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of the learners and inspiring flight – inspiring them toreach their full potential.Grade 12 Teacher’s GuideFor me, teaching is about applying myself in the livesThe accompanying Learner’s Book is written in accessible language and contains all the content your learners need to master.The exciting design and layout will keep their interest and make teaching a pleasure for you.We would love to hear your feedback. Why not tell us how it’s going by emailing us at maths@viaafrika.com? Alternatively,visit our teacher forum at www.viaafrika.com.Language: Englishwww.viaafrika.comVia Afrika Mathematics1. The series was written to be aligned with CAPS. See page 5 to see how CAPS requirements are met.2. A possible work schedule has been included. See page 6 to 9 to see how much time this could save you.3. Each topic starts with an overview of what is taught, and the resources you need. See page 31 to find out how this willhelp with your planning.4. There is advice on pace-setting to assist you in completing all the work for the year on time. Page 31 shows you how thisis done.5. Advice on how to introduce concepts and scaffold learning is given for every topic. See page 31 for an example.6. All the answers have been given to save you time doing the exercises yourself. See page 32 for an example.7. Also included is a CD filled with resources to assist you in your teaching and assessment. See the inside front cover.Grade 12 Study GuideM. Malan, L.J. Schalekamp, E.C. Brown, L. Bruce,G. du Toit, C.R. Smith, L.M. Botsane, J. Bouman, M. Pillay— Brynn Taylor, TeacherVia Afrika understands, values and supports your role as a teacher. You have the most important job in education, and werealise that your responsibilities involve far more than just teaching. We have done our utmost to save you time and makeyour life easier, and we are very proud to be able to help you teach this subject successfully. Here are just some of the thingswe have done to assist you in this brand-new course:Via AfrikaMathematics
M. MalanSt udy G u id eVia Afrika MathematicsGrade 12ISBN: 978-1-41546-335-2
Exponents and SurdsContentsIntroduction . 3Chapter 1 Number patterns, sequences and series . 4OVERVIEW . 4Unit 1 Arithmetic sequences and seriesUnit 2 Geometric sequences and seriesUnit 3 The sum to n terms(Sn): Sigma notationUnit 4 Convergence and sum to infinityMixed exercises . 8Chapter 2 Functions . 10OVERVIEW . 10Unit 1 The definitions of a functionUnit 2 The inverse of a functionUnit 3 The inverse of y ax qUnit 4 The inverse of the quadratic function y ax2Mixed exercises . 21Chapter 3 Logarithms . 24OVERVIEW . 24Unit 1 The definition of a logarithmUnit 2 Solving exponential equations using logarithmsUnit 3 The graph of y logbx where b 1 and 0 b 1Mixed exercises . 28Chapter 4 Finance, growth and decay . 29OVERVIEW . 29Unit 1 Future value annuitiesUnit 2 Present value annuitiesUnit 3 Calculating the periodUnit 4 Analysing investments and loansMixed exercises . 35Chapter 5 Compound angles . 37OVERVIEW . 37Unit 1 Deriving a formula for cos(𝛼 𝛽)Unit 2 Formula for cos(𝛼 𝛽) and 𝑠𝑖𝑖(𝛼 𝛽)Unit 3Unit 4Unit 5Unit 6Double anglesIdentitiesEquationsTrigonometric graphs and compound anglesMixed exercises . 46Chapter 6 Solving problems in three dimensions . 48OVERVIEW . 48Unit 1 Problems in three dimensionsUnit 2 Compound angle formulae in three dimensionsMixed exercises . 51 Via Afrika ›› Mathematics Grade 121
