero g ( ) = D = R = The domain consists of all real numbers because an real number can be cubed The range also consists of all real numbers because the cube of a positive real number is positive, the cube of a negative real number is negative and the cube of zero is zero h ( ) = 5+ 6 Continued on net page { : } D= { : } R= An value of is acceptable ecept for If =, however, the denominator of the epression is zero Since division b zero is undefined, B noting that 5+ 6 ( )( ) = =, we see that if were allowed to equal, would equal Therefore, can take on an real value ecept for Note the open circle on the graph of h This indicates that the ordered pair (,) is not part of the graph That is, it is not included in the set of ordered pairs making up the function Copright, Nick E Nolfi MCRUO Unit Characteristics of Functions COF-
Equation of Function Graph Domain and Range Eplanation f( ) = + 6 g ( ) = 6 5 h ( ) = + u pu ( ) = u 7u+ f( u) = u + Homework pp 78 79, # 5 Copright, Nick E Nolfi MCRUO Unit Characteristics of Functions COF-
Phsical Perspective Appling Functions to a Phsical Situation A cannonball is fired verticall into the air with an initial speed of 49 m/s Its height above the ground in metres, at a time t seconds after it is fired, is given b the function h defined b ht ( ) = 49t 49t Because this function is used to model a phsical situation, we must keep in mind that not all values of t make sense For eample, it is nonsensical to consider negative values of t because the timing begins at t= when the cannonball is fired Similarl, there is no point in considering values of t greater than the time that it takes to fall back to the earth This is summarized in the table given below Phsical Situation Algebraic Support Graph Domain and Range ht ( ) = 49t 49t = 49t + 49t A cannonball is fired verticall into the air = 49( t + t) with an initial speed of = 49( t + t+ 5 5 ) 49 m/s Its height above the ground in metres, at = 49( t + 5) + 49(5) D= { : } a time t seconds after it = 49( t + 5) + 5 R= : 5 is fired, is given b the function h defined b 49t 49t = ht ( ) = 49t 49t 49 t( t) = t = or t = t = or t = { } Since the function ht ( ) 49t 49t = is used to model a phsical situation, its domain and range are restricted Eercise Complete the following table Phsical Situation Algebraic Support Graphs Domain and Range A baseball is hit from a point m above the ground, at an angle of 7º to the ground and with an initial velocit of 4 m/s (Originall the initial speed given was m/s This produced ver unrealistic answers) The horizontal distance travelled b the ball is given b the function dt ( ) = (4cos7 ) tand the vertical distance travelled b the ball is given b the function ht ( ) = 49 t + (4sin7 ) t+ How far did the ball travel? What was the maimum height reached b the ball? Note that time is measured in seconds and distance is measured in metres Homework pp 8 8, #6, 9,,,, 4, 5, 6, 8, 9,, Copright, Nick E Nolfi MCRUO Unit Characteristics of Functions COF-4
TRANSFORMATIONS OF THE BASE FUNCTIONS The Base Functions Mathematicians stud mathematical objects such as functions to learn about their properties An important objective of such research is to be able to describe the properties of the objects of stud as concisel as possible B keeping the basic set of principles as small as possible, this approach greatl simplifies the daunting task of understanding the behaviour of mathematical structures Take, for eample, the goal of understanding all quadratic functions Instead of attempting to understand such functions all in one fell swoop, mathematicians divide the process into two simpler steps as shown below First, understand the base function f( ) Then, learn how to transform the base function = completel f( ) = into an other quadratic Complete the following table to ensure that ou understand the base functions of a few common families of functions Famil of Functions Base Function Linear f( ) = Domain and Range Table of Values Graph Intercepts f( ) Quadratic Cubic f( ) f( ) = = f( ) f( ) Copright, Nick E Nolfi MCRUO Unit Characteristics of Functions COF-5
Famil of Functions Base Function Domain and Range Table of Values Graph Intercepts Square Root f( ) = f( ) 4 6 9 4 6 Reciprocal f( ) = f( ) 6 4 4 4 4 6 Absolute Value f( ) = f( ) 7 4 4 7 Homework pp 8 8, #, 7, 8, 9 Copright, Nick E Nolfi MCRUO Unit Characteristics of Functions COF-6
TRANSFORMATION # - TRANSLATIONS OF FUNCTIONS What is a Translation? A vertical translation of a function is obtained b adding a constant value to each value of the dependent variable of the given function This results in the graph of the function sliding up or down, depending on whether the constant is positive or negative A horizontal translation of a function is obtained b adding a constant value to each value of the independent variable of the given function This results in the graph of the function sliding left or right, depending on whether the constant is positive or negative Eample Vertical Translations h f ( ) = + = ( ) + Eample Horizontal Translations g f ( ) = ( 7) = ( 7) f( ) = g f ( ) = 7 = ( ) 7 h f ( ) = ( + ) = ( + ) f( ) = Understanding Translations of Functions Complete the following table of values f( ) = g ( ) = + = f( ) + 4 4 Sketch the graphs of f, g, h and q h f ( ) = ( + 5) = ( + 5) q f ( ) = ( + 5) + = ( + 5) + Copright, Nick E Nolfi MCRUO Unit Characteristics of Functions COF-7
