11/9 5a graphs of functions - desert academyusers.desertacademy.org/balei/math/1718/sl/sl1... ·...

SL1.AlgFuncCh5Transforms.1718.notebook

1

November12,2017

11/95AGraphsoffunctions

Muchofmathematicsisbasedonjustafewfunctions.Therearegeneralprincipalsthatapplytoallfunctions.Byunderstandingthem,youcaninterpret,graph,andanalyzeaninfinitenumberofcommonsituations.

Someoftheprimaryfunctionfamiliesaregivenbelow.Theparentfunctionor(motherorfather)isthesimplestform,generallycenteredattheoriginwithunitcoefficients.Thefamilyconsistsofvariationsoftheparentfunction,obtainedbychangingvariouscoefficients.

Parent

Linear y=x

Quadratic f(x)=x2

SquareRoot

Cubic f(x)=x3

CubeRoot

Exponential f(x)=bxb>0

Logarithmic f(x)=logbxx>0

Reciprocal

Rational

Itisimportanttoknowandrecognizethemajorfeaturesofeachfunctionfamily.Thegeneralshapeofeachfamilyshouldbefamiliar.

Importantgraphicalfeaturesofanygraph: DomainandRange Axesintercepts Localmaximaandminima Zerosorroots Valuesofxforwhichthefunctionisundefined AsymptotesLet'sreviewtheseforthemainfunctionfamilies

Family Parent Domain* Range* Asymp?

Turn

Pts.?

XInt* YInt*

Linear y=x No No 1 1

Quadratic f(x)=x2 y0 No Yes 0,1,2 1

SquareRoot x0 y0 No No 1 1*

Cubic f(x)=x3 No Yes 1,2,3 1

CubeRoot No No 1 1*

Exponential f(x)=bxb>0 y>0 Yes No 0* 1

Logarithmic f(x)=logbxx>0 x>0 Yes No 1 0*

Reciprocal x0 y0 Yes No 0* 0*

Rational H&V.! No 0,1 0,1

*Thedomain,rangeandnumberofinterceptsisfortheparentfunction.Amemberofthefamilymayhavedifferentvalues,dependingonhowitisshiftedorstretched.

5A:#1,3,4,5(GraphingFunctions)Objectives1. Recognize and use function families for graphing

4H:#214even(Solvegrowth&decayusinglogs) VerybriefquestionsQB:#8,9,10,11,16(QBGrowth&Decay)

Exponent&LogTesttoday

Introtowhat'snext

5A:#1,3,4,5(GraphingFunctions)


2

November12,2017

11/135BTransformationsIntro

Startingfromanyfunction,f(x),onecantransformitinseveralways.Graphically,themostcommontransformationsare:

TranslationsshiftthegraphhorizontallyorverticallybysomeamountStretchesorcompressionsscalethegraphverticallyorhorizontallybysomefactorReflectionsflipthegraphoveragivenlineRotationsrotatethegrapharoundagivenpointbysomeangle.

Let'sexplorethechangesinthegraphandthechangesintherelatedfunction.

Beforeyouexploreonyourown,it'sworthreviewingtheideaoffunctioncomposition.Supposewehaveafunction,fthatsquareswhatevercomesintoit.Wewouldwritethatasf(x)=x2sincexrepresents"thethingweputintof".Tohelptalkaboutthis,weusethewordargumenttorepresentthe"expressionthatweputintoafunction".Sointhiscasexistheargumentoff.Ifweputt+5intof,theargumentoffist+5whichiswrittenf(t+5).

f(x)=(x+3)2

fx f(x)=x2

Squaretheinputx+3

Seeingthisideainreverseisveryhelpfulingraphingcertainfunctions.YouwillpracticefunctioncompositionmoreintheHWforthissection.Payattentiontowhattheresultingfunctionformslookliketheyaresimpletransformationsofacommonmotherfunction!

Let'slookatsomecompositionsofx2.Thiscanhelpusrecognizeformsthatarisefromtransformationsofit.

Objectives1. Understand the reasons behind the four basic

transformations.

5B:#2,3,4,5(Composition)

Thisisaveryshortunit.QuizonitwillbeMon,11/205A:#1,3,4,5(GraphingFunctions) Discussall

5B:#2,3,4,5(Composition)IntroGeogebrasetupsketches


3

November12,2017

11/145CTranslations

Objectives1. Understand translations in x and y.