Exponents and SurdsChapter 7 Polynomials . 53OVERVIEW . 53Unit 1 The Remainder TheoremUnit 2 The Factor TheoremMixed exercises . 57Chapter 8 Differential calculus . 58OVERVIEW . 58Unit 1 LimitsUnit 2 The gradient of a graph at a pointUnit 3 The derivative of a functionUnit 4 The equation of a tangent to a graphUnit 5 The graph of a cubic functionUnit 6 The second derivative (concavity)Unit 7 Applications of differential calculusMixed exercises . 70Chapter 9 Analytical geometry . 72OVERVIEW . 72Unit 1 Equation of a circle with centre at the originUnit 2 Equation of a circle centred off the originUnit 3 The equation of the tangent to the circleMixed exercises . 77Chapter 10 Euclidean geometry . 81OVERVIEW . 81Unit 1 Proportionality in trianglesUnit 2 Similarity in trianglesUnit 2 Theorem of PythagorasMixed exercises . 94Chapter 11 Statistics: regression and correlation . 97OVERVIEW . 97Unit 1 Symmetrical and skewed dataUnit 2 Scatter plots and correlationMixed exercises . 106Chapter 12 Probability . 108OVERVIEW . 108Unit 1 Solving probability problemsUnit 2 The counting principleUnit 3 The counting principle and probabilityMixed exercises . 112ANSWERS TO MIXED EXERCISES . 113EXEMPLAR PAPER 1 . 151EXEMPLAR PAPER 2 . 168 Via Afrika ›› Mathematics Grade 122
Exponents and SurdsIntroduction to Via Afrika Mathematics Grade 12 Study GuideWoohoo! You made it! If you’re reading this it means that you made it through Grade 11,and are now in Grade 12. But I guess you are already well aware of that It also means that your teacher was brilliant enough to get the Via Afrika MathematicsGrade 12 Learner’s Book. This study guide contains summaries of each chapter, and shouldbe used side-by-side with the Learner’s Book. It also contains lots of extra questions tohelp you master the subject matter.Mathematics – not for spectatorsYou won’t learn anything if you don’t involve yourself in the subject-matter actively. Dothe maths, feel the maths, and then understand and use the maths.Understanding the principles Listen during class. This study guide is brilliant but it is not enough. Listen to yourteacher in class as you may learn a unique or easy way of doing something. Study the notation, properly. Incorrect use of notation will be penalised in testsand exams. Pay attention to notation in our worked examples. Practise, Practise, Practise, and then Practise some more. You have to practiseas much as possible. The more you practise, the more prepared and confident youwill feel for exams. This guide contains lots of extra practice opportunities. Persevere. We can’t all be Einsteins, and even old Albert had difficulties learningsome of the very advanced Mathematics necessary to formulate his theories. If youdon’t understand immediately, work at it and practise with as many problems fromthis study guide as possible. You will find that topics that seem baffling at first,suddenly make sense. Have the proper attitude. You can do it!The AMA of MathematicsABILITY is what you’re capable of doing.MOTIVATION determines what you do.ATTITUDE determines how well you do it.“Pure Mathematics is, in its way, the poetry of logical ideas.” Albert Einstein Via Afrika ›› Mathematics Grade 123
