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Annex C Cover Page for Compilation by Topics Example : If NYJC compiled questions on Vectors, this Cover Page will show all the Vectors questions from the other colleges with answers. S/No Graphs/Curve Sketching Answers 1 AJC AJC/I/9 (i) (ii) 4 Translation (-1) unit in the x-direction Scaling, parallel to the x-axis, factor 1/2 Reflection about the x-axis Translation of 3 units in the y-direction AJC/II/4 A = 3 and B = -3 2 ACJC ACJC/I/8 6, 1 a b = =- Translation of 1 unit in direction of the positive x – axis, followed by scaling parallel to the x – axis with scale factor 1 3 unit, followed by translation of 2 unit in direction of the positive y – axis 2 y x ) ( x f y = 2 x -1 9 -2

Graphs and Curve Sketching 1

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Annex C Cover Page for Compilation by TopicsExample : If NYJC compiled questions on Vectors, this Cover Page will show all the Vectors questions from the other colleges with answers. S/No Graphs/Curve SketchingAnswers 1AJC AJC/I/9 (i) (ii) 4 Translation (-1) unit in the x-direction Scaling, parallel to the x-axis, factor 1/2 Reflection about the x-axis Translation of 3 units in the y-direction

AJC/II/4 A = 3 and B = -3 2ACJC ACJC/I/8 6,1 a b = = Translation of 1 unit in direction of the positive x axis, followed by scaling parallel to the x axis with scale factor 13 unit, followed by translation of 2 unit in direction of the positive y axis 2 y x ) (x f y =2 x -1 9 -2 ACJC/I/12 (i),2 y x a x = + =(ii)(2 2 , 2 5 ) a a + + ,(2 2 , 2 3 ) a a 16 132 y x 0 1 6 13 1xyx= 16 13 y x 0 1 26 13 1xyx= -1 22 22 a a y x a = +(2 2 , 2 5 ) a a + +(2 2 , 2 3 ) a a 2 x = 3CJC CJC/I/6(a) For no real roots,3 0.Hencesketchthegraphofy = f (x), wheref (x) is the first derivative function off (x) with respect to x.[3] (b)Sketch, on separate clearly labeled diagrams, the graphs of (i) y = f (|x|)(ii) y = f (2x + 1)(iii) y2 = f (x) [8] [Showonyourdiagrams,theequationsoftheasymptote,thecoordinatesofturning points and points of intersection with the coordinate axes.] 15SRJC/I/11 The curve C has the equation y =( ) f x where ( )2fax bx cxx d+ +=+, where a, b, c and d are constant integer values such that 20 ad bd c + .The line y = 3x 5 is an asymptote to the curve C. Byrewriting ( ) ( ) fkx Q xx d= ++where( ) Q x isapolynomialofxandkisaconstant,or otherwise, show that a = 3 and b = 3d 5. [3] (i)Show that C has two stationary points if and only if c > 5d. [3] (ii)Hence, using the minimum value of c and finding the value of b, sketch the curve when d = 1. Your sketch should include all asymptotes, stationary points and axial intercepts if any. [4] (iv)FindtheequationsofasymptoteswhenCundergoestransformationfrom( ) f x to ( ) 2f 3 x . [2] (-2,-3) -11-6 y = f (x) x O y 16TPJC/I/10 The diagram shows a sketch of ( )2x ayx b=+.The curve has a vertical asymptote1 = xand a minimum point at( ) 1, 0 . (i)Find the values ofa and b.Hence obtain the equation of the oblique asymptote.[4] (ii)Withouttheuseofgraphiccalculator,findthecoordinatesofthemaximumpointand deduce the set of values of k for which ( )2x akx b=+ has no real solutions.[3] (iii)Thecurveistranslatedpunitsalongy-axiswherepispositive.Theresultingcurve intersects the line6 y = at exactly onepoint. Show that the equation of theresulting curveis 21x cyx+=+,wherecistobedetermined.Hence,drawasketchof21x cyx+=, stating clearly the stationary points and equations of asymptotes. [4] x = -1yx1 17TJC/I/10 (a)Find all the asymptotes of ( )( )221x xyx x +=+ , where1 < .Hence, sketch the curve.(Coordinates of the turning points are not required.)[4] (b) The diagram shows a sketch of the graph of( ) f y x = . Sketch, on separate clearly labeled diagrams, the graphs of (i) ( )1fyx= ,[3] (ii)( ) f ' y x = .[3] 18VJC/I/6 Sketch, on separate diagrams, the graph ofk y kx 42 2= + , where (i)k = 4,[2] (ii)k < 0,[3] making clear the main relevant features of each curve. The graph in part (i) undergoes a single transformation (which is not a reflection) that leaves only the points on the xaxis unaffected. Give a possible description of the transformation.[1] VJC/I/11 y x O 5 1 1 ( )122, ) ( f x y =x 0 y 2 13 y = 3 x = 2 Thediagramshowsasketchofthecurve) ( f x y = .Thelineswithequationsx=2and y = 3 are asymptotes to the curve. The intersections of the curve with the x andyaxes have coordinates (1, 0), (3, 0) and (0, 2). On separate diagrams, sketch the graphs of (i)) ( f2x y = ,[2] (ii) ) ( f1xy = , given that0) 2 ( f1= , [4] (iii)) ( f x y = .[3] Whenever appropriate,your sketch should indicate clearly the equationsofanyasymptotes, intercept(s) and the coordinates of turning point(s). 19YJC/I/2 The diagram above shows the curve) ( f x y = .There is a maximum point at A(0, 3), a minimum point at B(2,0) and the curve also cuts the x-axis at the point C(2, 0).Sketch on separate diagrams, showing the corresponding points, if possible, the graphs of (i)) ( f2x y = ,[2] (ii) |||

\|=2fxy ,[2] (iii)) ( f x y = , where) ( f x is the derivative of) ( f x .[2] YJC/I/8 The curve C has equation 3 212++ +=xkx xywhere k is a positive constant. (i)Obtain the equations of the asymptotes of C.[3] (ii)Find the value of k for which the x-axis is a tangent to C.[3] (iii) Sketch C for the case k = 3. Hence, using graphical method, find the range of values of x that satisfy the inequalities6 4 1 32+ > + + x x x .[4] x y O B(2, 0) C(2, 0) A(0,3)