Complete the following table of values 4 4 f( ) = Sketch the graphs of f, g, h and q g ( ) = = f( ) h ( ) = = f( ) q ( ) = = f( ) Complete the following table of values f( ) 4 4 = g ( ) = + = f( ) + h ( ) = 4 = f( 4) q ( ) = 4 + = f( 4) + Sketch the graphs of f, g, h and q Copright, Nick E Nolfi MCRUO Unit Characteristics of Functions COF-8
Analsis and Conclusions Now carefull stud the graphs and the tables on the two previous pages Then complete the following table Base Function = f( ) Translations of Base Function = f( h), h = f( ) + k, k = f( h) + k, h, k Description of Translations = f( ) = f( + ) = f( ) + = f( ) Etremel Important Questions and Eercises So far we have considered both horizontal and vertical translations of functions Does it matter in which order these translations are performed? Complete the following table Function Base Function Translation(s) of Base Function Required to obtain Function There are two correct answers for this one (a) f( ) = + 6 (b) g ( ) = 5 + 6 (c) h ( ) = 6 + 5 (d) p ( ) = 5 + (e) q ( ) = + + + Copright, Nick E Nolfi MCRUO Unit Characteristics of Functions COF-9
Complete the following table Graph of Relation Relation or Function? Discrete or Continuous? Graph of Translated Relation Translate units up and to the left If the given relation is a function = f( ), then this is Translate units down If the given relation is a function = f( ), then this is Translate units down and to the left If the given relation is a function = f( ), then this is Copright, Nick E Nolfi MCRUO Unit Characteristics of Functions COF-
Graph of Relation Relation or Function? Discrete or Continuous? Graph of Translated Relation Translate units down and to the right If the given relation is a function = f( ) then this is Translate unit down and 4 to the right If the given relation is a function = f( ) then this is Translate units up and to the right If the given relation is a function = f( ) then this is Copright, Nick E Nolfi MCRUO Unit Characteristics of Functions COF-
4 Without using a table of values, graph the following functions on the same grid In addition, state the domain and range of each function Functions Graphs Domain and Rage = + = + = 4 = + 5 = + 5 = + 7 4 Copright, Nick E Nolfi MCRUO Unit Characteristics of Functions COF-
Functions Graphs Domain and Range = + = ( + ) = + ( 5) = 6 = 6 = + 5 7 Copright, Nick E Nolfi MCRUO Unit Characteristics of Functions COF-
5 Without graphing, state the domain, range, intercepts (if an) and asmptotes (if an) of each given function Function Domain and Range Intercepts Asmptotes f ( ) = + 5 g ( ) = h ( ) = 5 st ( ) = 5 t + p ( ) = 8 + 5 6 For each graph, state an equation that best describes it Homework pp 89 9, 5, 6f, 7e,f, 8e,f,,,,4, 6, 8 Copright, Nick E Nolfi MCRUO Unit Characteristics of Functions COF-4
TRANSFORMATION # - REFLECTIONS OF FUNCTIONS What is a Reflection? A reflection of a relation in a given line is produced b replacing each point in the given relation b a point smmetricall placed on the other side of the line Intuitivel, a reflection of a point in a line is the mirror image of the point about the line Think of what ou see when ou look at ourself in a mirror First of all, our reflection appears to be behind the surface of the mirror Moreover, its distance from the mirror appears to be eactl the same as our distance from the mirror Eamples of Reflections Relation 8 9 5 4 8 4 9 7 6 4 8 Reflection in -ais 8 9 5 4 8 4 9 7 6 4 8 Original Relation: Set of Blue Dots Refection in the -ais: Set of Red Dots Relation 8 9 5 4 8 4 9 7 6 4 8 Reflection in -ais 8 9 5 4 8 4 9 7 6 4 8 Original Relation: Set of Blue Dots Refection in the -ais: Set of Red Dots Relation 8 9 5 4 8 4 9 7 6 4 8 Reflection in = 9 8 4 5 8 4 9 7 4 6 8 Original Relation: Set of Blue Dots Refection in line = : Set of Red Dots The reflection of = (blue) in the -ais is = (red) The reflection of = ( 5) 4 (blue) in the -ais is = ( + 5) 4 (red) The reflection of in the line = (blue) = is = ± (red) Copright, Nick E Nolfi MCRUO Unit Characteristics of Functions COF-5
Important Questions Suppose that the point Q is the reflection of the point P in the line l Suppose further that the line segment PQ intersects the line l at the point A What can ou conclude about the lengths of the line segments PA and QA? Draw a diagram to illustrate our answer Is the reflection of a function in the -ais also a function? Eplain Is the reflection of a function in the -ais also a function? Eplain 4 Is the reflection of a function in the line = also a function? Eplain 5 Suppose that R represents a relation that is not a function Can a reflection of R be a function? If so, give eamples Investigation You ma use a graphing calculator or graphing software such as TI-Interactive to complete the following table Function Equation of f( ) Equation of f( ) Equation of f( ) Graph of = f( ) Graph of = f( ) Graph of =f( ) f( ) = f( ) = f( ) = Continued on net page Copright, Nick E Nolfi MCRUO Unit Characteristics of Functions COF-6