1Number patterns, sequences and seriesChapterOverviewUnit 1 Page 10 Arithmetic sequences andseriesFormula for an arithmeticsequenceUnit 2 Page 14Chapter 1 Page 8Number patterns,sequences andseriesGeometric sequences andseries Formula for the nth termof a sequence The sum to 𝑖 terms in anarithmetic sequenceThe sum to 𝑖 terms in ageometric sequenceUnit 3 Page 18The sum to 𝑖 terms (𝑆𝑛 ): Sigmanotation Unit 4 Page 28Convergence and sum to infinity ConvergenceREMEMBER YOUR STUDY APPROACH SHOULD BE:1Work through all examples in this chapter of your Learner’s Bok.2 Work through the notes in this chapter of this study guide.3 Do the exercises at the end of the chapter in the Learner’s Book.4 Do the mixed exercises at the end of this chapter in this study guide. Via Afrika ›› Mathematics Grade 124
1Number patterns, sequences and seriesChapterTABLE 1: SUMMARY OF SEQUENCES AND SERIESTYPEArithmetic Sequence (AS)(also named the linearsequence)Constantst1 differenceGENERAL TERM: 𝑇𝑛𝑇𝑛 𝑎 (𝑖 1)𝑑𝑎 𝑓𝑖𝑟𝑠𝑡 𝑡𝑐𝑟𝑚 𝑇1𝑑 𝑐𝑜𝑖𝑠𝑡𝑎𝑖𝑡 𝑑𝑖𝑓𝑓.𝑑 𝑇2 𝑇1𝑜𝑟 𝑇3 𝑇2 etc.SUM OF TERMS: 𝑆𝑛𝑆𝑛 or𝑖[2𝑎 (𝑖 1)𝑑]2𝑆𝑛 where𝑖[𝑎 𝑙]2𝑙 the last term ofthe sequenceEXAMPLESA)2 ; 5 ; 8 ; 11 ; . 3 3 3𝑑 𝑇𝑛 2 (𝑖 1)(3) 2 3𝑖 3 3𝑖 1B) 1 ; -4 ; -9 ; .𝑑 -5Geometric Sequence (GS)(also named exponentialsequence)ConstantratioQuadratic Sequence (QS)Constantnd2difference𝑇𝑛 𝑎𝑟 𝑛 1𝑎 𝑓𝑖𝑟𝑠𝑡 𝑡𝑐𝑟𝑚 𝑇1𝑟 ��𝑇2𝑇3𝑟 𝑜𝑟𝑇1𝑇2𝑇𝑛 𝑎𝑖2 𝑏𝑖 𝑐𝑆𝑛 Or𝑆𝑛 𝑎(𝑟 𝑛 1)𝑟 1𝑛𝑎(1 𝑟 )1 𝑟Or 𝑆 𝑎1 𝑟Where 1 𝑟 1(Converging series)-5𝑇𝑛 1 (𝑖 1)( 5) 1 5𝑖 5 5𝑖 6A)2 ; -4 ; 8 ; -16 ; .𝑟 x-2 x-2 x-2𝑇𝑛 2( 2)𝑛 1NOT CONVERGING as 𝑟 1B) 3 ;32;341𝑟 x23;; .81x21x21 𝑛 1𝑇𝑛 3 2CONVERGING as 1 𝑟 13 ; 8 ; 16 ; 27 ; .𝑓 1st difference𝑠 2nd difference𝑓:using simultaneousequations (seeexample)Setup three equations usingthe first three terms:𝑇1 3:Determine 𝑎, 𝑏 and 𝑐Alternatively:𝑎 𝑠 2𝑏 𝑓1 3𝑎𝑐 𝑇1 𝑎 𝑏where𝑓1 first term of firstdifferences Via Afrika ›› Mathematics Grade 12𝑠:5831133 𝑎 𝑏 𝑐𝑇2 8: (1)8 4𝑎 2𝑏 𝑐 (2)𝑇3 16:16 9𝑎 3𝑏 𝑐 (3)Solving simultaneously leadsto:𝑇𝑛 32𝑖2 12𝑖 15
1Number patterns, sequences and seriesChapterTYPES OF QUESTIONS YOUCAN EXPECTSTRATEGY TO ANSWER THIS TYPEOF QUESTIONEXAMPLE(S) OF THIS TYPE OFQUESTIONIdentify any of the followingthree types of sequences:Arithmetic (AS), Geometric(GS) and Quadratic (QS)Determine whether sequence has a constant 1st difference (AS) constant ratio (GS) constant 2nd difference (QS)See Table 1 aboveDetermine the formula for thegeneral term, 𝑇𝑛 , of AS, GSYou need to find: 𝑎 and 𝑑 for an ASSee Table 1 aboveDetermine any specific termSubstitute the value of 𝑖 into 𝑇𝑛See Text Book :Example 1, nr. 1 d and 2 d, p.8(AS)Example 1, nr. 1 b, 3 b, p.11(AS)Example 1, nr. 1, p. 15 (GS)Substitute all known variables intothe general term to get an equationwith 𝒔 as the only unknown. Solvefor 𝑖.ORSubstitute all known variables intothe 𝑆𝑛 -formula to get an equationwith 𝒔 as the only unknown. SolveSee Text Book:Example 1, nr.1 c, p.8Example 1, nr.1 c, p.11Example 1, nr. 3, p.15and QS (from Grade 11)for a sequence e.g. 