Strand 5 Of 5
3rd YearMathsHigher LevelStrand 5 of 5Topics:Functions & GraphsNo part of this publication may be copied, reproduced or transmitted in any form or by anymeans, electronic, mechanical, photocopying, recording, or otherwise, without prior writtenpermission from The Dublin School of Grinds. (Notes reference: 3-mat-h-Strand 5 of 5).
6-HOURCRASHCOURSESMAY & JUNE 2016 % 01.! čƫ * )% /ƫ0! !.ƫ *8*ƫ 1. 'The final push for CAO points. !ƫ 1 (%*ƫ (ƫ "ƫ .%* /ƫ %/ƫ .1**%*#ƫ ćġ 1.ƫ . / ƫ 1./!/ƫ 0ƫ 0 !ƫ !* ƫ "ƫ 5ƫ * ƫ 0 !ƫ !#%**%*#ƫ "ƫ 1*!ċƫ !/!ƫ 1./!/ƫ #%2!ƫ /01 !*0/ƫ 0 !ƫ !/0ƫ , //% (!ƫ2 *0 #!ƫ /ƫ 0 !5ƫ ,.!, .!ƫ " .ƫ 0 !ƫ ((ġ%), .0 *0ƫ 0 0!ƫ 4 )%* 0% */ċƫ *!ƫ ( /0ƫ 1* !ƫ "ƫ !"" .0ƫ 1( ƫ ) '!ƫ ((ƫ0 !ƫ %""!.!* !ċ !.!ƫ%/ƫ 3ƫ0 !/!ƫ 1./!/ƫ3%((ƫ!*!"%0ƫ5 1č They will offer students one last opportunity toavail of expert teaching before the StateExaminations They will provide students with a final boost ofconfidence before exam day They will give students an exam strategy plan tohelp them maximise their grade on the day 0!: At these courses our teachers will predict whatquestions are most likely to appear on your exam paper.These questions will be covered in detail and our teacherswill provide you with model A1 answers. čƫĺāćĀƫ ƫ To book, call us on 01ġ442 4442 or book online atwww.dublinschoolofgrinds.ie . / ƫ 1./!s Timetable6th YearSubjectDateTimeAccountingLevelHSunday 29th May9am - 3pmBiologyHSaturday 28th May9am - 3pmBusinessHSunday 29th May2pm - 8pmChemistryHSaturday 4th June9am - 3pmEconomicsHSaturday 28th May9am - 3pmEnglishHSunday 29th May9am - 3pmEnglishHSaturday 4th June9am - 3pmFrenchHSaturday 4th June9am - 3pmGeographyHSaturday 28th May9am - 3pmIrishHSaturday 4th June9am - 3pmMaths Paper 1HSaturday 4th June9am - 3pmMaths Paper 2HSunday 5th June9am - 3pmMathsOSaturday 28th May9am - 3pmMathsOSaturday 4th June9am - 3pmPhysicsHSaturday 28th May9am - 3pmSpanishHSunday 5th June9am - 3pmDateTime3rd YearSubjectLevelBusiness StudiesHSunday 5th June9am - 3pmEnglishHSunday 5th June9am - 3pmFrenchHSunday 29th May9am - 3pmIrishHSunday 29th May9am - 3pmMathsHSunday 29th May9am - 3pmScienceHSaturday 4th June9am - 3pmH HigherO Ordinary (! /!ƫ* 0!ƫ0 0ƫ ((ƫ 1./!/ƫ3%((ƫ0 '!ƫ,( !ƫ 0ƫ 1.ƫ ! .*%*#ƫ !*0.!ƫ 0ƫ !ƫ .%) .5ƫ (ƫ%*ƫ 0( * /Čƫ 0%(( .# *Čƫ ċƫ 1 (%*ċ
Strand 5 is worth 5 % to 16% of The Junior Cert.It appears on paper 1.ContentsFunctions and Graphs1)Understanding functions. 22)Graphing linear and quadratic functions . 63)Finding coefficients of functions . 94)Graphs crossing the x and y axes . 125)Maximum/minimum values . 166)Increasing/ Decreasing functions . 167)One graph below the other . 178)Axis of symmetry . 199)Quadratic real life graphs . 2010)Other real life graphs . 2811)Exponential functions . 3512)Past and probable exam questions . 3813)Solutions to Functions and Graphs . 66 The Dublin School of GrindsPage 1