Function Equation of f( ) Equation of f( ) Equation of f( ) Graph of = f( ) Graph of = f( ) Graph of =f( ) f( ) = f( ) = f( ) = Conclusions Given a function f, the graph of = f( ) is the of the graph of = f( ) in the the graph of = f( ) is the of the graph of = f( ) in the the graph of =f( ) is the of the graph of = f( ) in the followed b the of the graph of in the Important Terminolog Suppose that a transformation is applied to a point P to obtain the point Q Then, P is called the pre-image of Q and Q is called the image of P If Q is the image of P under some transformation and P = Q, then P is said to be invariant under the transformation Homework Read pp 96 pp 7, #,,, 4cd, 7, 9,,,, 4, 5, 7, 8, 9,, Copright, Nick E Nolfi MCRUO Unit Characteristics of Functions COF-7
Overview COMBINATIONS OF TRANSLATIONS AND REFLECTIONS Translations of Base Function Base Function: = f( ) (eg f( ) = ) Reflections of Base Function Horizontal Vertical In -ais (Horizontal) In -ais (Vertical) = g ( ) = f( h) (, ) ( + h, ) eg g ( ) = + (, ) (, ) Translations = g ( ) = f( ) + k (, ) (, + k) eg g ( ) = (, ) (, ) Horizontal combined with Vertical Combinations = g ( ) = f( ) (, ) (, ) eg g ( ) = (, ) (, ) Reflections = g ( ) = f( ) (, ) (, ) eg g ( ) = (, ) (, ) In -ais (vertical) combined with -ais (horizontal) = g ( ) = f( h) + k (, ) ( + h, + k) = g ( ) =f( ) (, ) (, ) eg g ( ) = + (, ) (, ) eg g ( ) = (, ) (, ) Translations and Reflections = g ( ) =f( ( h)) + k (, ) ( + h, + k) eg g ( ) =f( ( + )) = ( + ) (, ) (, ) To understand this combination of translations and reflections, it is necessar to treat the vertical and horizontal transformations separatel understand the order of operations eg g ( ) = ( + ) = ( ) ( )( + ) Vertical ( ) ( + ) Horizontal ( ) ( + ) ( + ) ( ) ( ) ( + ) Because of the order of operations, ( + ) must be multiplied b before is subtracted Therefore, the reflection in the -ais must be performed before the vertical translation down units ( )( + ) ( ) Keep in mind that ( + ) is the -value used as the input for the function f However, for the function g defined b g( ) = f( ( + )), the input is still This means that given the value of ( + ), we need to reverse the operations to find (in the order opposite the order of operations) Therefore, the transformations must be performed in the order opposite the order of operations First f is reflected in the -ais and then the resulting function is translated units left Note that division b is the same as multiplication b, which is wh the transformation is still a reflection Copright, Nick E Nolfi MCRUO Unit Characteristics of Functions COF-8
Deepening our Understanding A Machine View of g( ) = f( ( + )) If ou found the discussion on the previous page a little daunting, it ma be helpful to return to the machine analog We can think of the function g as a single machine that consists of several simpler machines If the input is given to g, through a series of small steps, the output f( ( + )) is eventuall produced g + + () ( + ) f f( ( + )) f f( ( + )) This diagram should help ou to understand wh everthing works backwards for horizontal transformations Although we must evaluate f at ( + ) to obtain the value of g(), g is still evaluated at Therefore, we must work our wa in reverse from ( + ) to to understand the horizontal transformations applied to f to produce g Summar To obtain the graph of = g( ) =f( ( h)) + k = ( ) f(( )( h)) + k from the graph of = f( ), perform the following transformations Vertical (Follow the order of operations) First reflect in the -ais Then translate k units up if k > or k units down if k < Horizontal (Reverse the operations in the order opposite the order of operations) First reflect in the -ais Then translate h units right if h > or h units left if h < Important Eercises Perform the specified transformations on the graph of each given function Graph of Function = f( ) Transformation of f Graph of Transformed Function =g( ) g ( ) =f( ) Eplain this transformation of f in words Summar (, ) ( + h, + k) g ( ) =f( ) + Eplain this transformation of f in words Copright, Nick E Nolfi MCRUO Unit Characteristics of Functions COF-9
g ( ) = f( ( + )) 4 Eplain this transformation of f in words g ( ) =f( ( )) + 6 Eplain this transformation of f in words Possible equations of f and g f() = g() = g ( ) =f( ( 5)) + 8 Eplain this transformation of f in words Possible equations of f and g f() = g() = g ( ) = f( 5) + Eplain this transformation of f in words (Be careful with this one) Possible equations of f and g f() = g() = Copright, Nick E Nolfi MCRUO Unit Characteristics of Functions COF-
TRANSFORMATION #: STRETCHES AND COMPRESSIONS OF FUNCTIONS Terminolog Dilatation, Dilation, Epansion or Stretch of a Function A transformation in which all distances on the co-ordinate plane are lengthened b multipling either all -co-ordinates (horizontal dilation) or all -co-ordinates (vertical dilation) b a common factor greater than Compression of a Function A transformation in which all distances on the co-ordinate plane are shortened b multipling either all -co-ordinates (horizontal compression) or all -co-ordinates (vertical compression) of a graph b a common factor less than Eamples g ( ) = f( ) = f( ) = Notice that the -co-ordinates of points ling on the graph of the function g are three times the corresponding -co-ordinates of points ling on f We sa that f is stretched verticall b a factor of to produce g g ( ) = f( ) = ( ) = 9 f( ) = Notice that the -co-ordinates of points ling on the graph of the function g are three times the -co-ordinates of corresponding points ling on f We sa that f is stretched horizontall b a factor of to produce g Investigation Sketch f( ) =, g ( ) f( ) ( ) 4 = = = and h ( ) = f( ) = ( ) = on the same grid Describe 4 how the graphs of g and h are related to the graph of f Sketch f( ) =, g ( ) = f( ) = and h ( ) = f( ) = on the same grid Describe how the graphs of g and h are related to the graph of f Copright, Nick E Nolfi MCRUO Unit Characteristics of Functions COF-
