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 th