Videolinktolesson:https://youtu.be/HgfzlHR1H5Q

UsingGeogebraonline,doPartIofthe"ExploringTransformations"activity.YouwillcompletethisaspartofyourHW.

Quizontransformations:Mon,11/20 (Oneweekfromtoday!)

5B:#2,3,4,5c&dforall(Composition) Present2d,3d,4d,5d

5C:#ExploringXformsPartI,#6,7(Investigation)5D:#ExploringXformsPartII,#,7(Investigation)

You will turn this HW in - it will be graded as a Quiz.

https://youtu.be/HgfzlHR1H5Q


4

November12,2017

11/145DDilations

Objectives1. Understand translations in x and y.

Video:https://youtu.be/5Dm4rW96CO4(Stretches)

UsingGeogebraonline,doPartIIofthe"ExploringTransformations"activity.YouwillcompletethisaspartofyourHW.

5C:#ExploringXformsPartI,#6,7(Investigation)5D:#ExploringXformsPartII,#,7(Investigation)

You will turn this HW in - it will be graded as a Quiz.

https://youtu.be/5Dm4rW96CO4


5

November12,2017

11/165EReflections

SummaryofDilations

SummaryofReflections

Video:https://www.youtube.com/watch?v=iFYX3MvLt4M(Reflections)

Objectives1. Understand reflections in x and y.

QuizonMon,11/20(Tuesdayschedule)

SummaryofDilations

SummaryofTranslations

UsingGeogebraonline,doPartIIIofthe"ExploringTransformations"activity.YouwillcompletethisaspartofyourHW.

You will turn in the Exploring Transformations work which will be graded as a Quiz.

5E:#ExploringXformsPartIII,#58(Reflections)5F:#18(MiscTransforms)QB:#17(QBTransformations)

https://www.youtube.com/watch?v=iFYX3MvLt4M


6

November12,2017

11/165FMiscTransforms

Thereisnonewinformationforsection5F.Justsomemorepracticelookingatavarietyoftransformationsandcombinationsoftransformations.

PracticeinpreparationforQuizThursday

QBProblems ConceptMap

Forextrapractice,seetheReviewSectionsA,B,andCinthebook.

Objectives1. Practice miscellaneous combinations of transformations.

30minQuizontransformationsnexttime....afterdiscussingproblemsandreview.

You will turn in the Exploring Transformations work which will be graded as a Quiz.

5E:#ExploringXformsPartIII,#58(Reflections)5F:#18(MiscTransforms)QB:#17(QBTransformations)

Attachments

Reciprocals.ggb

concept_map.pdf

Transformations.ggb

SLA3&4ExpLogPractice.pdf

SLA5TransformPractice.pdf

XformGen2.ggb

geogebra_thumbnail.png

geogebra_javascript.js

function ggbOnInit() {}

geogebra.xml

SMART Notebook

y = f(x)

y = pf(x)

y

x

A(180, -1)A(180, -1)

Example: Vertical stretch by a factor of 2

A (180, -2)0

y = f(x)

y = f(qx)

y

x

A(90, 0)A(90, 0)

Example: Horizontal stretch by a factor of 2

A (180, 0)0

y

x

Example: Inverse function reflect in y = x

y = f(x)

y = f-1(x)

y = xA(0.5, 3)A(0.5, 3)

B(-3, -4)B(-3, -4)

A (3, 0.5)0

B (-4, -3)0

y = f(x)

y

x

Example: Horizontal translation 2 units right

y = f(x-2)

A (2, 8)0

B (1, 1)0

A(0, 8)A(0, 8)

B(-1, 1)B(-1, 1)

y = f(x)y

x

Example: Vertical translation 3 units up

y = f(x) + 3

A (-2, 3)0

A(-2, 0)A(-2, 0)

y = f(x)

y

x

Example: Reflection in y-axis

A (-2, 4)0 A(2, 4)A(2, 4)

y = f(-x)

Example: Reflection in x-axisy

x

A(2, 4)A(2, 4)

A (2, -4)0

y = f(x)

y = -f(x)

Transformations

of graphs y = f(x)Reflection

Stretch

Translation

In y-axis

y = f(-x)

In x-axis

y = -f(x)

Horizontal

y = f(x-a)

Vertical

y = f(x)+b

Inverse

y = f-1(x)

Vertical

y = pf(x)

Horizontal

y = f(qx)

p > 0 q > 0

(

(ab

Return to text

SMART Notebook




geogebra.xml

SMART Notebook

IB Math Standard Level Year 1 Exponent and Logarithm Practice Alei - Desert Academy

C:\Users\Bob\Documents\Dropbox\Desert\SL1\1 - Algebra&Functions\Practice\SLA3&4ExpLogPractice.docx on 10/31/15 at 12:13 PM Page 1 of 4

Exponents & Logs Practice Problems

1. Solve the equation 9x1 =

Working:

Answer: ......................................................................