Sketch f( ) =, g ( ) f( ) = = and h ( ) f( ) graphs of g and h are related to the graph of f = = on the same grid Describe how the 4 Sketch f( ) =, g ( ) = f( ) = and h ( ) = f( ) = on the same grid Describe how the graphs of g and h are related to the graph of f 5 Sketch f( ) = and g ( ) = f( ) = Describe how the graph of g is related to the graph of f 6 Sketch f( ) = and g ( ) = f( ) = Describe how the graph of g is related to the graph of f Summar To obtain the graph of = g( ) = af ( ), a from the graph of = f( ) To obtain the graph of = g( ) = f ( b), b from the graph of = f( ) To obtain the graph of = g( ) = af ( b), a, b from the graph of = f( ) Etremel Important Question In the transformation = g( ) = af ( b), a, b of = f( ), what happens if a = or b =? What can ou conclude from this? Homework Read pp 8 pp 9, bdg, f,, 4iv, 6, 7, 8, 9, d, df,,, 4, 6 Copright, Nick E Nolfi MCRUO Unit Characteristics of Functions COF-
PUTTING IT ALL TOGETHER - COMBINATIONS OF TRANSLATIONS AND STRETCHES Summar B now we have acquired enough knowledge to combine all the transformations To obtain the graph of = g( ) = af ( b( h)) + k from the graph of = f( ), perform the following transformations Vertical (Follow the order of operations) First stretch or compress verticall b the factor a (, ) (, a) If a >, stretch b a factor of a ( a > is a short form for a > or a < ) If < a <, compress b a factor of a If a < (ie a is negative), the stretch or compression is combined with a reflection Then translate k units up if k > or k units down if k < (, ) (, + k) Horizontal (Reverse the operations in the order opposite the order of operations) First stretch or compress horizontall b the factor = b (, ) ( b, ) b If b >, compress b the factor b If < b <, stretch b the factor b If b < (ie b is negative), the stretch or compression is combined with a reflection Then translate h units right if h > or h units left if h < (, ) ( + h, ) Summar using Mapping Notation The following shows how an ordered pair belonging to f (pre-image) is mapped to an ordered pair belonging to g (image) under the transformation defined b g( ) = af ( b( h)) + k Eplanation pre-image image + + (, ) ( b h, a k) Vertical af( b ( h)) + k Horizontal b( h) f( b ( h)) a + k af( b ( h)) + k Because of the order of operations, f( b ( h)) must be multiplied b a before k is added Therefore, the vertical stretch b a factor of a must be performed before the vertical translation b k units b( h) b + h To find the output of g when the input is given, we look at the output of f when the input b ( h) is given Therefore, the transformations must be performed in the reverse order, opposite the order of operations First f is stretched or compressed horizontall b a factor of b - and then the resulting function is translated h units right if h> or h units left if h< (, ) (, a + k) (, ) ( b + h, ) Copright, Nick E Nolfi MCRUO Unit Characteristics of Functions COF-
Eample Given f ( ) ( ) =, sketch the graph of g ( ) = f( 6) + Solution First of all, we should rewrite the transformation so that it matches the standard form g( ) = af ( b( h)) + k This means that we need to factor 6 Then, g ( ) = f( 6) + = f(( )) + Net we list all the transformations, remembering to treat the vertical and horizontal transformations separatel a =, b =, h =, k = (, ) ( +, + ) Vertical (Follow order of operations) (i) Stretch verticall b a factor of a = (Stretch b a factor of, then reflect in the -ais) (ii) Shift up b k = units Horizontal (Reverse the order of operations) (i) Compress horizontall b a factor of b = (ii) Shift right b h = units Finall, we perform the transformations (At each step, watch how the verte is mapped from pre-image to image) (a) (b) = f( ) p ( ) = h ( ) + = h ( ) h ( ) = f( ) (, ) (, ) eg (, ) (, ( )) = (,6) (, ) (, + ) eg (,6) (,6 + ) = (,9) (c) (d) = p ( ) = q ( ) q ( ) = p( ) g ( ) = q ( ) (, ) (, ) eg (,9) (),9 =,9 (, ) ( +, ) eg 8,9 +,9 =,9 Copright, Nick E Nolfi MCRUO Unit Characteristics of Functions COF-4
Eample (a) Determine the zeros of f( ) = 9+ 5 (b) Determine the zeros of g ( ) = f( 7) Solution (a) 9+ 5 = (b) Algebraic Solution 4 5+ 5 = The zeros of g can be found b solving the equation g= ( ), that is, 4 () 5( ) = f( 7) = To this end, it is easier to use the factored form of f, f( ) = ()(4 5) ()(4 5) = = or 4 5 = g= ( ) f( 7) = 5 = or = [( 7) ][4( 7) 5] = 4 f( ) = 9+ 5 ( 6)( 88 5) = Recall that the horizontal ( 6)( 8 ) = transformations are done in the 6 = or 8 = reverse order This is wh the, image of is ( + 7), which = = or = corresponds to a shift 7 units 6 8 right followed b a 5 compression b a factor of, 4 Geometric Solution Since the given transformation does not involve a vertical shift, the zeros of g are simpl the images of the zeros of f under the given transformation g ( ) = f( 7) =, 8 44 78, Pre-image (, ) Image ( + 7), ( ), 5, 4 ( + 7), () =, 5 ( + 7), () =, 4 8 ( ) ( ) Notice that the roots of g= ( ) are the images of the roots of f( ) = Homework Read pp 9 pp 4 4, bef, cfh, 4bh, 5ef, 6adf, 7bdfh, 8f, 9df, 4, 5, 6, 7, 8, 9, Copright, Nick E Nolfi MCRUO Unit Characteristics of Functions COF-5