Functions and GraphsFunctions and Graphs is worth 5 % to 16% of The Junior Cert.It appears on Paper 1.1) Understanding functionsA function is a rule that produces one output for each input.For example, if I say “pick a number, then add 3, then multiply by 5”:If you start with an input of 10, then your output would be 65.If you start with an input of 6, your output would be 45.If you start with an input of 100, your output would by 515and so on.The set of inputs in called the ‘domain’.The set of outputs is called the ‘range’.The fancy way to write the above rule is:𝑓(𝑥) 5(𝑥 3) 5𝑥 15Instead of using 𝑓(𝑥) , we can use 𝑓: 𝑥 or 𝑦 Example 1If 𝑓(𝑥) 3𝑥 7, findi)𝑓(2)ii)𝑓(0)iii)𝑓( 8)i)Well, if 𝑓(𝑥) 3𝑥 7 𝑓(2) 3(2) 7 6 7 1ii)Well, if 𝑓(𝑥) 3𝑥 7 𝑓(0) 3(0) 7 0 7 7iii)Well, if 𝑓(𝑥) 3𝑥 7 𝑓( 8) 3( 8) 7 24 7 31Example 2If 𝑓(𝑥) 6𝑥 4, solve 𝑓(𝑥) 38.Now, be careful here, we aren’t asked for 𝑓(38), we’re actually being told 𝑓(𝑥) 38 6𝑥 4 386𝑥 38 46𝑥 4242𝑥 6𝑥 7 The Dublin School of GrindsPage 2
If there is more than one function in a question, the Examiner usually calls the second one 𝑔(𝑥) or 𝑔: 𝑥, although he canactually use any letter.Example 3f and g are two functions such that 𝑓: 𝑥 𝑥 2 2 and 𝑔: 𝑥 17 2𝑥.Find the values of x for which 𝑓(𝑥) 𝑔(𝑥).Well, we’re told they’re equal, so we put them equal:𝑥 2 2 17 2𝑥2𝑥 2𝑥 2 17 0𝑥 2 2𝑥 15 0𝑎 1 𝑏 2 𝑐 15 𝑏 𝑏 2 4𝑎𝑐𝑥 2𝑎 (2) (2)2 4(1)( 15)𝑥 2(1) 2 64𝑥 2 2 8𝑥 2 2 8 2 8𝑥 𝑜𝑟𝑥 226 10𝑥 𝑜𝑟𝑥 22𝑥 3𝑜𝑟𝑥 5Example 421If 𝑓: 𝑥 1 , , find the value of k if 𝑓 ( ) 𝑘𝑓(2)Solution𝑥322 𝑘 [1 ]1231 6 𝑘[1 1]7 𝑘[2]7 2𝑘7 𝑘21 Question 1.1𝑔(𝑥) 5𝑥 2, 𝑥 ℕ. Find 𝑔(2).Give your answer in the form 𝑎 𝑎, 𝑎 ℕ. The Dublin School of GrindsPage 3
Question 1.2Here is a number machine.InputAdd 5Multiply by 4a) Find the output when the input is 10b) Work out the input when the output is 28c) Work out the input when the output is -8d) Find an expression, in terms of x, for the output when the input is x. The Dublin School of GrindsPage 4Output
Question 1.3If 𝑓(𝑥) 𝑥 2 3𝑥 7, show that the 𝑓(𝑥 1) 3 𝑥 2 5𝑥. The Dublin School of GrindsPage 5
2) Graphing linear and quadratic functionsFunctions without squared things in them represent lines. These are called ‘linear functions’.For example 𝑓(𝑥) 3𝑥 7 would represent a line and is called a linear function.Functions with squared things in them represent U or shaped curves. These are called quadratic functions.For example 𝑓(𝑥) 𝑥 2 2𝑥 3 would be U shaped and is called a quadratic function.𝑓(𝑥) 𝑥 2 2𝑥 3 would be shaped and can be called a quadratic function.How do I know if it’s U shaped or shaped?!Well, if the 𝑥 2 is positive we get a happy face: UIf the 𝑥 2 is negative we get a sad face: Example 1It’s handy to make out a table when you’re asked to work with numerous values. On a graph 𝑓: 𝑥 or 𝑓(𝑥) or 𝑔(𝑥)stand for y.X𝒚 𝟐𝒙𝟐 𝟏𝟒𝟎𝒙 𝑦 00𝑦 2(0)2 140(0) 𝑦 120010𝑦 2(10)2 140(10) 𝑦 200020𝑦 2(20)2 140(20) 𝑦 240030𝑦 2(30)2 140(30) 𝑦 240040𝑦 2(40)2 140(40) 𝑦 200050𝑦 2(50)2 140(50) 𝑦 120060𝑦 2(60)2 140(60) 𝑦 070𝑦 2(70)2 140(70)Note: There is a shortcut way to create the above table on your calculator. However, students usually makemistakes so we will do it using the table as above. The Dublin School of GrindsPage 6
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Question 2.2Let f be the function 𝑓: 𝑥 5𝑥 4 and g be the function 𝑔: 𝑥 3𝑥 1. Draw the graph of f and the graph of g,for 0 𝑥 5, 𝑥 𝑹. The Dublin School of GrindsPage 8