𝑇30Determine the number ofterms in a sequence, 𝑖, for anAS, GS and QS orthe position, 𝑖, of a specificgiven term or when the sumof the series is givenWhen given two sets ofinformation, make use ofsimultaneous equations tosolve:𝒂 and 𝒅 (for an AS)𝒂 and 𝒓 (for a GS)Determine the value of avariable (𝑥) when given asequence in terms of 𝑥. 𝑎 and 𝑟 for a GS𝑎, 𝑏 and 𝑐 for a QSfor 𝑖.Example 2, nr.3, p.20Example 3, nr. 2, p.24For each set of information given,See Text Book:Example 1, nr. 3, p.11 (AS)Example 1, nr.2, p.15 (AS)Example 3, nr.3, p.24 (GS)Remember:𝑖 must be a natural number(not negative, not a fraction)substitute the values of 𝑖 and 𝑇𝑛 or𝑖 and 𝑆𝑛 .You then have 2 equations whichyou can solve simultaneously (bysubstitution)For AS use constant difference:𝑇3 𝑇2 𝑇2 𝑇1For GS use constant ratio:𝑇2 𝑇3 𝑇1 𝑇2 Via Afrika ›› Mathematics Grade 12The first three terms of an ASare given by2𝑥 4; 𝑥 3; 8 2𝑥Determine 𝑥:8 2𝑥 (𝑥 3) 𝑥 3 (2𝑥 4) 𝑥 56
1Number patterns, sequences and seriesChapterFor a series given in sigmanotation: Determine the number oftermsRemember:The “counter” indicates the numberof terms in the series 𝑛𝑘 1 𝑇𝑘 has 𝑖 terms(counter 𝑘 runs from 1 to 𝑖) 𝑛𝑘 0 𝑇𝑘 has (𝑖 1) terms(counter runs from 0 to 𝑖; soone term extra) 𝑛𝑘 5 𝑇𝑘 has (𝑖 4) terms( four terms not counted ) Determine the value ofthe series, in other words,𝑆𝑛 .Write a given series in sigmanotation.Determine the sum, 𝑺𝒔 , of anAS and a GS (when thenumber of terms are given ornot given)Determine whether a GS isconverging or notDetermine 𝑆 for aconverging GSDetermine the value of avariable (𝑥) for which a serieswill converge,e.g. (2𝑥 1) (2𝑥 1)2 Apply your knowledge ofsequences and series on anapplied example (ofteninvolving diagram/s)Remember the expression next tothe -sign is the general term, 𝑇𝑛 .See Text Book:Example 1, p.19Determine the general term, 𝑇𝑘 andnumber of terms, 𝑖 and substituteinto 𝑛𝑘 1 𝑇𝑘Example 1, p.19This will help you to determine 𝑎and 𝑑 or 𝑟.In some cases you have to firstdetermine the number of terms, 𝑖using 𝑇𝑛 .Substitute the values of 𝑎, 𝑖 and 𝑑/𝑟into the formula for 𝑆𝑛See Text Book:Substitute vales of 𝑎 and 𝑟Into formula for 𝑆 See Text Book:Example 1, nr. 1, p.29Determine 𝑟 in terms of 𝑥and use 1 𝑟 1See Text Book:Example 1, nr. 3, p.29Generate a sequence of terms fromthe information given. Identify thetype of sequence.See Text Book:Exercise 5, nr. 6, p.30Converging if 1 𝑟 1 Via Afrika ›› Mathematics Grade 12Example 2, nr.1 & 2, p.20Example 3, nr. 1, p.247