INVERSES OF FUNCTIONS Introduction The Notion of an Inverse On an intuitive level, the inverse of a function is simpl its opposite The following table lists some common operations and their opposites Some Operations and their Inverses Eample Observations Conclusion + + 5 = 8 8 5 = + : 8 : 8 The inverse of an 5 = 5 5 5 = : 5 : 5 square a number square root 5 = 5 5 = 5 : 5 5 : 5 5 operation gets ou back to where ou started It undoes the operation A Classic Eample of a Function and its Inverse With the eception of the United States and perhaps a ver small number of other countries, the Celsius scale is used to measure temperature for weather forecasts and man other purposes In the United States, however, the Fahrenheit scale is still used for most non-scientific purposes The following shows ou how the two scales are related C = degrees Celsius, F = degrees Fahrenheit f = function that outputs the Fahrenheit temperature when given the Celsius temperature C as input A Function f that Converts Celsius Temperatures into Fahrenheit Temperatures F Boiling Point of Water at Sea Level The Inverse of f, written f, converts Fahrenheit Temperatures into Celsius Temperatures C C Freezing Point of Water at Sea Level 9 f( C) = C+ 5 F Absolute Zero C f f( C ) F f f ( F) To obtain the equation of f, appl the inverse operations in the reverse order: Forward: C (9 5) + F Reverse: F (9 5) C (Note that dividing b 9/5 is the same as multipling b 5/9) C = f ( F) = ( F ) 9 5 Understanding the Inverse of a Function from a Variet of Perspectives It is critical that ou understand that the inverse of a function is its opposite That is, the inverse of a function must undo whatever the function does The inverse of a function is denoted f It is important to comprehend that the in this notation is not an eponent The smbol f means the inverse of the function f, not f The notation f( ), called mapping notation, can be used to conve the same idea as a function machine Copright, Nick E Nolfi MCRUO Unit Characteristics of Functions COF-6
Eample Does f( ) = have an inverse? If so, what is the inverse function of f( ) =? Solution B eamining the si different perspectives of functions that we have considered throughout this unit, we can easil convince ourselves that f( ) = does have an inverse, namel f ( ) = f f f = {(, 7),(, 8),(,8),(,7), } {( 7, ),( 8, ),(8,),(7,), } f = Note that the ordered pairs of f are reversed That is, the and -coordinates of the ordered pairs of f are interchanged to obtain the ordered pairs of f f( ) 7 8 8 7 f ( ) 7 8 8 7 7 8 8 7 7 8 8 7 f( ) = f ( ) = = Eample Does f( ) = have an inverse? If so, what is the inverse function of f( ) =? Solution In the last eample we learned that the inverse of a function f is obtained b interchanging the and -co-ordinates of the ordered pairs of f Let s tr this on a few of the ordered pairs of the function f( ) = {(, ):,, } {,(,9),(,4),(,),(,),(,),(,4),(,9), } f= = = The inverse of f should be the following relation: {(, ):(, ) f} = {,(9, ),(4),(, ),(,),(,),(4,),(9,), } It is readil apparent that there is something wrong, however This relation is not a function Therefore, not have an inverse function unless we restrict its domain f( ) = does Copright, Nick E Nolfi MCRUO Unit Characteristics of Functions COF-7
f( ) = f ( ) =, f ( ) =, f ( ) =, f ( ) =, The relation obtained b interchanging the and co-ordinates fails the vertical line test It is not a function The inverse of f is a function if the domain of f is restricted to the set of all real numbers Clearl, f ( ) = Alternativel, the domain of f can be restricted to the set of all real numbers In this case, the inverse is f ( ) = Observations f( ) = is one-to-one and has inverse function f ( ) = f( ) = is man-to-one; the inverse of f is not a function unless its domain is restricted to a piece of f that is one-to-one (eg or ) Summar We can etend the results of the above eamples to all functions The inverse function f of a function f eists if and onl f is one-to-one (Technicall, f must be a bijection For our purposes, however, it will suffice to require that f be one-to-one) The inverse relation of a man-to-one function is not a function However, if the domain of a man-to-one function is restricted in such a wa that it is one-to-one for a certain set of -values, then the inverse relation defined for this piece is a function Important Question: How are the Domain and Range of a Function related to the Domain and Range of its Inverse? Write our answer to this question in the following space Copright, Nick E Nolfi MCRUO Unit Characteristics of Functions COF-8