3) Finding coefficients of functionsOne of the Examiners favourite Junior Cert questions is to give you a graph and ask you to find values. Just follow thesesteps:Step 1: Sub in the points given.Step 2: Do simultaneous equationsExample 1SolutionNow we know 𝑓: 𝑥 𝑥 2 𝑏𝑥 𝑐 can be re-written as 𝑦 𝑥 2 𝑏𝑥 𝑐.Step 1: We know two points on the graph:( 3, 0)0 ( 3)2 𝑏( 3) 𝑐0 9 3𝑏 𝑐3𝑏 𝑐 9 ①(1, 0)0 (1)2 𝑏( 1) 𝑐0 1 1𝑏 𝑐𝑏 𝑐 1 ②Step 2: Now this is just an Algebra question where we can solve by using simultaneous equations:①3𝑏 𝑐 9②𝑏 𝑐 14𝑏 88𝑏 4𝑏 2Subbing back into ①:3(2) 𝑐 96 𝑐 96 9 𝑐 3 𝑐 The Dublin School of GrindsPage 9
Question 3.1Let 𝑓 be the function 𝑓: 𝑥 4𝑥 2 𝑏𝑥 𝑐, 𝑥 ℝ and 𝑏, 𝑐 ℤ.The points (2, 6) and ( 1, 0) lie on the graph of 𝑓, as shown in the diagram.Find the value of b and the value of c. The Dublin School of GrindsPage 10
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4) Graphs crossing the x and y axesThe Examiner can ask you where a graph crosses the x-axis or y-axis.Rule:On the x-axis: y 0On the y-axis: x 0Note:Where a graph crosses the x-axis is known as the root(s) of the function.Example 1SolutionWell, A and B are on the x-axis, so y 0 0 𝑥 2 2𝑥 8𝑎 1 𝑏 2 𝑐 8 𝑏 𝑏 2 4𝑎𝑐𝑥 2𝑎 ( 2) ( 2)2 4(1)( 8)𝑥 2(1)2 36𝑥 22 6𝑥 22 62 6𝑥 𝑜𝑟𝑥 228 4𝑥 𝑜𝑟𝑥 22𝑥 4𝑜𝑟𝑥 2 𝐵 (4, 0)𝑎𝑛𝑑𝐴 ( 2, 0)Note: You can just tell which one is which by looking at the diagram.Now, C is on the y-axis, so x 0 𝑦 (0)2 2(0) 8𝑦 0 0 8𝑦 8 𝐶 (0, 8) The Dublin School of GrindsPage 12
Question 4.1Note: If you’re asked where 2 graphs intersect simply put them equal to each other and tidy up the algebra to find thex-value (or values if there’s more than one intersection).To find the y-values, you simply sub the x-values you foundback into either of the two functions. The Dublin School of GrindsPage 13
Question 4.2Find where the graph intersects the graph by:i)Reading your graph.ii)Using algebra.State one advantage and one disadvantage of each method. The Dublin School of GrindsPage 14
Question 4.3 The Dublin School of GrindsPage 15
5) Maximum/minimum valuesThe maximum (or minimum) value of a graph is the highest (or lowest) y-value.For example, in the following graph: the minimum value is 4 (it has no maximum.)Note the way we talk about y-values here , not x-values.6) Increasing/ Decreasing functionsA function is increasing when the graph is ‘going uphill’.A function is decreasing when the graph is ‘going downhill’In the graph below: the function is increasing from x 1 to x 2.The function is decreasing from x 4 to x 1.Note the way we talk about x-values here, not y-values. The Dublin School of GrindsPage 16
7) One graph below the otherIf you were given the following graph:f(x)g(x)and asked where is 𝑓(𝑥) 𝑔(𝑥)?This simply means: where is f(x) below g(x)?i.e.: Where is the curve below the line?The answer is “between x 1 and x 3.”The fancy way you might see this written is 1 𝑥 3, but you can just write it in words like we did above, as this getsfull marks.What if you were asked where is f(x) 0?This simply means: where is f(x) lower than the x-axis?The answer is “between 3 and 2”, which can be written as 3 𝑥 2.Note:If you’re not sure whether to use or , just use what’s given in the question. The Dublin School of GrindsPage 17
Question 7.1Write the range of the values of x for whichi)ii)𝑔(𝑥) 𝑓(𝑥)𝑓(𝑥) 0g(x)f(x) The Dublin School of GrindsPage 18