(Total 4 marks) 2. Solve the equation 43x1 = 1.5625 102.

Working:

Answer: ......................................................................

(Total 4 marks) 3. If loga 2 = x and loga 5 = y, find in terms of x and y, expressions for

(a) log2 5; (b) loga 20. Working:

Answers: (a) .................................................................. (b) ..................................................................

(Total 4 marks)

4. Let log10P = x , log10Q = y and log10R = z. Express 2

10 3logP

QR

in terms of x , y and z.

Working:

Answer: ....................................................................

(Total 4 marks)

.31 2x



5. Solve the equation log9 81 + log919

+ log9 3 = log9 x.

Working:

Answer: .......................................................................

(Total 4 marks) 6. Consider the following statements

A: log10 (10x) > 0. B: 0.5 cos (0.5x) 0.5.

C: arctan2 2

x .

(a) Determine which statements are true for all real numbers x. Write your answers (yes or no) in the table below. Statement (a) Is the statement true for all

real numbers x? (Yes/No) (b) If not true, example

A B C

(b) If a statement is not true for all x, complete the last column by giving an example of one value of x for which the statement is false.

Working:

(Total 6 marks) 7. Given that log5 x = y, express each of the following in terms of y.

(a) log5 x2

(b) log5

(c) log25 x Working:

Answers: (a) .................................................................. (b) .................................................................. (c) ..................................................................

(Total 6 marks) 8. A population of bacteria is growing at the rate of 2.3% per minute. How long will it take for the size

of the population to double? Give your answer to the nearest minute. Working:

Answer: ......................................................................

(Total 4 marks)

x1



9. Initially a tank contains 10 000 litres of liquid. At the time t = 0 minutes a tap is opened, and liquid then flows out of the tank. The volume of liquid, V litres, which remains in the tank after t minutes is given by

V = 10 000 (0.933t). (a) Find the value of V after 5 minutes.

(1) (b) Find how long, to the nearest second, it takes for half of the initial amount of liquid to flow out

of the tank. (3)

(c) The tank is regarded as effectively empty when 95% of the liquid has flowed out. Show that it takes almost three-quarters of an hour for this to happen.

(3) (d) (i) Find the value of 10 000 V when t = 0.001 minutes.

(ii) Hence or otherwise, estimate the initial flow rate of the liquid. Give your answer in litres per minute, correct to two significant figures.

(3) (Total 10 marks)

10. A group of ten leopards is introduced into a game park. After t years the number of leopards, N, is modelled by N = 10 e0.4t. (a) How many leopards are there after 2 years? (b) How long will it take for the number of leopards to reach 100? Give your answers to an

appropriate degree of accuracy. Give your answers to an appropriate degree of accuracy. Working:

Answers: (a) .................................................................. (b) ..................................................................

(Total 4 marks) 11. Each year for the past five years the population of a certain country has increased at a steady rate of

2.7% per annum. The present population is 15.2 million. (a) What was the population one year ago? (b) What was the population five years ago? Working:

Answers: (a) .................................................................. (b) ..................................................................

(Total 4 marks) 12. Michele invested 1500 francs at an annual rate of interest of 5.25 percent,

compounded annually. (a) Find the value of Micheles investment after 3 years. Give your answer to the nearest franc.

(3) (b) How many complete years will it take for Micheles initial investment to double in value?

(3) (c) What should the interest rate be if Micheles initial investment were to double in value in 10

years? (4)

(Total 10 marks)



13. Solve the equation log27 x = 1 log27 (x 0.4). Working:

Answer: ......................................................................

(Total 6 marks) 14. Consider functions of the form y = ekx

(a) Show that ( )1

0

1 1kx ke dx ek

= . (3)

(b) Let k = 0.5 (i) Sketch the graph of y = e0.5x, for 1 x 3, indicating the coordinates of the

y-intercept. (ii) Shade the region enclosed by this graph, the x-axis, y-axis and the line x = 1. (iii) Find the area of this region.

(5)

(c) (i) Find dydx

in terms of k, where y = ekx.