1Number patterns, sequences and seriesChapterMixed Exercise on sequences and series1Consider the following sequence:5; 9; 13; 17; 21; aDetermine the general term.bWhich term is equal to 217?2abT5 of a geometric sequence is 9 and T9 is 729. Determine the constant ratio.Determine T10.3The following is an arithmetic sequence: 2 x 4 ; 5 x ; 7 x 445aDetermine the value of x .bDetermine the first 3 terms.Consider the following sequence:aDetermine the general term.bWhich term is equal to 260?2 ; 7 ; 15 ; 26 ; 40 ; How many terms are there in the following sequence?17 ; 14 ; 11 ; 8 ; ; -27856Tom links balls with rods in arrangements as shown below:Arrangement 1 Arrangement 21 ball, 4 rods 4 balls, 12 rods7Arrangement 3Arrangement 49 balls, 24 rods16 balls 40 rodsaDetermine the number of balls in the nth arrangement.bDetermine the number of rods in the nth arrangement.Determine the following:30a k 110(8 5k)b k 21 5 9 2114(2)𝑘 18Write the following in sigma notation:9The 5th term of an arithmetic sequence is zero and the 13th term is equal to 12.Determine:athe constant difference and the first term.bthe sum of the first 21 terms. Via Afrika ›› Mathematics Grade 128
1Number patterns, sequences and seriesChapter10The first two terms of a geometric sequence are: (𝑥 3) and (𝑥 2 9)abFor which value of 𝑥 is this a converging sequence?Calculate the value of 𝑥 if the sum of the series to infinity is 13.11Calculate the value of:12𝑆𝑛 3𝑖2 2𝑖. Determine 𝑇9 .1399 97 95 1299 297 295 201The first four terms of a geometric sequence are 7; 𝑥 ; 𝑦 ; 189.abDetermine the values of 𝑥 and 𝑦.If the constant ratio is 3, make use of a suitable formula to determine the number ofterms in the sequence that will give a sum of 206 668. Via Afrika ›› Mathematics Grade 129
2FunctionsChapterOverviewUnit 1 Page 40 The definition of a functionRelations and functionsType of relationsWhich relations arefunctions?Definition of a function?Function notation Unit 2 Page 44Chapter 2 Page 36FunctionsThe inverse of a function The concept of inversesby studying sets ofordered number pairs Graphs of 𝑓and 𝑓 1 onthe same set of axesUnit 3 Page 46The inverse of 𝑦 𝑎𝑥 𝑞Unit 4 Page 48The inverse of the quadraticfunction 𝑦 𝑎𝑥 2 Restricting the domain ofthe parabolaREMEMBER YOUR STUDY APPROACH SHOULD BE:1Work through all examples in this chapter of your Learner’s Book.2 Work through the notes in this chapter of the study guide.3 Do the exercises at the end of the chapter in the Learner’s Book.4 Do the mixed exercises at the end of this chapter in the study guide. Via Afrika ›› Mathematics Grade 1210
2FunctionsChapterTYPES OF RELATIONS BETWEEN TWO VARIABLESTYPEDESCRIPTIONNON-FUNCTIONS One-to-manyFUNCTIONSOne-to-onePROPERTIESTYPICAL EXAMPLES One 𝑥-value in domain has MORETHAN ONE 𝑦-value Does NOT pass vertical line test Inverse of a parabola(See Unit 4) Each 𝑥-value has a unique 𝑦value Straight line graphand its inverse Hyperbola and itsinverse Exponential graphand its inverse, thelogarithmic function No 𝑥-value appears more thanonce in domain More than one 𝑥-value maps ontothe same 𝑦-value Passes VERTICAL line test Parabola Graph of the cubicfunction Trigonometric graphs No 𝑥- or 𝑦-value appear morethan once in domain or range Passes VERTICAL line testMany-to- oneREVISION OF THESTRAIGHT LINE GRAPH 𝒎Standard form: 𝒚 𝒎𝒙 𝒄Gradient of lineIndicates “steepness”and direction of line: 𝑚 0 ( )𝑚 0( )𝑚 0 𝒄𝑦-interceptWhere 𝑥 0𝑦 𝑦𝑚 𝑥2 𝑥121 Via Afrika ›› Mathematics Grade 1211