8) Axis of symmetryThe axis of symmetry is simply a straight line that the graph could fold over onto itself.The Examiner can ask you to draw or write the equation of the axis of symmetry.For example, the straight line below shows the axis of symmetry:Question 8.1( 1, 4). The Dublin School of GrindsPage 19
9) Quadratic real life graphsThe Examiner loves to relate quadratic graph questions to real life.Example 1Solutiona) We’re asked to find the width.Let’s call it y.Now, we’re told the perimeter is 24metres 𝑥 𝑦 𝑥 𝑦 242𝑥 2𝑦 242𝑦 24 2𝑥24 2𝑥𝑦 2𝑦 12 𝑥b) (i) Inner length 𝑥 2Inner width 𝑦 2 (12 𝑥) 2 10 𝑥(ii) Area (𝑙𝑒𝑛𝑔𝑡ℎ)(𝑤𝑖𝑑𝑡ℎ) (𝑥 2)(10 𝑥) 10𝑥 𝑥 2 20 2𝑥 𝑥 2 12𝑥 20 The Dublin School of GrindsPage 20
c) (i)x2345678910𝒚 𝒙𝟐 𝟏𝟐𝒙 𝟐𝟎𝑦 (2)2 12(2) 20𝑦 (3)2 12(3) 20𝑦 (4)2 12(4) 20𝑦 (5)2 12(5) 20𝑦 (6)2 12(6) 20𝑦 (7)2 12(7) 20𝑦 (8)2 12(8) 20𝑦 (9)2 12(9) 20𝑦 (10)2 12(10) 20 𝑦 0 𝑦 7 𝑦 12 𝑦 15 𝑦 16 𝑦 15 𝑦 12 𝑦 7 𝑦 0(ii) We’re asked to find the max possible area. But area 𝑥 2 12𝑥 20 (from part (b))But 𝑓: 𝑥 𝑥 2 12𝑥 20 (from part (c)) area f:xBut f:x is y area ySo, really, we’re being asked to find the max possible y-value. From the graph this is clearly 16.i.e.: max possible area 16m2 The Dublin School of GrindsPage 21
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10) Other real life graphsWe meet ‘distance-time’ graphs in the Arithmetic chapter.However, the Examiner may use a different scenario.Example 1Solutioni)ii)iii)We can see the temperature is 0 degrees after 6 minutes.The temperature was -3 degrees at the start. At 10 minutes it was 2 degrees, so therefore the rise in thefirst 10 minutes was 5 degrees.Here, we’re being asked to find a missing coefficient, so, according to Section 3, we must sub in a pointon the graph.Now, there are loads of points we could pick, let’s just pick (6, 0), where 6 is the time (t) and 0 is thetemperature (C)1𝐶 (𝑡 𝑘)210 (6 𝑘)2 2: 0 6 𝑘 6 𝑘 The Dublin School of GrindsPage 28
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Another type of potential exam question involves water filling containers.If I took two containers: and I put a water hose in each of them. The height of the water in container 1 would increase far quicker thancontainer 2:Now, the increase in height may not always be linear (a straight line)For example, the following shape:would have a graph like this:What about if you fill up a spherical flask:The graph for this would be: The Dublin School of GrindsPage 32
Question 10.2The following shapes are all containers that are to be filled with water from a hose pipe. The water flows at a steadyrate all of the time:Sketch a graph for each container, representing the height of the water over time The Dublin School of GrindsPage 33
Question 10 .3For each graph of height versus volume/time below, sketch a container that could result in such a graph when filled at aconstant rate. It may be that it’s not possible to match a container to some graphs – in which case you should explainwhy a match can’t be found. The Dublin School of GrindsPage 34