The point P(1, 0.8) lies on the graph of the function y = ekx. (ii) Find the value of k in this case. (iii) Find the gradient of the tangent to the curve at P.

(5) (Total 13 marks)

15. $1000 is invested at 15% per annum interest, compounded monthly. Calculate the minimum number of months required for the value of the investment to exceed $3000. Working:

Answer: ......................................................................

(Total 6 marks) 16. The mass m kg of a radio-active substance at time t hours is given by

m = 4e0.2t. (a) Write down the initial mass. (b) The mass is reduced to 1.5 kg. How long does this take? Working:

Answers: (a) .................................................................. (b) ..................................................................

(Total 6 marks)

IB Math Standard Level Year 1 Exponent and Logarithm Practice - Markscheme Alei - Desert Academy


Exponents & Logs Practice Problems - Markscheme

1. 9x1 =

32x2 = 32x (M1) (A1) 2x 2 = 2x (A1)

x = (A1) (C4)

[4] 2. 43x1 = 1.5625 102

(3x 1)log10 4 = log10 1.5625 2 (M1)

3x 1 = 1010

log 1.5625 2log 4

(A1)

3x 1 = 3 (A1)

x = (A1) (C4)

[4]

3. (a) log2 5 = (M1)

= (A1) (C2)

(b) loga 20 = loga 4 + loga 5 or loga 2 + loga 10 (M1) = 2 loga 2 + loga 5 = 2x + y (A1) (C2)

[4]

4. log10 = 2log10 (M1)

2log10 = 2(log10P log10(QR3)) (M1)

= 2(1og10P log10Q 3log10R) (M1) = 2(x y 3z) = 2x 2y 6z or 2(x y 3z) (A1)

[4] 5. METHOD 1

log9 81 + log9 + log9 3 = 2 1 + (M1)

= log9 x (A1)

x = (M1) x = 27 (A1) (C4)

METHOD 2 log 81 + log9 + log9 3 = log9 (M2)

= log9 27 (A1) x = 27 (A1) (C4)

[4]

x2

31

21

32

2log5log

a

a

xy

2

3

QRP

3QR

P

3QR

P

91

21

23

23

9

91

3

918 1



6. Statement (a) Is the statement true for all

real numbers x? (Yes/No) (b) If not true, example

A No x = l (log10 0.1 = 1) (a) (A3) (C3) B No x = 0 (cos 0 = 1) (b) (A3) (C3) C Yes N/A

Notes: (a) Award (A1) for each correct answer. (b) Award (A) marks for statements A and B only if NO in column (a). Award (A2) for a correct counter example to statement A, (A1) for a correct counter example to statement B (ignore other incorrect examples). Special Case for statement C: Award (A1) if candidates write NO, and give a valid reason (eg

arctan 1 = ).

[6] 7. (a) log5 x2 = 2 log5 x (M1)

= 2y (A1) (C2)

(b) log5 = log5 x (M1)

= y (A1) (C2)

(c) log25 x = (M1)

= (A1) (C2) [6]

8. 1.023t = 2 (M1)

t = (M1)(A1)

= 30.48... 30 minutes (nearest minute) (A1) (C4)

Note: Do not accept 31 minutes. [4]

9. Note: A reminder that a candidate is penalized only once in this question for not giving answers to 3 sf

(a) V(5) = 10000 (0.9335) = 7069.8 = 7070 (3 sf) (A1) 1

(b) We want t when V = 5000 (M1) 5000 = 10000 (0.933)t 0.5 = 0.933t (A1)

log(0.5) ln(0.5) or log(0.993) ln(0.993)

t =

9.9949 = t After 10 minutes 0 seconds, to nearest second (or 600 seconds). (A1) 3

(c) 0.05 = 0.933t (M1) log(0.5) 43.197 minutes

log(0.993)t= = (M1)(A1)

3/4 hour (AG) 3

45

x1

25loglog

5

5 x

y21

0 2 3.1ln2ln



(d) (i) 10000 10000(0.933)0.001 = 0.693 (A1)

(ii) Initial flow rate = where t = 0, (M1)

0.6930.001

dvdt

= = 693

= 690 (2 sf) (A1) OR

dvdt

= 690 (G2) 3

[10] 10. (a) At t = 2, N = 10e0.4(2) (M1)

N = 22.3 (3 sf) Number of leopards = 22 (A1)

(b) If N = 100, then solve 100 = 100e0.4t

10 = e04t ln 10 = 0.4t

t = ~ 5.76 years (3 sf) (A1)

