RI C8B Curve Transformation Add Prac (Soln)

RAFFLES INSTITUTION H2 Mathematics (9758) 2025 Year 5 ________________________________________________________ Additional Practice Questions for Chapter 8B: Curve Transformation Page 1 of 27 Additional Practice Questions for Chapter 8B: Curve Transformation Transformations 1 The diagram shows the graph of f( )yx . The curve passes through the points (1 , 1 )A  and (1, 0)B , and has a maximum point at (2,1)C . The curve has asymptotes 0x and 0y . Sketch, on separate diagrams, the graphs of (i) f( 2 1 )yx , (ii) f( 1 )yx  , (iii) 2f (2 ) 1yx . In each case, state the coordinates of the points corresponding to A, B, and C, and the equations of the asymptotes, whenever applicable. [9] (i) (ii) x O y x y 0 0y  x O y

Raffles Institution H2 Mathematics 2025 Year 5 __________________________________________________________________________________________ _______________________________________________________ Additional Practice Questions for Chapter 8B: Curve Transformation Page 2 of 27 (iii) 2 State precisely how each of the following curves may be obtained by applying a sequence of geometrical transformations from the given starting curve. (a) 23exy  (starting curve : exy  ), (b) 12 1yx  (starting curve : yx ). [7] (a) exy  Replace x with 2x 2exy  Replace y with 3y 23exy  Translate the graph of exy  by 2 units in the negative x-direction to obtain the graph of 2exy  . Scale the graph of 2exy  parallel to the y-axis by a factor of 3 to obtain the graph of 23exy  . y x 0

Raffles Institution H2 Mathematics 2025 Year 5 __________________________________________________________________________________________ _______________________________________________________ Additional Practice Questions for Chapter 8B: Curve Transformation Page 3 of 27 (b) yx Replace x with 1x 1yx Replace x with x 1yx Translate the graph of yx by 1 unit in the negative x-direction to obtain the graph of 1yx . Reflect the graph of 1yx in the y- axis to obtain the graph of 1yx . Replace y with y 1yx  Replace y with 2y 21yx  Reflect the graph of 1yx in the x-axis to obtain the graph of 1yx  . Scale the graph of 1yx  parallel to the y-axis by a factor of 2 to obtain the graph of 21yx  Replace y with 1y 1yx  Alternative for the first 2 transformations for (b): (1 ) 1yx y x y x x         Description: Reflect the graph of yx in the y-axis to obtain the graph of yx , followed by translation of 1 unit in the positive x-direction to obtain the graph of (1 ) 1yx x  . Alternative for the last 3 transformations for (b): Translate the graph of 1yx by 12 units in the negative y direction to obtain the graph of 1 12yx   , followed by reflection in the x-axis to obtain 1 12yx and a scaling by factor 2 parallel to the y-axis to obtain 121yx  121yx  Translate the graph of 21yx  by 1 unit in the positive y-direction.

Raffles Institution H2 Mathematics 2025 Year 5 __________________________________________________________________________________________ _______________________________________________________ Additional Practice Questions for Chapter 8B: Curve Transformation Page 4 of 27 3 The transformations , a n d A BC are as follows: :A A translation of 1 unit in the negative y -direction; :B A stretching parallel to the y -axis (with x-axis invariant) by a factor of 2; :C A reflection about the x-axis. A curve undergoes in succession, the transformations , a n d A BC , and the equation of the resulting curve is 226yx  . (i) Determine the equation of the curve before the transformations were effected. [3] Suppose the curve undergoes in succession, th e above transformations in the reverse order (i.e. , a n d CB A ). (ii) Determine the equation of the resulting curve. [3] (iii) Suggest one amendment to the above tran sformations for the reverse order such that the resulting equation is still 226yx  . [1] (i)  22 just before just before 2 2 22 just before 2626 1226 26 1 26 32 CB A y xyxyx yx yxx            2 2 13 4 yx yx    Thus, the equation of the curve before the transformations were effected is 2 4yx . Alternatively, f ( ) 1f ( ) 1f ( ) 1f ( ) , i . e . 2 [ 1 f ( ) ]22yyAB Cyx y x x x y x               So 22 22 6 2 [ 1 f( ) ] 1 f( ) 3 f( ) 4xx x x x x          Thus, the equation of the curve before the transformations were effected is 2 4yx . (ii) 22 2 2 2 14 4 4 4 , i . e . 2 9 22yyCBAyx yx x x y x                  The equation of the resulting curve is 229yx  . (iii) Amend A to a translation of 2 units in the positive y -direction.

Raffles Institution H2 Mathematics 2025 Year 5 __________________________________________________________________________________________ _______________________________________________________ Additional Practice Questions for Chapter 8B: Curve Transformation Page 5 of 27 4 9758/2021/01/Q6 A curve C has equation 2 1 , 4 y ax x   where a > 0. (a) Sketch C and give the equations of any asymptotes, in terms of a where appropriate. [4] (b) Find the smallest possible value of y in terms of a. [1] (c) Describe the transformatio n that maps the graph of C onto the graph of 22 1 . 4 y ax   [3] (a) 2 11 (4 )4 y x axax x   , (b) When 2,xa 12y a . The smallest possible value of y is 1 2y a . 2 2 22 22 22 1(c) 4 1 (4 ) 1 (2 ) ( 2 ) 1 4( 2 ) Replace by ( 2 ) 1 4 y ax x xa x xa a ax a xx a y ax             Translate the graph of C 2a units in the negative x-direction. O y x

Raffles Institution H2 Mathematics 2025 Year 5 __________________________________________________________________________________________ _______________________________________________________ Additional Practice Questions for Chapter 8B: Curve Transformation Page 6 of 27 Transformations involving Modulus Functions 5 The graph of f2 1yx is shown. Sketch on separate diagrams, the graphs of (i) 2f 2yx , (ii) fyx , (iii) fyx . Your sketch should show clearly the coordinates corresponding to A , B and C , and the equations of asymptotes (if any). [8] (i) (ii) (iii) x A (2, 2) y B (4, 0) y = –1 y C x y y = 0 O    x y y = 0 O    x y y = 0 O     

Raffles Institution H2 Mathematics 2025 Year 5 __________________________________________________________________________________________ _______________________________________________________ Additional Practice Questions for Chapter 8B: Curve Transformation Page 7 of 27 6 The diagram shows the graph of f( )yx . Sketch, on separate diagrams, the graphs of (i) f( )yx , (ii) f( )yx , (iii) f1yx , (iv) f1 2yx . [12] (i) (ii) (iii) (iv) x y 1 3 2 f( )yx f( )yx

Raffles Institution H2 Mathematics 2025 Year 5 __________________________________________________________________________________________ _______________________________________________________ Addition