RI C8B Curve Transformation Tut (Qn)

RAFFLES INSTITUTION H2 Mathematics (9758) 2025 Year 5 ______________________________ Tutorial 8B : Curve Transformation Page 1 of 7 Tutorial 8B: Curve Transformation Section A (Basic Questions) 1 The diagram above shows the graph of f( )yx= . The curve passes through the points (0, 1)A , (2, 2)B and (3, 0)C . Sketch on separate diagrams, the graphs of (i) f ( 1)yx= − , (ii) f( ) 1yx= − , (iii) f (2 )yx= , (iv) f 2xy =  (v) 3f ( )yx=− , (vi) f( )yx= , (vii) f( )yx= , showing in each case, the coordinates of points corresponding to A, B and C. x y O

Raffles Institution H2 Mathematics 2025 Year 5 __________________________________________________________________________________________ _____________________________ Tutorial 8B: Curve Transformation Page 2 of 7 x y 2 2 O 2 For each of the following graphs with equation y = f(x), sketch, on separate diagrams, the graphs of (i) ( )fyx ′= , (ii) ( ) 1 fy x= (a) The function f is a semi -circle with domain being [ −2, 2]. (0, 2) is a maximum turning point. (b) Domain of f is ( −∞, ∞). Maximum gradient occurs at x = 0 and 2y=± are horizontal asymptotes. (c) Domain of f is 1\ 2− . 11 and are asymptotes.22yx= =− x y O 2 y x O 1

Raffles Institution H2 Mathematics 2025 Year 5 __________________________________________________________________________________________ _____________________________ Tutorial 8B: Curve Transformation Page 3 of 7 Section B (Discussion Questions) Transformations 1 Sketch the graph of 222 50xy+= making clear its main relevant features. Describe fully a sequence of two transformations which would transform this curve onto the curve 2 2 ( 1)( 6 16) 0 2yxx −+−+ = . State the equation of one line of symmetry of this curve. [ 3 or 1xy= −= ] 2 The graph of ( f)yx= which passes through the origin has stationary points A(2, –2) and B(6, 2) as shown in the diagram below. The vertical and horizontal asymptotes of the graph are given by x = –2 and y = –4 respectively. Sketch the following graphs, labelling clearly the equations of asymptotes (if any) and stating the coordinates of points corresponding to A and B after the transformation. (a) f( ) 2yx= −+ , (b) f (3 2)yx= + , (c) 1 f (2 ).yx= − 3 (a) Describe a sequence of transformations which transforms the graph of 1y x= on to the graph of 211 xy x= −− . [3] (b) A curve L undergoes the following transformations: A: translation of 2 units in the negative x-direction, B : scaling parallel to the y-axis by a factor of 3, C: translation of 4 units in the positive y-direction. The equation of the resulting curve is 229 4 8 40 0xyy− +−= . Determine the equation of the curve L. [4] [( ) ( ) 2 22 114x y− −+= or 22 4 4 8 40xx yy−− −− = ] y = −4 A(2, −2) x y B(6, 2) x = −2

Raffles Institution H2 Mathematics 2025 Year 5 __________________________________________________________________________________________ _____________________________ Tutorial 8B: Curve Transformation Page 4 of 7 Graphs of Modulus Functions 4 The diagram shows the graph of f(2 )yx= − . The curve passes through the origin, and has asymptotes 1x=− , 2x= and 0y= . Sketch, on separate diagrams, the graphs of (i) f ( 2)yx= + , (ii) f( )yx= , ( iii) ( )f3yx= , (iv) ( )f1yx= + . 5 The sketches below show the graphs of f( )yx= and ( )fyx= for a certain function f which is a polynomial of degree four. Sketch the graph of f( )yx= , showing clearly the co-ordinates of turning point(s) and intersections with the axes. y = |f(x)| y O (4, 0) (3, 27) x y = f(|x|) (−4, 0) (4, 0) (3, 27) x y O (−3, 27) x y O

Raffles Institution H2 Mathematics 2025 Year 5 __________________________________________________________________________________________ _____________________________ Tutorial 8B: Curve Transformation Page 5 of 7 x y O Graphs of Derivative Functions & Reciprocal Functions 6 For each of the functions whose graphs are shown below, sketch on separate diagrams, the graphs of (i) the derivative functions, (ii) the reciprocal functions. [The equations of any asymptotes should be stated, together with the coordinates of any stationary point(s).] (a) (b) (c) (d) (e) O x y 4 O y (3, 1) (5, 6) x O y x O x y 0x= 0y= 0x= 0y= f( )yx=

Raffles Institution H2 Mathematics 2025 Year 5 __________________________________________________________________________________________ _____________________________ Tutorial 8B: Curve Transformation Page 6 of 7 7 (a) The diagram shows the graph of ( )gyx= with a stationary point of inflexion at ( )0,3 and a maximum point at ( )4,5 . The asymptotes of the graph are 2x= and 3.y= On a separate diagram, sketch the graph of g( )yx ′= , labelling the equations of any asymptotes and the coordinates of any points where the curve crosses the axes. [3] (b) The diagram shows the graphs of ( )hyx= and ( )1hyq xp= +−− , where p and q are constants. The curve ( )hyx= cuts the axes at ( )1, 0− and has asymptotes 0x= and 2y=− . The curve ( ) 1 hyq xp= +−− has asymptotes 2x=− and 1y= . Find the values of p and q. [3] [ 3, 1.5pq= = ] x y O O x y x y O

Raffles Institution H2 Mathematics 2025 Year 5 __________________________________________________________________________________________ _____________________________ Tutorial 8B: Curve Transformation Page 7 of 7 8 The diagram below shows the graph of f( )yx= with asymptotes 2, 3xx= −= and 1yx= + . The curve intersects the x -axis at points B and D, and has turning points at points A, B, C and E. The coordinates of A , B , C , D and E are ( )3, 3−− , ( )1, 0− , 33,,24 ( )2, 0 and ( )5,8 respectively. (i) By showing clearly the equations of asymptotes and the coordinates of any turning points and the points where the curve crosses the axes, where possible, sketch, on separate diagrams, the graphs of (a) f( )yx= , [4] (b) ( ) 1 fy x= . [4] (ii) By drawing another suitable graph on the same diagram in part (i)(b), determine the number of solutions to the equation ( ) 2 2 1 136 16 fx x+=  . [2] B C D O y x 1yx= + A E