[4]

11. (a) = 14.8 million (M1)(A1) (C2)

(b) 515.2

(1.027) = 13.3 million (M1)(A1)(C2)

OR

414.8

(1.027)= 13.3 million (M1)(A1)(C2)

[4] 12. (a) Value = 1500(1.0525)3 (M1)

= 1748.87 (A1) = 1749 (nearest franc) (A1) 3

(b) 3000 = 1500(1.0525)t 2 = 1.0525t (M1)

t = log 2

log1.0525= 13.546 (A1)

It takes 14 years. (A1) 3 (c) 3000 = 1500(1 +r)10 or 2(1 +r)10 (M1)

= 1 + r or log 2 = 10 log (1 + r) (M1)

r = 1 or r = 1 (A1) r = 0.0718 [or 7.18%] (A1) 4

[10] 13. log27 (x(x 0.4)) = l (M1)(A1)

x2 0.4x = 27 (M1) x = 5.4 or x = 5 (G2) x = 5.4 (A1) (C6)

Note: Award (C5) for giving both roots. [6]

tV

dd

4.010ln

027.12.15

10 2

10 2 102log

10



14. (a) 11

00

1kx kxe dx ek

= (A1)

= ( )01 ke ek (A1)

= ( )1 1kek (A1)

= ( )1 1 kek (AG) 3

(b) k = 0.5 (i)

(A2)

Note: Award (A1) for shape, and (A1) for the point (0,1). (ii) Shading (see graph) (A1)

(iii) Area =1

0

kxe dx for k = 0.5 (M1)

= (1 e0.5)

= 0.787 (3 sf) (A1) OR Area = 0.787 (3 sf) (G2) 5

(c) (i) dydx

= kekx (A1)

(ii) x = 1 y = 0.8 0.8 = e k (A1) ln 0.8 = k k = 0.223 (A1)

(iii) At x = 1 dydx

= 0.223e0.223 (M1)

= 0.179 (accept 0.178) (A1) OR dydx

= 0.178 or 0.179 (G2) 5

[13]

15. 15% per annum = % = 1.25% per month (M1)(A1)

Total value of investment after n months, 1000(1.0125)n > 3000 (M1) => (1.0125)n > 3

n log (1.0125) > log (3) => n >log3

log1.0125 (M1)

Whole number of months required so n = 89 months. (A1) (C6)

1 0 1 2 3

1(0,1)

y

x

5.01

1215



Notes: Award (C5) for the answer of 90 months obtained from using n 1 instead of n to set up the equation. Award (C2) for the answer 161 months obtained by using simple interest. Award (C1) for the answer 160 months obtained by using simple interest.

[6] 16. (a) Initial mass t = 0 (A1)

mass = 4 (A1) (C2) (b) 1.5 = 4e0.2t (or 0.375 = e0.2t) (M2)

ln 0.375 = 0.2t (M1) t = 4.90 hours (A1) (C4)

SMART Notebook

IB Math Standard Level Year 1 Transformation Practice Alei - Desert Academy

/Users/Shared/Dropbox/Desert/SL1/1 - Algebra&Functions/Practice/SLA5TransformPractice.docx on 11/14/15 at 8:16 PM Page 1 of 5

Transformations Practice Problems 1. Three of the following diagrams I, II, III, IV represent the graphs of

(a) y = 3 + cos 2x (b) y = 3 cos (x + 2) (c) y = 2 cos x + 3.

Identify which diagram represents which graph.

Working:

Answers: (a) .................................................................. (b) .................................................................. (c) ..................................................................

(Total 4 marks)

x

12

12

32

y

2

1

1

2

x

12

12

32

y

3

2

1

3

y

x

4

2

12

12

32

x

12

12

32

5

4

3

2

1

y

I

III

II

IV



2. The diagrams show how the graph of y = x2 is transformed to the graph of y = f (x) in three steps. For each diagram give the equation of the curve.

Working: Answers:

(a) .................................................................. (b) .................................................................. (c) ..................................................................

(Total 4 marks) 3. The diagram shows the graph of y = f (x), with the x-axis as an asymptote.

(a) On the same axes, draw the graph of y =f (x + 2) 3, indicating the coordinates of the images

of the points A and B. (b) Write down the equation of the asymptote to the graph of y = f (x + 2) 3.

Working:

Answer: (b) ....................................................................