11) Exponential functionsFunctions which have a variable power are called exponential functions.For example 𝑓(𝑥) 2𝑥 is an exponential function.When graphed, these have weird shapes, similar to a skateboard ramp.𝑓(𝑥) 2𝑥 would look like this:Example 1Graph the function 𝑓(𝑥) 4.2𝑥 in the domain 4 𝑥 1, where 𝑥 𝑅.Note: The dot here means multiply, not decimal.SolutionJust like other graphs, we do up a table to help us:𝒚 (𝟒). 𝟐𝒙x𝑦 (4).2 4 𝑦-4 0.25 𝑦-3𝑦 (4).2 3 0.5 𝑦 1-2𝑦 (4).2 2𝑦 (4).2 1 𝑦 2-1 𝑦 40𝑦 (4).20𝑦 (4).21 𝑦 81So our graph would look like:Note that the graph crosses the y-axis at 4. This is no coincidence! The graph will always cut the y-axis at whatevernumber your function is multiplied by. The Dublin School of GrindsPage 35
For example 7.2x would cross at y 7-8.3x would cross at y -82x would cross at y 1 (because it is 1.2x) and so on. In the above diagram, what if I asked you to find f(-2)? This simply means: find the height of the graph at x -2.The answer is 1.1Similarly 𝑓( 3) , 𝑓( 1) 2 , and so on. And if I asked you to estimate the value of x for which 𝑓(𝑥) 1.1?This simply means: find which x value gives a height of 1.1?The answer is - 1.9.2Question 11.1a) Sketch the graph of 𝑓(𝑥) 3𝑥 in the domain 2 𝑥 2, where 𝑥 𝑅.b) Where is 𝑓(𝑥) 9?c) On the same diagram, sketch 𝑔(𝑥) 2.3𝑥 The Dublin School of GrindsPage 36
Question 11.2Identify each function as linear, quadratic, or exponentialQuestion 11.3Give an example ofi)ii)iii)A linear relationshipA quadratic relationshipAn exponential relationship The Dublin School of GrindsPage 37
12) Past and probable exam questionsQuestion 1(a) The Dublin School of GrindsPage 38
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Question 2(a)Let 𝑓 be the function 𝑓: 𝑥 7𝑥 𝑥 2 .Draw the graph of 𝑓 for 0 𝑥 7, 𝑥 ℝ. The Dublin School of GrindsPage 40
(b)(a): The Dublin School of GrindsPage 41
Question 3(a) The Dublin School of GrindsPage 42
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(b)Use your graph from part (a) to estimate The Dublin School of GrindsPage 46
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13)Solutions to Functions and GraphsQuestion 1.1𝑔(𝑥) 5𝑥 2𝑔(2) 5(2) 2 10 2 8 2 2Question 1.2a) 10 5 1515 4 60Output 60b) (x 5) 4 2828 x 5 74 x 5 7x 2c) (x 5) 4 – 8 x 5 –2 x –7d) 4(x 5)Question 1.3Substitute (x 1) for x(x 1)2 3(x 1) – 7 3 x2 2x 1 3x 3 – 7 3 x2 5xQuestion 2.1x7x – x2y07(0) – 02017(1) –1267(2) –22107(3) –32127(4) –421257(5) –521067(6) – 62677(7) – 720234 The Dublin School of GrindsPage 66
Question 2.2x𝑓: 𝑥 5𝑥 45x – 4y05(0) – 4–415(1) – 4125(2) – 4635(3) – 41145(4) – 41655(5) – 421x𝑔(𝑥) 3𝑥 13x 1y03(0) 1113(1) 1423(2) 1733(3) 11043(4) 11353(5) 116Question 3.1f(x) 4x2 bx c(2,6)4(2)2 b(2) c 616 2b c 6 2b c –10(–1,0) 4(–1)2 b(–1) c 04–b c 0 – b c –42b c –10 ( 1)– b c –4 ( 2)2b c –10– 2b 2c –83c –18 c –6– b c –4– b – 6 –4–b 2 b –2 The Dublin School of GrindsPage 67
Question 3.2i)(–1,0)(2,0)(–1)2 b(–1) c 01–b c 0 –b c –1(2)2 b(2) c 0 2b c –4–b c –1 ( 2)2b c –4 ( 1)–2b 2c –22b c –43c –6 c –2– b c –1– b – 2 –1–b 1 b –1 x2 – x – 2 0ii)k2 – k – 2 – k 14k2 – 16 0(k 4)(k – 4) 0 k 4ork –4Question 4.1i)ii)f(x) x2 – 4x 3x2 – 4x 3 0(x – 3)(x – 1) 0 x 3 or x 1A (1,0) and B (3,0)A (1,0) which is also a point on g(x)g(x) x k (sub in (1,0))0 1 k k –1 The Dublin School of GrindsPage 68
