DHS Graphs & Transformations 2 (9758) Topical Revision

1.2 Graphs and Transformations 1 1.2 Graphs and Transformations 2 Transformation of Curves 1 The graph of ( )fyx = below cuts the x- axis at x = a and x = b. It has a turning point at (h, k). (i) State the range of values of x for which the graph of ( )f x (a) is strictly increasing, (b) concaves upwards. [3] (ii) State the value(s) of x for the stationary point(s) of f( x), specifying clearly the nature of each stationary point. [2] 2 The diagram shows the graph of ( )fyx = (not drawn to scale) with x-intercepts a, c, e and stationary points at x = b, x = d and x = e . (i) State the x-coordinates and nature of the stationary points of the graph of ( )fyx= . [3] (ii) Sketch a possible graph of ( )fyx= indicating clearly the x-coordinates of the stationary points. [2] 3 ACJC/2010Promo/12 The diagram below shows the graph of ( )fyx= . It has a maximum point at (0, 2 ),Aa − a minimum point at 12 , 6B a a−− , x-intercept at a− , and asymptotes at 2 ax= . On separate diagrams, sketch the graphs of (i) f ( )y x a=+ , [3] (ii) 1 f ( ) y x = , [3] (iii) f '( )yx= [3] showing clearly the asymptotes and the coordinates of the turning points. (h, k) a b 0 x y x y a b c d e 0 y = f (x) y x -a A B 2

1.2 Graphs and Transformations 2 4 NYJC/2010Promo/11 The graph of ( )fyx= is shown above, where the lines ya= and xa= , , are asymptotes to the curve and (−3, 0) is a minimum point. The graph intersects the y-axis at (0, a). On separate diagrams, sketch the graphs of (i) f ( )y a x=− , [2] (ii) ( ) 1 fy x= , [3] showing clearly the equations of asymptotes, coordinates of any points of intersection with the x and y-axes and the coordinates of the stationary point(s). 5 DHS/2013MYE JC1/12(a) (a) The curves 1C and 2C are given by the equations 2 1xy+= and ( ) 2 2 2 1+ + =x y k respectively, where k is a positive constant. (i) Describe clearly a sequence of transformations that maps 1C to 2C [2] (ii) Given that 3k = , sketch the graph of 2C , indicating the coordinates of the intercepts and the vertex. [2] 6 RI/2010Promo/5 (a) The graph of ( )fyx= passes through the origin, has a maximum turning point at ( 2,2)A − and a minimum turning point at 32, 2B − , as shown in the diagram on the right. 1a 0 y y = a x = a y = f (x) O x x (−3, 0)

1.2 Graphs and Transformations 3 Sketch, on separate diagrams, the graphs of (i) f ( 1)yx=+ , [2] (ii) f (1 | |)yx=− , [3] stating clearly the coordinates of the points corresponding to A and B (if any). (b) The graph of ( ) ( ) 22 2x p y p p− + − = , 0p underwent exactly two simple transformations, involving either reflection, stretching or translation only, to become the following ellipse. (i) Write down the equation of the ellipse. [1] (ii) Describe clearly (in words) a sequence of the two transformations. [2] 7 SRJC/2010Promo/8a(part) (a) The diagram shows the graph of f ( )yx= . The points A, B, C have coordinates (2,0) , ( )4, 2− , ( )7,0 respectively. The equations of the asymptotes are 0x= and 2y= . Sketch the graph of 1 .f ( )y x= [2] State clearly the equations of all the asymptotes and show the coordinates of the points corresponding to A, B and C. O 0 A(2, 0) B(4, -2) C(7, 0)