Important Eercise Complete the following table The first two rows are done for ou Hint: To find the inverse of each given function, appl the inverse operations in the reverse order Function Notation f( ) Function Mapping Notation = One-to- One or Man-to- One? one-to-one Inverse Function State an Restrictions to Domain of f Function Mapping Notation Notation f ( ) = Domain and Range of f D = R = Domain and Range of f - D = R = f( ) = man-toone f ( ) =, provided that { : } D= { : } R= { : } D= { : } R= f( ) = + f ( ) = 7 f( ) = f ( ) = 4 f ( ) = + + (This one is a bit trick Look ahead to page 4 if ou need help) Copright, Nick E Nolfi MCRUO Unit Characteristics of Functions COF-9
Geometric View of the Inverse of a Function B now it should be clear that finding the inverse of a function is equivalent to appling the transformation (, ) (, ) The geometric effect of this transformation can be seen quite easil from the diagram at the right It is nothing more than a reflection in the line = To obtain the graph of = f ( ), reflect the graph of = f( ) in the line = (,7) (7,) Eample Consider the function f( ) = 5+ 6 (a) Sketch the graphs of both f and f (b) Find the inverse of f State an necessar restrictions to the domain of f (c) State the domain and range of both f and f Solution (a) When f( ) = 5+ 6 is reflected in the line =, the resulting relation is not a function Therefore, the domain of f needs to be restricted in such a wa that f is one-to-one for the restricted set of -values f ( ) = 5 + 6 f( ) = 5+ 6, 5 f (b) Since appears in two terms, it is not possible to appl the inverse operations in the reverse order directl Once again then, our good friend completing the square comes to our rescue f ( ) = 5 + 6 5 5 = + + 5 6 5 5 4 = + 4 4 5 = 4 5 f( ) = 5+ 6= 4 The inverse relation is not a function! Operations Performed to to obtain f() 5 square 4 Reversal of the Operations 5 + + 4 f( ) 5 4 f( ) 5 4 Therefore, to go from to f ( ) ( f ( ) in mapping notation), we must do the following: f + 4 + 5 ( ) Thus, 5 f ( ) = + + 4 Copright, Nick E Nolfi MCRUO Unit Characteristics of Functions COF-4
Alternative Method for Finding f ( ) Appl the transformation (, ) (, ) (reflection in the line = ) Algebraicall this involves interchanging the and variables Solve for in terms of f( ) = 5+ 6 5 4 + If we restrict the domain of f to = ± = 5+ 6 5 D= Now interchange and and solve for 5 :, then clearl we should = ± 4 + 5 5 5+ 6 = choose = + + = + +, which 5 4 4 Now we can appl the quadratic formula = ± 4 + 4 agrees with the answer that we obtained using with a =, b = 5 and c = 6 the method of reversing the operations 5 ( 5) ± ( 5) 4()(6 ) = ± 4 + 5 = 4 Therefore, f ( ) = + + 4 () 5 ± 5 4 + 4 = 5± 4 + = (c) The domain and range of f and Domain and Range of f f 5 = ± () + 4 5 = ± + 4 can be determined b using a combination of geometr and algebra Domain and Range of f 5 5 f( ) = 5+ 6 =, 4 5 f ( ) = + + 4 From the graph it is clear that the values of and need to be restricted To ensure that f be one-to-one, we were forced to impose the restriction 5 B eamining the equation of f, it is clear that 4is the lowest possible value for f() Therefore, { : 5} and R= { : 4} D= Etremel Important Follow-up Questions Clearl, the domain and range will be the opposite of the domain and range of f Therefore, { : 4} and R= { : 5} D= If f( ) =, we have discovered that f ( ) = Evaluate f( f ( )) and f ( f( )) For an function f, evaluate f( f ( )) and f ( f( )) (Hint: It does not matter what f is All that matters is that is defined at and at f( ) In addition, keep in mind that an inverse of a function undoes the function) The slope of f ( ) = m + b is m What is the slope of f? Homework pp 5,, bdfi, 5h, 6f, 7e,f, 8, 9,, 4v, 5c, 6viii, 7,,,, 4, 7,, 4, 6, 8, 9 f Copright, Nick E Nolfi MCRUO Unit Characteristics of Functions COF-4
APPLICATIONS OF QUADRATIC FUNCTIONS AND TRANSFORMATIONS Introduction Prerequisite Knowledge The following table summarizes the criticall important core knowledge that ou must have at our fingertips if ou hope to be able to solve a significant percentage of the problems in this section Relations and Functions Quadratic Functions Transformations Idea of Mathematical Relationship Definition of Relation Definition of Function Si Perspectives Set of ordered pairs, function machine, table of values, mapping diagram, graphical (geometric), algebraic Discrete and Continuous Relations Function Notation independent variable f() dependent variable Mapping Notation input output f( ) Domain and Range D < D = D > Functions and their Inverses Solving Quadratic Equations First write the quadratic equation in the form a + b + c = Tr factoring first If the quadratic does not factor, use the quadratic formula Onl use the method of completing the square if ou are asked to! The nature of the roots can be determined b calculating the discriminant D = b 4ac General Form of Quadratic Function f ( ) = a + b + c Verte Form of Quadratic Function f( ) = a ( h) + k Rate of Change of Quadratic Functions Quadratic functions are used to model quantities whose rate of change is linear That is, the first differences change linearl and the second differences are constant Transformations Studied Horizontal and Vertical - Translations (Shifts) - Reflections in Aes - Stretches & Compressions A reflection in the -ais is the same as a vertical stretch b A reflection in