2FunctionsChapterPARALLEL ANDPERPENDICULAR LINESLet 𝑦 𝑚1 𝑥 𝑐1 and𝑦 𝑚2 𝑥 𝑐2 be two lines.If the lines are PARALLEL, then:𝑚1 𝑚2If the lines are PERPENDICULAR,then:𝑚1 𝑚2 1TO DETERMINE THE EQUATION OF A STRAIGHT LINEGIVEN:1. Gradient and a point2. 𝒚-intercept and a point3. Two points on the lineEXAMPLES12A line has a gradient of and goes through the point (4;1):1𝑚 21Substitute point (4;1) into 𝑦 𝑥 𝑐211 (4) 𝑐2𝑐 11𝑦 𝑥 12A line has a 𝑦-intercept 3 and goes through the point (-2;1):𝑐 3Substitute point (-2;1) into 𝑦 𝑚𝑥 31 𝑚( 2) 3𝑚 1𝑦 𝑥 3A line goes through the points (4;-3) and (2;1).𝑦 𝑦1 ( 3)𝑚 𝑥2 𝑥1 4 (2) 224. A point or 𝒚-intercept plusinformation regardingrelationship to another line1Substitute any one of the two points into 𝑦 2𝑥 𝑐1 2(2) c𝑐 3𝑦 2𝑥 3a) A line is parallel to the line 𝑦 𝑥 3 and goesthrough the point (5;-2).Parallel lines have same gradients; so 𝑚 1Sub (5;-2) into 𝑦 𝑥 𝑐 2 (5) 𝑐𝑐 3b) A line is perpendicular to the line 𝑦 2𝑥 1 and has a𝑦-intercept of 4. Via Afrika ›› Mathematics Grade 1212
2FunctionsChapterPerpendicular lines have gradients with a product of 𝟏.1𝑚 2 1 𝑚 2 1𝑦 𝑥 42REVISION OF THE PARABOLAEQUATION IN STANDARD FORM𝒚 𝒂𝒙𝟐 𝒃𝒙 𝒄(𝒂 𝟎)𝒂𝒄 Indicates shape of parabola𝑦-intercept 𝒂 0 ( )Where 𝑥 0Concave up𝒃 Remember:Positive( ) people smile! Affects the axis of symmetry andturning point (TP)𝒂 0( ) Concave down Equation of axis of symmetry: 𝑥 𝑏Coordinates of TP 2𝑎 ;Remember:Negative ( ) people are sad! Via Afrika ›› Mathematics Grade 124𝑎𝑎 𝑏2 4𝑎𝑏2𝑎𝒙-intercepts Also called roots/zeroes Substitute 𝑦 013
2FunctionsChapterEQUATION INTURNING POINT FORM𝒚 𝒂(𝒙 𝒑)𝟐 𝒒 (𝒂 𝟎)𝒂𝒑 and 𝒒Indicates shape of parabola𝒂 0 ( )Concave up Remember:Positive( ) people smile!𝒂 0( ) Equation of axis of symmetry 𝒙 𝒑Coordinates of turning point (𝒑; 𝒒)InterceptsConcave down Remember: 𝑥-intercepts (make 𝑦 0)𝑦-intercept (make 𝑥 0)Negative ( ) people are sad!DOMAIN: 𝑥 𝑅RANGE:𝑦 ( ; 𝑞) Via Afrika ›› Mathematics Grade 12𝑦 (𝑞; )14
2FunctionsChapterDETERMINE THE EQUATION OF A PARABOLAGIVEN: 2 ROOTS (𝒙-INTERCEPTS) PLUS 1 POINTGIVEN: TURNING POINT PLUS 1 POINTFORM OF EQUATION:FORM OF EQUATION:𝒚 𝒂(𝒙 𝒙𝟏 )(𝒙 𝒙𝟐 )𝒙𝟏 and 𝒙𝟐 are the roots𝑦 𝑎(𝑥 𝑝)2 𝑞(𝑝; 𝑞) is die turning point of the parabolaEXAMPLE:EXAMPLE:yy5(2;6)( 1;2)x 13x𝒙𝟏 𝟏𝒚 𝒂(𝒙 𝒙𝟏 )(𝒙 𝒙𝟐 )𝒚 𝒂(𝒙 ( 𝟏))(𝒙 𝟏)𝒚 𝒂(𝒙 𝟏)(𝒙 𝟏)𝒙𝟐 𝟏Now substitute the other point (𝟐; 𝟔):𝟔 𝒂(𝟐 𝟏)(𝟐 𝟏)𝟔 𝒂(𝟏)( 𝟏)𝟔 𝟏𝒂 𝟐 𝒂𝒚 𝟐(𝒙 𝟏)(𝒙 𝟏)𝒚 𝟐 𝒙𝟐 𝟐𝒙 𝟏 𝒚 𝟐𝒙𝟐 𝟒𝒙 𝟔 (standard form) Via Afrika ›› Mathematics Grade 12(𝑝; 𝑞) ( 1; 2)𝑦 𝑎(𝑥 𝑝)2 𝑞2𝑦 𝑎 𝑥 ( 1) 2𝑦 𝑎(𝑥 1)2 2Now substitute the point (0;5):5 𝑎(0 1)2 25 𝑎 23 𝑎𝑦 3(𝑥 1)2 2𝑦 3(𝑥 2 2𝑥 1) 2𝑦 3𝑥 2 6𝑥 3 2𝑦 3𝑥 2 6𝑥 5 (standard form)15
2FunctionsChapterREVISION OF THE HYPERBOLA𝒂𝒂𝒚 𝒙 𝒑 𝒒𝒒Horizontal asymptote𝒚 𝒒Indicates shape of hyperbola(with respect to asymptotes)𝒂 0 ( )𝒂 0( )𝒑InterceptsVertical asymptote𝒙 𝒑 𝑥-intercept (make 𝑦 0) 𝑦-intercept (make 𝑥 0)Domain: 𝑥 𝑅; 𝑥 𝑝Range: 𝑦 𝑅; 𝑦 𝑝2EXAMPLE: 𝑦 𝑥 1 2Axes of symmetry:Axes of symmetry (AS) Two axes of symmetry AS go through intersect of asymptotes (𝑝; 𝑞)Equations: 𝑦 𝑥 𝑘1 and𝑦 𝑥 𝑘2Substitute the point (𝑝; 𝑞) tocalculate 𝑘1 and 𝑘2 Via Afrika ›› Mathematics Grade 12𝑦-intercept:2𝑦 2 4 1𝑥- intercept:20 2 ;𝑥 2Substitute(1; 2) into𝑦 𝑥 𝑘1 𝑎𝑖𝑑 𝑦 𝑥 𝑘2𝑥 1Asymptotes:𝑥 1 𝑎𝑖𝑑 𝑦 2 2 1 𝑘1 𝑎𝑖𝑑 2 1 𝑘2𝑘1 3 𝑎𝑖𝑑 𝑘2 1𝑦 𝑥 3 𝑎𝑖𝑑 𝑦 𝑥 1yx12 2 416