(Total 4 marks)

y

y

y

y

0

0

0

0

x

xx

xy=x2

4

1

1 1

1

3

7

(a)

(b) (c)

A(5, 4)

B(5, 4)

y

x



4. The following diagram shows the graph of y = f (x). It has minimum and maximum points at

(0, 0) and ( ).

(a) On the same diagram, draw the graph of y = f(x 1) + 32 .

(b) What are the coordinates of the minimum and maximum points of y = f(x 1) + 32 ?

Working:

Answer: (b) ................................................................

(Total 4 marks) 5. The sketch shows part of the graph of y = f (x) which passes through the points A(1, 3), B(0, 2),

C(l, 0), D(2, 1) and E(3, 5).

A second function is defined by g (x) = 2f (x 1).

(a) Calculate g (0), g (1), g (2) and g (3). (b) On the same axes, sketch the graph of the function g (x).

Working:

Answers: (a) .................................................................. ..................................................................

(Total 6 marks)

2

1,1

2 1 0 1 2 3

3.5

3

2.5

2

1.5

1

0.5

1

1.5

0.5

2

2.5

y

x

8

7

6

5

4

3

2

1

0

1

2

4 3 2 1 1 2 3 4 5

A

B

C

D

E



6. (a) The diagram shows part of the graph of the function f (x) = The curve passes through

the point A (3, 10). The line (CD) is an asymptote.

Find the value of

(i) p; (ii) q.

(b) The graph of f (x) is transformed as shown in the following diagram. The point A is

transformed to A (3, 10).

Give a full geometric description of the transformation.

Working:

Answers: (a) (i) ........................................................... (ii) ........................................................... (b) .................................................................. ..................................................................

(Total 6 marks)

. pxq

15

10

5

-5

-10

-15

C

A

D

y

x15105051015

y

x

15

15

10

10

5

50

5

5

10

10

15

15

A

C

D



7. The diagram shows parts of the graphs of y = x2 and y = 5 3(x 4)2.

The graph of y = x2 may be transformed into the graph of y = 5 3(x 4)2 by these transformations.

A reflection in the line y = 0 followed by a vertical stretch with scale factor k followed by a horizontal translation of p units followed by a vertical translation of q units.

Write down the value of (a) k; (b) p; (c) q.

Working:

Answers: (a) .................................................................. (b) .................................................................. (c) ..................................................................

(Total 4 marks)

2

4

6

8

2 0 2 4 6

y

x

y = x2

2y x= 5 3( 4)



Transformations Practice Problems - Markscheme 1. (a) I

(b) III (c) IV

Note: Award (C4) for 3 correct, (C2) for 2 correct, (C1) for 1 correct. [4]

2. (a) y = (x 1)2 (A2)(C2) (b) y = 4(x 1)2 (A1) (C1) (c) y = 4(x 1)2 + 3 (A1) (C1)

Note: Do not penalize if these are correctly expanded. [4]

3. (a) Correct vertical shift (A1) Coordinates of the images (see diagram) (A1) (A1)

(b) Asymptote: y = 3 (A1)

[4] 4. (a)

(A2)(C2)

(b) Minimum: (A1) (C1)

Maximum: (2, 2) (A1) (C1) [4]

5. (a) g (x) = 2 f (x l) x 0 1 2 3

x 1 1 0 1 2 f (x 1) 3 2 0 1

g (0) = 2 f (1) = 6 (A1)(C1) g (1) = 2 f (0) = 4 (A1)(C1) g (2) = 2 f (l) = 0 (A1)(C1) g (3) = 2 f (2) = 2 (A1)(C1)

A(5, 4)

B(5, 4)

y

x

A(7, 7)

B(3, 1)

2 1 0 1 2 3

0.5

1

1.5

2

2.5

0.5

1

1.5

2

2.5

3

3.5

y

x

(1, )12

23,1



(b) Graph passing through (0, 6), (1, 4), (2, 0), (3, 2) (A1)

Correct shape. (A1)

(C2)

[6] 6. (a) (i) p = 2 (A2)(C2)

(ii) 10 = (or equivalent) (M1)

q = 10 (A1) (C2) (b) Reflection, in x-axis (A1)(A1) (C2)

[6] 7.

q = 5 (A1)(C1)

k = 3, p = 4 (A3)(C3) [4]

8

7

6

5

4

3

2

1

0

1

2

4 3 2 1 1 2 3 4 5

A

B

C

D

E

x

y

23q

2 0 2 4 6

2

4

6

8y = x

y = x53( 4)2

2

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geogebra.xml

SMART Notebook

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