the -ais is the same as a horizontal stretch b Mapping Notation Horizontal and Vertical Translations (, ) ( + h, + k) Reflections in Vertical and Horizontal Aes (, ) (, ) Horizontal and Vertical Stretches and Compressions (, ) ( a, b) Combination of Stretches and Translations (, ) ( a + h, b + k) Equivalent Transformations in Function Notation Horizontal and Vertical Translations g ( ) = f( h) + k Reflections in Vertical and Horizontal Aes g ( ) =f( ) Horizontal and Vertical Stretches and Compressions g( ) = af ( b ) = af (( b) ) Combination of Stretches and Translations g( ) = af ( b ( h)) + k The inverse of a relation is found b appling the transformation (, ) (, ), which is a reflection in the line = If f is a function and its inverse is a function, the inverse function is denoted The inverse of a function undoes the function That is, f f( f ( )) f ( f( )) = = The inverse of a function can be found b appling the inverse operations in the reverse order The inverse of a function can also be found b appling the transformation (, ) (, ) (interchanging and ) and then solving for in terms of A one-to-one function alwas has an inverse function A man-to-one function, on the other hand, must have its domain restricted in order for an inverse function to be defined Copright, Nick E Nolfi MCRUO Unit Characteristics of Functions COF-4
Problem Solving Activit You are to do our best to tr to solve each of the following problems without assistance Although answers can be found in the Activit Solutions document for unit, ou should not consult them until ou have made a concerted, independent effort to develop our own solutions! The word function is often used to epress relationships that are not mathematical Eplain the meaning of the following statements: (a) Crime is a function of socioeconomic status (b) Financial independence is a function of education and hard work (c) Success in school is a function of students work habits, qualit of teaching, parental support and intelligence Investigate, both graphicall and algebraicall, the transformations that affect the number of roots of the following quadratic equations (a) = (b) + 4= (c) + + 56 = (d) 6 5 = Although the method of completing the square is ver powerful, it can involve a great deal of work If our goal is to find the maimum or minimum of a quadratic function, ou can use a simpler method called partial factoring Eplain how partiall factoring f( ) = 6+ 5 into the form f( ) = ( ) + 5 helps ou determine the minimum of the function 4 Find the maimum or minimum value of each quadratic function using three different algebraic methods Check our answers b using a graphical (geometric) method (a) ht ( ) = 49t + 49t+ 744 (b) g( ) = 6 5 5 5 Given the quadratic function f( ) = ( ) and g( ) = af ( b( h)) + k, describe the effect of each of the following If ou need assistance, ou can use a slider control in TI-Interactive (to be demonstrated in class) (a) a = (b) b = (c) a > (d) a < (e) < a < (f) < a < (g) b > (h) b < (i) < b < (j) < b < (k) k is increased (l) k is decreased (m) h is increased (n) h is decreased (o) a =, b =, h =, k = (p) a =, b =, h =, k = 6 The profit, P(), of a video compan, in thousands of dollars, is given b P ( ) = 5 + 55 5, where is the amount spent on advertising, in thousands of dollars Determine the maimum profit that the compan can make and the amounts spent on advertising that will result in a profit of at least $4 7 Suppose that a quadratic function f ( ) = a + b + c has -intercepts r and r Describe transformations of f that produce quadratic functions with the same -intercepts That is, describe transformations of f under which the points ( r,) and ( r,) are invariant 8 Determine the equation of the quadratic function that passes through (, 5) if the roots of the corresponding quadratic equation are 5 and + 5 9 Determine, through investigation, the equations of the lines that have a slope of and that intersect the quadratic function f( ) = (6 ) (a) once (b) twice (c) never At a wild part, some inquisitive MCRU students performed an interesting eperiment h The obtained two containers, one of which was filled to the brim with a popular part beverage while the other was empt The full container was placed on a table and the h ( t ) = 49 t 5478 t+ 5 empt container was placed on the floor right net to it Then, a hole was poked near the bottom of the first container, which caused the part liquid to drain out of the first container and into the other B carefull collecting and analzing data, the students h ( t) = t determined two functions that modelled how the heights of the liquids (in metres) varied with time (in seconds): h ( t ) = 49 t 5478 t+ 5 and h ( t) = t (a) Which function applies to the container with the hole? Eplain (b) At what time were the heights of the liquids equal? (c) Eplain wh the height of the liquid in one container varied linearl with time while the height in the other container varied quadraticall with time t Copright, Nick E Nolfi MCRUO Unit Characteristics of Functions COF-4