2FunctionsChapterREVISION OF THEEXPONENTIAL GRAPH𝒚 𝒂𝒙 𝒑 𝒒𝒂Indicates shape of hyperbola𝒂 𝟏 𝟎 𝒂 1 𝒒Horizontal asymptote: 𝑦 𝑝Indicates that the graph 𝑦 𝑎 𝑥 wastranslated (shifted) vertically up/down𝒒 0: shifted upwards𝒒 0: shifted downwards𝒑𝑥Indicates that the graph 𝑦 𝑎 wastranslated (shifted) horizontally left/right𝒑 0: shifted left𝒑 0: shifted rightEXAMPLE:𝑦 2𝑥 1 1Asymptote: 𝑦 1𝑥-intercept (𝑦 0): 2𝑥 1 1 0 𝑥 10 1𝑦-intercept: (𝑥 0): 𝑦 2 1 1yIntercepts1 𝑥-intercept (make 𝑦 0)𝑦-intercept (make 𝑥 0)x 1 1Domain: 𝑥 𝑅Range: 𝑦 (𝑞; ) Via Afrika ›› Mathematics Grade 1217
2FunctionsChapterEXAMPLES OF SYMMETRICALEXPONENTIAL GRAPHSSYMMETRICAL IN THE 𝑦 axisy𝑥𝑦 3𝑥1𝑦 3 𝑥3xSYMMETRICAL IN THE 𝑥 axisy𝑦 3𝑥x𝑦 3𝑥 Via Afrika ›› Mathematics Grade 1218
2FunctionsChapterINTERSECTS OF TWO GRAPHSTo determine the coordinates of thepoint where two graphs INTERSECT:Use SIMULTANEOUS EQUATIONSEXAMPLEDetermine the coordinates of the points ofintersection of 𝑓(𝑥) 3𝑥 6 and𝑔(𝑥) 2𝑥 2 3𝑥 14Equate the two equations and solve for 𝒙:3𝑥 6 2𝑥 2 3𝑥 142𝑥 2 8 0𝑥2 4 0(𝑥 2)(𝑥 2) 0𝑥 2 or 𝑥 2Substitute 𝒙-values back into one of equations(choose the easier one):If 𝑥 2 then 𝑦 3(2) 6 12So one point of intersection is (2; 12).If 𝑥 2 then 𝑦 3( 2) 6 0The other point of intersection is ( 2; 0) whichis also the 𝑥-intercept of both graphs. Via Afrika ›› Mathematics Grade 1219
2FunctionsChapterTHE INVERSE OF A FUNCTION The inverse of a function, 𝑓, is denoted by 𝑓 1 . 𝑓 1 is a reflection of 𝑓in the line 𝑦 𝑥 To determine the equation of 𝑓 1 , swop 𝑥 and 𝑦 in theequation of 𝑓 The 𝑥-intercept of 𝑓 is the 𝑦-intercept of 𝑓 1FUNCTION 𝒇Straight lineINVERSE OFFUNCTION, 𝒇 𝟏Straight line𝒇: 𝒚 𝒎𝒙 𝒄Exponentialgraph𝒙𝒇: 𝒚 𝒂Parabola𝒇: 𝒚 𝒂𝒙𝟐EXAMPLESDIAGRAM𝑓: 𝑦 2𝑥 3yfInverse: 2𝑦 3 𝑥𝑓Logarithmicfunction𝑓 1: 𝑦 log 𝑎 𝑥The inverse of aparabola is NOT AFUNCTIONNB: The DOMAIN ofthe parabola has tobe RESTRICTED to𝑥 0 or 𝑥 0 sothat 𝑓 1 is also afunction Via Afrika ›› Mathematics Grade 12 1:𝑦 x1 3𝑥 2 2𝑓: 𝑦 3𝑥yfInverse:𝑓 1 : 𝑦 log 3 𝑥𝑓: 𝑦 2𝑥 2xyInverse:𝑥 2𝑦 21𝑦2 𝑥2x1𝑓 1 : 𝑥220
2FunctionsChapterMixed Exercise on Functions12Determine the coordinates of the intercept of the following two lines:2𝑥 3𝑦 173𝑥 𝑦 15abcDetermine the equation of line 𝑓.f3Determine the equation of line 𝑔.PDetermine the co-ordinates of point, P,1where the two lines intersect.dy4x 4Are these two lines perpendicular? 34(2;-1)Give a reason for your answer.eWrite down the equation of the linegwhich is parallel to line 𝑔 with a 𝑦-intercept 4of -2.3The diagram shows the graphs of y x 2 2 x 3yand y mx c .aDetermine the lengths of OA, OB and OC.bDetermine the co-ordinates of the turning point D.cDetermine m and c of the straight line.dUse the graph to determine for which valuesABxCof k for which the equation x 2 2 x k 0 wouldDhave only one real root.C4The diagram shows the graph of f ( x) 2( x 1) 8 .2EDC is the turning point.E is the mirror image of the y-intercept of 𝑓.Determine:athe length of AB.bthe co-ordinates of C.cthe length of DE. Via Afrika ›› Mathematics Grade 12AB21x