1.2 Graphs and Transformations 4 8 VJC/2010Promo/6 The diagram shows the graph of ( )f.yx= On separate axes, sketch the curves of (i) ( ) 1 fy x= [3] (ii) ( )d fdyx x=  [3] 9 MJC/2010Promo/4 The curve C1 has equation 2 2 2 2k x y k+= , where 1k  . (i) Sketch C1, indicating the axial intercepts, asymptotes and stationary points, if any. [2] (ii) Hence, or otherwise, sketch 2 2 2 2 1 4k x y k+= , where 1k  , indicating the axial intercepts, asymptotes and stationary points, if any. [1] C1 undergoes a single transformation to become C2. (iii) Given that C2 has a line of symmetry 2y= and passes through the origin, state the value of k. [1] (iv) Describe a geometrical transformation by which C2 may be obtained from C1. [1] (v) Write down the equation of C2. [1] 10 DHS/2013Promo/9 (a) Show that the equation 2236 36 4 27y y x− − = can be expressed as ( ) 2 221 1.49 y x− −= [2] Hence, (i) state precisely a sequence of transformations by which the graph of 2236 36 4 27y y x− − = may be obtained from the graph of 22 149 yx−= . [2] (ii) sketch the graph of 2236 36 4 27y y x− − = , indicating the coordinates of the intercepts and the equations of the asymptotes. [2] (b) The diagram shows the graph of ( )fyx= . The curve passes through the x-axis at ( )1,0a−− and has a maximum point at ,12 a−− , where a is a positive constant. The curve also has horizontal and vertical asymptotes 1 2y= and xa=− respectively. On separate diagrams, sketch the graphs of (i) ( ) 1 fy x= , [3] (ii) ( )f,y x a=− − [3] indicating clearing the equations of any asymptotes, and the coordinates of turning points and axial intercepts. x y x O  1 2y= xa=− ( )fyx= 1a−− ,12 a−−

1.2 Graphs and Transformations 5 11 PJC/2013Promo/9 (a) Describe a sequence of transformations which transforms the graph of 1y x= to the graph of 1 23y x= + . [2] (b) The diagram above shows the graph of . The curve passes through the point and has turning point at 11, 3B  . The lines 0x= , 2x= , 0y= , and 1 2y= are asymptotes to the curve. On separate diagrams, sketch the graphs of (i) , [2] (ii) , [2] stating, in each case, the axial intercepts, asymptotes and the coordinates of any turning points. 12 SAJC/2013Promo/9b State the sequence of transformations which transform the graph of 1y x to the graph of 27 25 xy x . [3] Sketch the graph of 27 25 xy x += + , giving the equations of any asymptotes and the coordinates of any points of intersection with the x- and y-axes. [3] By adding an additional curve on the same sketch, determine the number of real roots of the equation ( ) ( ) 2 22 2 7 9 2 5 1 4 xxx + = + −  . [3] ( )fyx= ( )2,0A − ( ) 1 fy x= ( )fyx=−

1.2 Graphs and Transformations 6 13 HCI/2013Promo/4 The diagram below shows the graph of ( )fyx= . It cuts the axes at the points ( )0, 1 , ( )1.5, 0 and ( )3, 0 . It has a minimum point at ( )2.5, 0.5− . The horizontal, vertical and oblique asymptotes are 0, 7y x a== and y x a=− + respectively, where a is a positive constant. On separate diagrams, sketch the graphs of (i) ( ) 1 fy x= , [3] (ii) ( )f'yx= , [3] showing clearly the axial intercepts, the stationary points and the equations of the asymptotes where applicable. 14 VJC/2012Promo/4 The diagram shows the graph of ( )f2yx= . The curve passes through the points ( )1, 0A − , ( )0, 2B , ( )1, 0C and ( )3, 3D − . Sketch, on separate clearly labelled diagrams, the graphs of (i) ( )f2yx=− , [1] (ii) ( )f 2 2yx=− , [2] (iii) ( )1 f22yx= . [2] x y y O 1.5 x 3 1 ( )2.5, 0.5− y x a=− + 7xa=

1.2 Graphs and Transformations 7 Answers Transformation of Curves 1(i)(a) or x a x b , (b) xh (ii) Maximum point at x = a, minimum point at x = b 2(i) x = a (Max), x = c (Min), x = e (Stationary point of inflexion)