Video games depend heavil on transformations The following table gives a few eamples of simple transformations ou might see while plaing Super Mario Bros Complete the table Before Transformation After Transformation Nature of the Transformation Reflection in the vertical line that divides Mario s bod in half Use in the Video Game Complete the following table Before Transformation After Transformation Nature of the Transformation Found Guilt of being Phsicall Present but Mentall Absent! Copright, Nick E Nolfi MCRUO Unit Characteristics of Functions COF-44
For each quadratic function, determine the number of zeros (-intercepts) using the methods listed in the table Quadratic Function f ( ) = 6 Graph Discriminant D = b 4ac a=, b=, c= 6 D = b 4ac = ( ) 4()( 6) = + 4 = 5 Factored Form 6= ( )( + ) = = or + = = or = Conclusion Number of Zeros Since there are -intercepts, f has two zeros Since D >, f has two zeros Since 6= has two solutions, f has two zeros gt t t () = + 6 Pt t t ( ) = 9 5 qz z z ( ) = 9 6 + f ( ) = 49 Copright, Nick E Nolfi MCRUO Unit Characteristics of Functions COF-45
4 Use the quadratic formula to eplain how the discriminant allows us to predict the number of roots of a quadratic equation 5 How can the discriminant be used to predict whether a quadratic function can be factored? Eplain 6 The following table lists the approimate accelerations due to gravit near the surface of the Earth, moon and sun Earth Jupiter Saturn 987 m/s 595 m/s 8 m/s The data in the above table lead to the following equations for the height of an object dropped near the surface of each of the celestial bodies given above In each case, ht () represents the height, in metres, of an object above the surface of the bod t seconds after it is dropped from an initial height h Earth Jupiter Saturn ht ( ) = 494t + h ht ( ) = 97t + h ht ( ) = 554t + h In questions (a) to (d), use an initial height of m for the Earth, m for Saturn and m for Jupiter (a) On the same grid, sketch each function (b) Eplain how the Jupiter function can be transformed into the Saturn function (c) Consider the graphs for Jupiter and Saturn Eplain the phsical meaning of the point(s) of intersection of the two graphs (d) State the domain and range of each function Keep in mind that each function is used to model a phsical situation, which means that the allowable values of t are restricted 7 Have ou ever wondered wh an object that is thrown up into the air alwas falls back to the ground? Essentiall, this happens because the kinetic energ of the object (energ of motion) is less than its potential energ (the energ that the Earth s gravitational field imparts to the object) As long as an object s kinetic energ is less than its potential energ, it will either fall back to the ground or remain bound in a closed orbit around the Earth On the other hand, if an object s kinetic energ eceeds its potential energ, then it can break free from the Earth s gravitational field and escape into space mv GMm v GM The function E, defined b the equation E () v = = m, gives the difference between the r r kinetic energ and potential energ of an object of mass m moving with velocit v in the gravitational field of a bod of mass M and radius r In addition, G represents the universal gravitational constant and is equal to 66749 m kg - s - (a) Use the data in the table to determine E for the Earth, Jupiter and Saturn for an object with a mass of kg Planet Radius (m) Mass (kg) E 6 Earth 68 7 Jupiter 75 7 Saturn 6 4 598 7 9 6 568 (b) Sketch the graph of E for the Earth, Jupiter and Saturn (for an object of a mass of kg) (c) An object can escape a bod s gravitational field if its kinetic energ eceeds its potential energ Using the graphs from part (b), determine the escape velocit for each of the given planets (d) Does the escape velocit of an object depend on its mass? Copright, Nick E Nolfi MCRUO Unit Characteristics of Functions COF-46
8 Suppose that f( ) 5 = and that g ( ) = f ( ) + = f ( ( 9)) + (a) The following table lists the transformations, in mapping notation, applied to f to obtain g Give a verbal description of each transformation Mapping Notation Vertical (, ) (,+ ) Verbal Description Horizontal (, ) (+ 9, ) Other (, ) (, ) (b) As ou alread know, horizontal and vertical transformations are independent of each other, which means that the results are the same regardless of the order in which the are applied Does it matter at what point in the process the transformation (, ) (, ) is applied? Eplain (c) Sketch the graph of g (d) State an equation of g Summar of Problem Solving Strategies used in this Section Problem Strategies Determine the number of zeros of a quadratic function (This is equivalent to finding the number of -intercepts) Predict whether a quadratic function can be factored Find the maimum or minimum value of a quadratic function (This is equivalent to finding the verte of the corresponding parabola) Find the equation of a quadratic function that passes through a given point and whose -intercepts are the same as the roots of a corresponding quadratic equation Intersection of a Linear Function and a Quadratic Function Copright, Nick E Nolfi MCRUO Unit Characteristics of Functions COF-47
APPENDIX - ONTARIO MINISTRY OF EDUCATION GUIDELINES - UNIT Copright, Nick E Nolfi MCRUO Unit Characteristics of Functions COF-48
Copright, Nick E Nolfi MCRUO Unit Characteristics of Functions COF-49
Copright, Nick E Nolfi MCRUO Unit Characteristics of Functions COF-5