Question 4.2i)x–3–2–10125 – 3x – 2x25 – 3(–3) – 2(–3)25 – 3(–2) – 2(–2)25 – 3(–1) – 2(–1)25 – 3(0) – 2(0)25 – 3(1) – 2(1)25 – 3(2) – 2(2)2y–43650–9x–3–2–1012– 2x –1–2(–3) –1–2(–2) –1–2(–1) –1–2(0) –1–2(1) –1–2(2) –1y531–1–3–5Using Algebra5 – 3x – 2x2 –2x – 1 2x2 x - 6 0(2x – 3)(x 2) 02x – 3 0orx –23x orx –223322If x , then y 2( ) 1 43( , 4)2If x –2, then y –2(–2) – 1 3(–2, 3)3Points of intersection are (–2, 3) and ( , 4).2Graph:Advantage It’s easy to read straight from the graphDisadvantage It may not be accurateAlgebraAdvantage It will give precise points of intersectionDisadvantage It is more difficult The Dublin School of GrindsPage 69
Question 4.3a) f(x) 2x2 x – 6 0(2x – 3)(x 2) 02x – 3 0 or x 2 03 x or x –22g(x) x2 – 6x 9 0(x – 3)(x – 3) 0 x 3h(x) x2 – 2x 0x2 – 2x 0x(x – 2) 0 x 0 or x 2b) h(x) Diagram 2f(x) Diagram 3g(x) Diagram 5Question 7.1i)ii)–2 x 1–4 x 2Question 8.1Axis of Symmetryx –1Question 9.1a) 2x 2y 18 2y 18 – 2x y 9–xArea length width x(9 – x) 9x – x2b) Area function:x1234567c)i)ii)iii)iv)9x – x29(1) – (1)29(2) – (2)29(3) – (3)29(4) – (4)29(5) – (5)29(6) – (6)29(7) – (7)2Area when x 2.7 17m2x 3.3 and 5.4Max possible area 20.25 m2Length Breadth 4.5 The Dublin School of GrindsPage 70y8141820201814
Question 9.2a) 2x 2y 14 2y 14 – 2x y 7–xb) x(7 – x) 7x – x2x012345677x – x27(0) – (0)27(1) – (1)27(2) – (2)27(3) – (3)27(4) – (4)27(5) – (5)27(6) – (6)27(7) – (7)2c)i)ii)iii)Area when width is 1.5 8.5 m2Max possible area 12.2 m20.5m or 6.4m The Dublin School of GrindsPage 71y061012121060
Question 10.1a)0:45 20 :45 40 :45 60 :45 80 :45 100 :45 Table710710710710710710(0) 12001(20) (40) (60) (80) 2001200120012001(100) Time is seconds, tAltitude in km, h(0)2 45(20)2 57(40)2 65(60)2 69(80)2 69200(100)2 65045205740656069806910065b)c) 69.5 kmd) 26 sece) h 45 f)45 710710𝑡 (26) 12001200(26)2 59.82𝑡 2 9110200 200(45) 200( ) 𝑡 200 (t2(multiply everything by 200)72) 𝑡 9(200)9,000 140t – 1800 t2 – 140t – 7200 0(t – 180)(t 40) 0 t 180ort - 40 (not possible since t must be greater than 0) t 180 secQuestion 10.2N.B. 2nd part of d is parallel to b The Dublin School of GrindsPage 72
Question 10.3Types of containers:1.2.3.4.Container no. 5 is impossible to draw since the height of water reduces. This is only possible if liquid is removedwhich is not the case.Question 11.1a)𝒙-2𝒚 𝟑𝒙𝑦 3 2-1𝑦 3 1012𝑦 30𝑦 31𝑦 32𝒙-2𝒚 (𝟐). 𝟑𝒙𝑦 (2).3 2-1𝑦 (2).3 1012𝑦 (2).30𝑦 (2).31𝑦 (2).31191 𝑦 3 𝑦 1 𝑦 3 𝑦 9 𝑦 b) 𝑓(𝑥) 9 when 𝑥 2.c) The Dublin School of Grinds292 𝑦 3 𝑦 2 𝑦 6 𝑦 18Page 73 𝑦
Question 11.2Question 11.3i)ii)iii)Age in yearsAngle of the sun in the skyBacterial growth The Dublin School of GrindsPage 74
12) Past & Probable Exam QuestionsQuestion 1(a)i)x–3–2–10123ii)iii)Max value 10.1x –1.8or10 – x – 2x10 – (–3) – 2(–3)210 – (–2) – 2(–2)210 – (–1) – 2(–1)210 – (0) – 2(0)210 – (1) – 2(1)210 – (2) – 2(2)210 – (3) – 2(3)2x 1.2(b)f(x) 3x – 43k – 4 11 3k 11 4 3k 15 k 5 The Dublin School of GrindsPage 75y–5491070–11
Question 2(a)x7x – x2y07(0) – 02017(1) – 12627(2) –22107(3) –32127(4) –421257(5) –521067(6) – 62677(7) – 72034(b)i)ii)iii)Max height 12.2m6.8 – 0.3 6.5 secondsTake a point on the graph, for example (6,0)f(x) ax – x2and substitute in (6,0)0 a(6) – 620 6a – 36 6a 36 a 6 The Dublin School of GrindsPage 76
