RI C8B Curve Transformation Lect Notes

RAFFLES INSTITUTION H2 Mathematics (9758) 2025 Year 5 _____________________________ Chapter 8B: Curve Transformation Page 1 of 24 Chapter 8B: Curve Transformation SYLLABUS INCLUDES  Effect of transformations on the graph of f( )yx as represented by f( )ya x , f( ) ,yx a f( )yx a and f( )ya x , and combinations of these transformations  Relating the graphs of f( )yx , f( )yx , and 1f( )y x to the graph of f( )yx  Relating the graph of f( )yx  to the graph of f( )yx CONTENT 1 Standard Transformations 1.1 Translation 1.2 Scaling (Stretching) 1.3 Reflection about axes 1.4 Combination of standard transformations 2 More Transformations 2.1 Graphs of modulus functions 2.2 Graphs of derivative functions 2.3 Sketching the graph of 1f( )y x from the graph of f( )yx

Raffles Institution H2 Mathematics 2025 Year 5 ______________________________________________________________________________________________________ _____________________________ Chapter 8B: Curve Transformation Page 2 of 24 In mathematics, the study of functions and th eir graphical representa tions provides a powerful tool for understanding the dynamic relationships between variables. Transformations of graphs play a pivotal role in this exploration, allowing mathematicians to analyze how a function evolves and adapts when subjected to various modifica tions. These transformations encompass a range of operations such as translations, reflections, scaling, each imparting distinct changes to the original graph. Through the lens of transforma tions, mathematicians ga in insights into the behaviour of functions and unravel the intricate patterns that underlie their geometric alterations. 1 STANDARD TRANSFORMATIONS Consider the curve given by 12xy x   . We notice that the shape of the graph is similar to the graph of the reciprocal function 1f( ) .x x In fact, we can write 13 11 3 f 222xyx xx   . From the form  13 f 2yx  , we observe that there is a sequence of transformations that would transform the curve 1y x onto the curve 12xy x   . We will learn more about these in the following sections.

Raffles Institution H2 Mathematics 2025 Year 5 ______________________________________________________________________________________________________ _____________________________ Chapter 8B: Curve Transformation Page 3 of 24 1.1 Translation Translating a graph in the direction of an axis is to move the graph in the direction of the axis, without changing its shape or size. Consider 2f( )x x . (A) Compare the graphs of 2f( )yx x , 2f( 2 ) ( 2 )yx x   and 2f( 2 ) ( 2 )yx x  . f( )yx Transformed equation f( ) , 0yx a a  Replacement Description of transformation Remarks 2yx  2 2yx Replace x by 2x Translation of the graph 2yx by 2 units in the positive x-direction. NO change in y values  2 2yx Replace x by 2x Translation of the graph 2y x by 2 units in the negative x-direction. (B) Compare the graphs of 2f( )yx x , 2f( ) 2 2yx x   and 2f( ) 2 2yx x   . f( )yx Transformed equation f( ) , 0yx a a Replacement Description of transformation Remarks 2y x 2 2yx Replace y by 2y Translation of the graph 2yx by 2 units in the positive y- direction. NO change in x values 2 2yx Replace y by 2y Translation of the graph 2y x by 2 units in the negative y- direction.

Raffles Institution H2 Mathematics 2025 Year 5 ______________________________________________________________________________________________________ _____________________________ Chapter 8B: Curve Transformation Page 4 of 24 1.2 Scaling (Stretching) Scaling/stretching a curve parallel to one axis (with the other axis being invariant) changes the size of the curve. Consider f( ) s i nxx , 02 x  . (C) Compare the graphs of f( ) s i nyx x and f( 2 ) s i n2yx x . f( )yx Transformed equation f, 0ya x a Replacement Description of transformation Remarks sinyx sin 2yx Replace x by 2 1 2 xx  Scaling parallel to the x-axis of the graph of sinyx by a factor of 1 2 . NO change in y values (D) Compare the graphs of f( )yx and 2f ( ) 2sinyx x . f( )yx Transformed equation f, 0yaxa Replacement Description of transformation Remarks sinyx 2sinyx Replace y by 2 y Scaling parallel to the y-axis of the graph of sinyx by a factor of 2. NO change in x values O x y

Raffles Institution H2 Mathematics 2025 Year 5 ______________________________________________________________________________________________________ _____________________________ Chapter 8B: Curve Transformation Page 5 of 24 1.3 Reflection about axes Consider f( ) e xx  . (E) Compare the graphs of f( ) e xyx and f( ) e xyx  . f( )yx Transformed equation fyx Replacement Description of transformation Remarks exy e xy  Replace x by x Reflection of the graph of exy in the y-axis. NO change in y values (F) Compare the graphs of f( ) e xyx and f( ) e xyx  . f( )yx Transformed equation fyx Replacement Description of transformation Remarks exy exy Replace y by y Reflection of the graph of exy in the x-axis. NO change in x values 1 O 1 O 1 f( )y x f( )yx

Raffles Institution H2 Mathematics 2025 Year 5 ______________________________________________________________________________________________________ _____________________________ Chapter 8B: Curve Transformation Page 6 of 24 In general, the effects of tr ansformations on the graph of f( )yx are summarised in the table below. Note that a is taken to be a positive constant. Replacement of x / y Transformed Equation Desc ription of transformation Replace x by x a f( )yx a Translation of a units in the positive x- direction Replace x by x a f( )yx a Translation of a units in the negative x-direction Replace y by ya f( )ya x i.e. f ( )yx a Translation of a units in the positive y- direction Replace y by ya f( )ya x i.e. f ( )yx a Translation of a units in the negative y-direction Replace x by x a f xy a  Scaling parallel to the x-axis by factor of a (in terms of coordinates, no change in corresponding y-coordinate) Replace y by ya f( )y xa  i.e. f( )ya x Scaling parallel to the y-axis by factor of a (in terms of coordinates, no change in corresponding x-coordinate) Replace x by x f( )yx Reflection in the y-axis Replace y by y f( )yx i.e. f ( )yx Reflection in the x-axis

Raffles Institution H2 Mathematics 2025 Year 5 ______________________________________________________________________________________________________ _____________________________ Chapter 8B: Curve Transformation Page 7 of 24 Example 1 The diagram below shows the graph of f( )yx . The curve passes through the origin and has a maximum turning point at (1, 1). The line 0y  is a horizontal asymptote to the curve. Sketch, on separate diagrams, the graphs of (a) f( 2 )yx , (b) 2f( )yx , (c) f( )yx , labelling clearly any intersections with the axes, turning points and asymptotes. Solution (a) f( ) Replace f( 2 ) yx yx    by 2xx  (b) f( ) Replace 2f( ) yx yx    by 2yy (c) f( ) Replace f( ) yx yx    by xx  * Answ