In order to master the techniques explained here it is vital that you undertake plenty of practice exercises so that they become second nature.

Size: px

Start display at page:

Download "In order to master the techniques explained here it is vital that you undertake plenty of practice exercises so that they become second nature."

Antonia Horn
7 years ago
Views:

1 Linear functions mc-ty-linearfns Some of the most important functions are linear. This unit describes how to recognize a linear function,andhowtofindtheslopeandthe y-interceptofitsgraph. In order to master the techniques eplained here it is vital that you undertake plenty of practice eercises so that they become second nature. Afterreadingthistet,and/orviewingthevideotutorialonthistopic,youshouldbeableto: recognise when a rule describes a linear function, beabletoplotthegraphofpartofalinearfunction, findtheslopeandthe y-interceptofthegraphofalinearfunction. Contents 1. Linear functions 2 2. Linear functions not written in standard form 1 c mathcentre 2009

In order to master the techniques eplained here it is vital that you undertake plenty of practice eercises so that they become second nature.

2 1. Linear functions Alinearfunctionisafunctionoftheform f() = a + b,where aand barerealnumbers.here, a represents the gradient of the line, and b represents the y-ais intercept(which is sometimes called the vertical intercept). Whatdoyouthinkwillhappenifwefi bandvary a? Letustrysomeeamples. Sincethe functionswearelookingatarelinear,thegraphwillbeastraightline. Soweonlyneedtwo pointstobeabletodrawtheline.however,wegenerallychoosethree,andthethirdpointisa goodcheckthatwehaven tmadeamistake. (a) f() = + 2 : f(0) = 2, f(1) = 3, f(2) = (b) f() = : f(0) = 2, f(1) =, f(2) = 6 (c) f() = + 2 : f(0) = 2, f(1) = 0, f(2) = (d) f() = + 2 : f(0) = 2, f(1) = 1, f(2) = (a) (b) (c) (d) Youcanclearlyseefromourdiagramthatthegraphsofallofthefunctionscrossthe y-aisat y = 2.Thisisbecause bisfiedas2,and brepresentsthe y-aisintercept.youcanseethatif a > 0thenthestraightlinegoesupas increases,andthebigger aisthefasterthelinegoes up. Similarly,if a < 0thenthelinegoesdownas increases,andthebigger aisinabsolute terms, the faster the line goes down. 2 c mathcentre 2009

Sincethe functionswearelookingatarelinear,thegraphwillbeastraightline. Soweonlyneedtwo pointstobeabletodrawtheline.

3 Whatdoyouthinkwillhappenifwefi aandvary b?againwecantrysomeeamples. (a) f() = : f(0) = 3, f(1) = 5, f(2) = 7 (b) f() = : f(0) = 1, f(1) = 3, f(2) = 5 (c) f() = 2 3 : f(0) = 3, f(1) = 1, f(2) = 1 2 (a) (b) (c) Whenyoulookatthegraphsofthefunctions,youcanseestraightawaythattheyallhavethe samegradient. Thisisbecause aisfiedas2,andrepresentsthegradient. Youshouldalso notice that b represents the y-ais intercept(that is, the vertical intercept) in each case. Nowweknowwhathappenswhen aand barepositiveornegative,butwhathappensifeitherof themiszero?supposethat a = 0.Thenwewouldhavefunctionsoftheform f() = bwhere b isconstant,foreample f() = 2or f() = 3. = 2 2 = 3 Youcanseethatagradientofzeroalwaysgivesahorizontalline,andthatthelinecutsthe y-aisat b. 3 c mathcentre 2009

Whenyoulookatthegraphsofthefunctions,youcanseestraightawaythattheyallhavethe samegradient. Thisisbecause aisfiedas2,andrepresentsthegradient.