Question 3i)–x – 4x 5 0 0 x 4x – 5(x 5)(x – 1) 0x –5 or x 1ii)x2 4x – 5 (x 1)2 4(x 1) – 5x2 4x – 5 x2 2x 1 4x 4 – 5x2 2x 1 4x 4 – 5 – x2 – 4x 5 02x 5 02x –55 x –2iii)f(x) 5x – 125a – 12 a 5a – a 124a 12 a 3iv)9k 8 449k 36 k 4Question 4i)x–2–101234ii)2x2 – 4x 52(–2)2 – 4(–2) 52(–1)2 – 4(–1) 52(0)2 – 4(0) 52(1)2 – 4(1) 52(2)2 – 4(2) 52(3)2 – 4(3) 52(4)2 – 4(4) 5x –0.4 or x 2.4 The Dublin School of GrindsPage 77y21115351121
Question 5a)x01234567b)i)ii)iii)35x – 5x235(0) – 5(0)235(1) – 5(1)235(2) – 5(2)235(3) – 5(3)235(4) – 5(4)235(5) – 5(5)235(6) – 5(6)235(7) – 5(7)2Max height 61mHeight after 5.5seconds 40m0.7 seconds and 6.3 seconds The Dublin School of GrindsPage 78y03050606050300
Question 6i)f:x x2 bx c(2, –6)22 2b c –6 2b c –10(0,6)02 b(0) c 6 c 6If c 6 : 2b 6 –10 2b –16 b –8x2 – 8x 6 0ii)Sub (k, –k) into x2 – 8x 6k2 – 8k 6 –kk2 – 7k 6 0(k – 1)(k – 6) 0 k 1ork 6Question 7a) 2x l 140l 140 – 2xb)i)Length width (140 – 2x)(x) 140x – 2x2 –2x2 140xx–2x2 140xy0–2(0)2 140(0)010–2(1)2 140(1)12002030–2(2)2 140(2) 140(3)2000240040–2(4)2 140(4)240050–2(5)2 140(5)200060–2(6)2 140(6)120070–2(7)2 140(7)0–2(3)2c)i)ii)Max possible area 2450m2Area when road frontage is 30m longL 30 30 140 – 2x30 – 140 –2x110 2x x 55When x 55, y Area 1600m2 The Dublin School of GrindsPage 79
Question 8a)b) f(x) : Roots 0 and –4g(x) : Roots 1 and 5(just find where the graph cuts the x–axis)c) Find the roots(x – 1)(x – 1) – 4 x2 – 2x 1 – 4 x2 – 2x – 3(x – 3)(x 1) 0x 3 or x –1 it cuts the x–axis at –1 and 3(–1,0) and (3,0) are on the graph of h(x)Looking for other points:If x 0 y 02 – 2(0) – 3 –3(0, –3) is on the graphIf x 1 y 12 – 2(1) – 3 –4(1, –4) is on the graphIf x 2 y 22 – 2(2) – 3 –3(2, –3) is on the graphd) (x – h)(x – h) – 2 x2 – 10x 23x2 – 2xh h2 – 2 x2 – 10x 23 –2xh –10x h 5e) In this Question f(x) x2 – 10x 23 (x – 5)2 – 2We can see from f(x), g(x) and h(x) that the axis of symmetry can be read from the function if it is in the form(x – h)2 a, and it is x h. the axis of symmetry of f(x) is x 5Another way to find this is as follows:𝑏Axis of symmetry is x –2𝑎Using x2– 10x 23 10 Axis of symmetry 2x 5 is axis of symmetry The Dublin School of GrindsPage 80
Question 9a) g(3) 23–3 20 1b)i)h(t) t2 – 3th(2t 1) (2t 1)2 – 3(2t 1) 4t2 4t 1 – 6t – 3 4t2 – 2t – 22ii)t – 3t 4t2 – 2t – 2 3t2 t – 2 0(3t – 2)(t 1) 03t 2 or t –12 t ort –13c)i)To find A and B just solve the equation:x2 – 2x – 8 0(x – 4)(x 2) 0x 4 or x –2 (4,0) and (–2,0) are A and BTo find C, let x 002 – 2(0) – 8 –8 C (0,–8)ii)–2 x 4Question 10(a)(b) The Dublin School of GrindsPage 81
Question 11(i)(ii)(iii)(iv)(v) The Dublin School of GrindsPage 82
Question 12(a)(b) The Dublin School of GrindsPage 83
Question 13(i)(ii)(iii)(iv)(v) The Dublin School of GrindsPage 84
Question 14(i)(ii)(iii)(iv)(v) The Dublin School of GrindsPage 85
Question 15(a)𝑓(7) 3(7) 5 26(b)𝑓(𝑘) 3𝑘 5(c)3𝑘 5 𝑘2𝑘 55𝑘 2Question 16(a)Cellulon𝑐(𝑥) 4𝑥𝑐(0) 4(0) 0𝑐(700) 4(700) 2800Mobil𝑚(𝑥) 1000 2𝑥𝑚(0) 1000 2(0) 1000𝑚(700) 1000 2(700) 2400(b) Answer: CellulonReason: From the graph you can see that the cost is zero when no data is downloaded.(c) (500, 200)(d) If Fergus is going to use less than 500MB of data he should choose Cellulon, but if he is going to use more heshould choose Mobil. The Dublin School of GrindsPage 86
Question 13(a) & (b)𝑦 𝑓(𝑥) 2 we need to add 2 to the y-axis.𝑦 𝑓(𝑥) we need to invert the graph. The Dublin School of GrindsPage 87