2FunctionsChapter5abc61 𝑥2Consider the function 𝑔(𝑥) 2.Make a neat drawing of 𝑔. Clearly show the asymptote and intercepts with the axes.Determine the domain of 𝑔.For which values of 𝑥 would 𝑔(𝑥) 0.The graph of 𝑓(𝑥) 𝑎𝑥y; 𝑥 0 is shown.𝐴( 2; 2) is a point on the graph where it cuts the line 𝑦 𝑥.abcd7Determine the value of 𝑎 .A( 2;2)xWrite down the coordinates of B.fBGraph 𝑓 is translated 2 units up and 1 unit right.Write down the equation of the new graph.The graphs of the following are shown :1𝑓(𝑥) 𝑥 2 2𝑥 8 and 𝑔(𝑥) 𝑥 12Determine:athe coordinates for Abthe coordinates for B and Ccthe length of CDdthe length of DE which is parallel to the 𝑦-axisefgERfthe length of GH which is parallel to the 𝑦-axisithe 𝑥-values for which 𝑓(𝑥) 𝑔(𝑥) 0.xHthe 𝑥-value for which RS would have a maximum length.the maximum length of RS.AFthe length of AF which is parallel to the 𝑥-axishygBGDCSy8The diagram alongside shows the graphs of the functions ofabcdef𝑎 𝑞𝑥 𝑝Write down the equation of the asymptote of 𝑓.𝑓(𝑥) 𝑏 𝑥 𝑐 𝑎𝑖𝑑 𝑔(𝑥) Determine the equation of 𝑓.Write down the equations of the asymptotes of 𝑔.Determine the equation of 𝑔.Determine the equations of the axes of symmetry of 𝑔 .For which values of 𝑥 is 𝑓(𝑥) 𝑔(𝑥)? Via Afrika ›› Mathematics Grade 12(2 ;5 )fB( 6 ;0 ) 2x 1gA(0 ; 3 ) 422
2FunctionsChapter9The graph of 𝑓(𝑥) 2𝑥 2 is given.ab10yDetermine the equation of 𝑓 1 inthe form 𝑓 1 : 𝑦 How can one restrict the domain of 𝑓 sothat 𝑓 1 will be a function?f(x) 2x²xThe graph of 𝑓(𝑥) 𝑎 𝑥 is given.The point A (-1; 3) lies on the graph.3y21abcd11Determine the equation of 𝑓.Determine the equation of 𝑓 1 in the form 𝑓 1 : 𝑦 Make a neat drawing of the graph of 𝑓 1 .Determine the domain of 𝑓 1 .x 3 2 1123 1 2 3A straight line graph has an 𝑥-intercept of -2 and a 𝑦-intercept of 3. Write down thecoordinates of the 𝑥- and 𝑦-intercepts of 𝑓 1 . Via Afrika ›› Mathematics Grade 1223
3LogarithmsChapterOverviewUnit 1 Page 60The definition of a logarithm Changing exponents tothe logarithmic formProofs of the logarithmiclawsUnit 2 Page 64Chapter 3 Page 58LogarithmsSolve exponential equationsusing logarithms Using logarithms Inverse of𝑦 𝑓(𝑥) 2𝑥Inverse of the function1 𝑥𝑦 𝑓(𝑥) 2Unit 3 Page 66The graph of 𝑦 log 𝑏 𝑥 where𝑏 1 and 0 𝑏 1 REMEMBER YOUR STUDY APPROACH SHOULD BE:1Work through all examples in this chapter of your Learner’s Bok.2 Work through the notes in this chapter of this study guide.3 Do the exercises at the end of the chapter in the Learner’s Book.4 Do the mixed exercises at the end of this chapter in this study guide. Via Afrika ›› Mathematics Grade 1224
3LogarithmsChapterDefinition of logarithmIf log 𝒃 𝒙 𝒚, then 𝒃𝒚 𝒙.EXAMPLES: Converting from one form to anotherLogarithmic formExponential form𝟏𝟓 𝟐𝟒𝟏𝟎, 𝟓𝟏 𝟎, 𝟏𝟐𝟓𝟏𝟎𝟏 𝟏𝟎𝟎𝟎log 𝟏 𝟐𝟒𝟏 𝟓log 𝟎,𝟓 𝟎, 𝟏𝟐𝟓 𝟏log 𝟏𝟎 𝟏𝟎𝟎𝟎 𝟏𝟏log 𝟏 𝟏 𝟐LOGARITHMIC LAWLaw 1: 𝐥𝐥𝐥 𝒎 𝑨. 𝑩 𝐥𝐥𝐥 𝒎 𝑨 𝐥𝐥𝐥 𝒎 𝑩𝑨Law 2: 𝐥𝐥𝐥 𝒎 𝑩 𝐥𝐥𝐥 𝒎 𝑨 𝐥𝐥𝐥 𝒎 𝑩𝟏𝟏𝟐 𝟏 Law 3: 𝐥𝐥𝐥 𝒙 𝑷𝒚 𝒚 𝐥𝐥𝐥 𝒙 𝑷𝐥𝐥𝐥 𝒂Law 4: 𝐥𝐥𝐥 𝒃 𝒂 𝐥𝐥𝐥 𝒃 EXAMPLESlog 𝑘 𝑎𝑏𝑐 log 𝑘 𝑎 log 𝑘 𝑏 log 𝑘 𝑐l
Grade 12 Teacher’s Guide Via Afrika Mathematics Via Afrika understands, values and supports your role as a teacher. You have the most important job in education, and we realise that your respon