4 Supposeinsteadthatb=0.Thenwewouldhavefunctionsoftheform f() = a,foreample f() = 2or f() = 3. = 2 2 = 3 The y-aisinterceptwouldbeequaltozero,andsothegraphsofallthesefunctionspassthrough theorigin,andthegradientofthelinedependsupon a. 2. Linear functions not written in standard form Finally,letusconsiderwhathappenswhenwehavelinearfunctionsthatarenotintheform f() = a + b.forsimplicity,weshallput y = f()here. Supposethatwehavetheequation 3y = 2.Togetthisintotherequiredformweneedto make y the subject: = 2 + 3y 2 = 3y 2 = y. 3 3 Then,since y = f(),thisequationrepresentsthefunction f() = Thegraphofthe functionwillhaveagradientof 3 anday-aisinterceptof 3. Supposethatwehavetheequation 2 + 8y 1 = 0.Togetthisintotherequiredformweneed tomake ythesubject: 2 + 8y = 1 8y = 1 2 y = = Then,since y = f(),thisequationrepresentsthefunction f() = Thegraphofthe functionwillhaveagradientof 1 anday-aisinterceptof 1 8. Supposethatwehavetheequation y = (13 8)/5. Although yisalreadythesubject,thisis stillnotintherequiredform. Weneedtodealwiththedivisionby5: so y = Then, 5 5 since y = f(),thisequationrepresentsthefunction f() = 13 8.Thegraphofthefunction 5 5 willhaveagradientof 13anday-aisinterceptof c mathcentre 2009

forsimplicity,weshallput y = f()here. Supposethatwehavetheequation 3y = 2.Togetthisintotherequiredformweneedto make y the subject: = 2 + 3y 2 = 3y 2 = y.

5 Key Point Functions of the form f() = a + b are linear, and they are represented graphically by straight lines. The number a represents the gradient of the line, and the number b represents the y-ais intercept. Eercises 1.Whatisalinearfunction? 2.Bydrawingupatableofvalues,plotthefollowinglinearfunctionsonthesameaes: (a) f() = (b) f() = 3 2 (c) f() = 3 (d) f() = 2 3. Find the gradient and the vertical intercept for each of the following linear functions by rearrangingthemintotheform f() = a + b(note: y = f()). (a) 2y + = 12 (b) 5 y = 9 (c) 3 = 1 y (d) 2 y/3 = (e) 3 = 3y/ 2/3 (f) 12 = y/ Write down three different functions in which all the graphs are represented by parallel lines. 5. Write down three different functions in which all the graphs have the same vertical intercept. Answers 1.Alinearfunctionisafunctionthatcanbeepressedintheform f() = a + b,where aand b represent real numbers (a) (b) (c) (d) 5 c mathcentre 2009

Bydrawingupatableofvalues,plotthefollowinglinearfunctionsonthesameaes: (a) f() = 2 + 1 (b) f() = 3 2 (c) f() = 3 (d) f() = 2 3.

6 3. (a) 2y + = 12 2y = 12 y = 6 2 y = + 6 f() = + 6 Gradient =, vertical intercept = 6. (b) 5 y = 9 5 = 9 + y 5 9 = y y = 5 9 f() = 5 9 Gradient = 5, vertical intercept = 9. (c) 3 = 1 y y 3 = 1 y = y = f() = Gradient = 3,verticalintercept = 1. (d) 2 y 3 = 2 = + y 3 Gradient = 3, vertical intercept = 6. 2 = y 3 y 3 = 2 y = 6 3 f() = c mathcentre 2009

(c) 3 = 1 y y 3 = 1 y = 3 + 1 y = 3 + 1 f() = 3 + 1 Gradient = 3,verticalintercept = 1.

7 (e) 3 = 3y = 3y = 3y = y y = f() = Gradient = 8,verticalintercept =. 9 (f) 12 = y Gradient = 36, vertical intercept = = y = y f() = Parallellineshavethesamegradient,sowemaytakeanythreefunctionswhere ahasthe same value, for eample f() = 2 1, f() = 2 + 5, f() = Wemaytakeanythreefunctionswhere bhasthesamevalue,foreample f() = 5, f() = 2 5, f() = c mathcentre 2009

Parallellineshavethesamegradient,sowemaytakeanythreefunctionswhere ahasthe same value, for eample f() = 2 1, f() = 2

Many common functions are polynomial functions. In this unit we describe polynomial functions and look at some of their properties.

Polynomial functions mc-ty-polynomial-2009-1 Many common functions are polynomial functions. In this unit we describe polynomial functions and look at some of their properties. In order to master the techniques