Carl Brien3rd Year Maths Higher LevelHaving worked for The State ExaminationsCommission, Carl brings his reputation as anauthority on the Maths Syllabus to The DublinSchool of Grinds.As a member of the Irish Mathematics TeachersAssociation, he is a popular teacher amongststudents due to his mastery of the Project MathsSyllabus and is well-known for his ability tostimulate students interest in Maths.OUR EXPERT TEACHERS
Maths O Saturday 28th May 9am - 3pm Maths O Saturday 4th June 9am - 3pm Physics H Saturday 28th May 9am - 3pm Spanish H Sunday 5th June 9am - 3pm English H Sunday 5th June 9am - 3pm French H Sunday 29th May 9am - 3pm Irish H Sunday 29th May 9am - 3pm Maths H Sunday 29th May 9am -
(B) Note that the rope is constructed of eight strands arranged in four strand pairs. Four strands (two strand pairs) rotate to the right, shown here in gray. Rotate 900 (with the standing part as the axis) and note the remaining four strands (two strand pairs) rotate to the left, shown below in white. (C) In making the splice, it is important to remember that the right-laid strand pairs of .
Interpolation Cl B dIt ltiClump Based Interpolation - Each interpolated strand is defined by 2D offset that is added to the guide strand in the direction of its coordinate frame. Pre-computed and stored in constants Cl di hi h h l h l h f h idClump radius which changes along the length of the guide strand Multi Strand InterpolationMulti Strand Interpolation
Learner's Book - one for each of the Our World Our People curriculum. This precedes the beginning of contents under each strand. Header labels Strand: This feature indicates the particular strand from which the lessons are developed. Sub-strand: These are larger groups of related owop topics to be studied under each strand.
The Church & the Crusades . school handout) . Strand 2: History and Nature of Science Strand 3: Science in Personal and Social Perspective Strand 4: Life Science Structure and Function in Living Systems Populations of Organisms in an Ecosystem Strand 5: Physical Science
This sequence ends at Year 5: strong English: /strong Sequence of content F- strong 6: Strand: /strong Language: v8.1 Australian Curriculum www.australiancurriculum.com.au December 2015 Page 3: Sub-strand Foundation Year 1 Year 2 Year 3 Year 4 Year 5 Year 6: Expressing and developing ideas sub-strand: Sentences and clause-level grammar: What a
A wire rope is a structure composed of many individual wires. A typical wire rope is composed of two major structural elements, as shown in figure 1. One is the strand that is formed by helically winding or laying wires around a central wire or a strand core. Different shapes of strand may beFile Size: 324KBPage Count: 32
3 S Region 63 TEKS Overview Strand 1 Strand 2 Strand 3 Strand 4 3.6 3.6 A 3.6 B 3.6 C 3.6 D 3.6 E 3.7 3.7 A 3.7 B 3.7 C 3.7 D 3
Strand Tucks The method for the next series of tucks is to go over a strand pair then under a strand pair. In this splice, when you go over a strand pair you need to follow the braid of the rope (as shown in photo). So, essentially you are doubling up the braid from two strands to four. Perform this second
