TMJC H2 Chapter 2 Transformation of Curves Learning Package 2024

Tampines Meridian Junior College 2024 H2 Mathematics (9758) Chapter 2 Transformation of Curves Learning Package Resources  Core Concept Notes  Discussion Questions  Extra Practice Questions SLS Resources  Recordings on Core Concepts  SLS Activity for Learning Experience (Exploring the three Basic Transformations)  Quick Concept Check

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Chapter 2 Transformation of Curves TMJC 2024 Page 1 H2 Mathematics (9758) Chapter 2 Transformation of Curves Core Concept Notes Success Criteria: Surface Learning Deep Learning Transfer Learning  Identify the replacement of variable involved in translation of a graph (i.e. ( )fy x a=+ & ( )fy x a=+ )  Draw the graph after applying translation (i.e. ( )fy x a=+ & ( )fy x a=+ ) and label the corresponding characteristics such as asymptotes, turning points and intersections with the axes after translation.  Identify the replacement of variable involved in stretching of a graph (i.e. ( )fyax= & f xy a =  )  Draw the graph after applying stretching (i.e. ( )fyax= & f xy a =  ) and label the corresponding characteristics after stretching.  Identify the replacement of variable involved in reflecting a graph either in the x-axis or y-axis (i.e. f ( )yx=− & f ( )yx=− ).  Draw the graph after applying reflection (i.e. f ( )yx=− & f ( )yx=− ). and label the corresponding characteristics after reflection.  Draw the graph of f ( )yx= and label the characteristics after transformation.  Identify the correct sequence of transformations for graphs with 2 or more transformations.  Draw the graph with 2 or more transformations and label the corresponding characteristics after transformations.  Describe a sequence of transformations given the original and resulting equations.  Determine the resulting equation of graph given a sequence of transformations.  Draw the graph of ( )fyx= and label the characteristics after transformation.  Draw the graph of 1 f ( )y x= and label the corresponding characteristics such as asymptotes, turning points and intersections with axes after transformation. Definition of Learning (John Hattie): The process of developing sufficient surface knowledge to then move to deeper understanding such that one can appropriately transfer this learning to new tasks and situations.

Chapter 2 Transformation of Curves TMJC 2024 Page 2 Basic Transformations §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. 1.1 Translation in the y-direction Let 2f ( )y x x== . Using GC, sketch the graphs of f ( )y x a=+ for 2a= and 3a=− . How does the graph of 2f ( )y x x== change with different values of a? f ( )yx= Transformed equation f ( )y x a=+ Replacement Geometrical description of transformation 2yx= 2 2yx=+ Replace y by 2y− Translation of the graph 2yx= by 2 units in the positive y-direction. i.e. y is replaced by ( y − 2), thus we add 2 to all the original y-coordinates 2 3yx=− Replace y by ( )33yy− − = + Translation of the graph 2yx= by 3 units in the negative y-direction. i.e. y is replaced by ( y – (−3)), thus we add −3 to all the original y-coordinates. To get ( )fy x a=+ from the graph of ( )fyx= , where 0a , • Replace y by ya− • ( ) ( ) ( )f f fy x y a x y x a= ⎯⎯ → − =  = + • The graph of f ( )y x a=+ is a translation of the graph of f ( )yx= by a units in the positive y-direction. • ( )( ) ( )( ), f , fi i i ix x x x a ⎯⎯ → + To get ( )fy x a=− from the graph of ( )fyx= , where 0a , • Replace y by ( )y a y a− − = + • ( ) ( ) ( )f f fy x y a x y x a= ⎯⎯ → + =  = − • The graph of f ( )y x a=− is a translation of the graph of f ( )yx= by a units in the negative y-direction. • ( )( ) ( )( ), f , fi i i ix x x x a ⎯⎯ → − y x (0,0) O y x (0,2) O y x (0,–3)

Chapter 2 Transformation of Curves TMJC 2024 Page 3 Example 1 The graph of ( )fyx= is shown. Sketch the graph of ( )f1yx=− . Solution: ( )f1yx=− Replace y by 1y+ x = 1 O y = f(x) y = 0 C (2, −2) y x A1 x = 1 y x O −1 y = f(x) −1 y = −1 Notice that all y-values have decreased by 1 unit, i.e. minus 1 to all y-values.

Chapter 2 Transformation of Curves TMJC 2024 Page 4 1.2 Translation in the x-direction Let 2f ( )y x x== . Using GC, sketch the graphs of ( )fy x a=+ for 1a=− and 2a= . How does the graph of 2f ( )y x x== change with different values of a? f ( )yx= Transformed equation f ( )y x a=+ Replacement Geometrical description of transformation 2yx= ( ) 2 1yx=− Replace x by 1x− Translation of the graph 2yx= by 1 unit in the positive x-direction. i.e. x is replaced by (x −1), thus we add  to all the original x-coordinates. ( ) 2 2yx=+ Replace x by ( )22xx− − = + Translation of the graph 2yx= by 2 units in the negative x-direction. i.e. x is replaced by (x + 2), thus we add −2 to all the original x-coordinates. To get ( )fy x a=− from the graph of ( )fyx= , where 0a , • Replace x by xa− • ( ) ( )ffy x y x a= ⎯⎯ → = − • The graph of f ( )y x a=− is a translation of the graph f ( )yx= by a units in the positive x-direction. • ( )( ) ( )( ), f , fi i i ix x x a x →+ To get ( )fy x a=+ from the graph of ( )fyx= , where 0a , • Replace x by ( )x a x a− − = + • ( ) ( )ffy x y x a= ⎯⎯ → = + • The graph of f ( )y x a=+ is a translation of the graph f ( )yx= by a units in the negative x-direction. • ( )( ) ( )( ), f , fi i i ix x x a x ⎯⎯ → − y x (0,0) O y x (1,0) O y x (–2,0)

Chapter 2 Transformation of Curves TMJC 2024 Page 5 Example 2 The graph of ( )fyx= is shown. Sketch the graph of f ( 2)yx=− . Solution: f ( 2)yx=− Replace x by 2x− x = 1 O y = f(x) y = 0 C (2, −2) y x A1 y x y = f(x−2) O x = 3 y = 0 ( )2 2,1A Notice that all x-values have increased by 2 unit, i.e. add 2 to all x-values.

Chapter 2 Transformation of Curves TMJC 2024 Page 6 §2 Stretch/Scale 2.1 Stretch parallel to the y-axis Let ( )f siny x x== . Using GC, sketch the graphs of ( )fyax= for 3a= and 1 2a= . How does the graph of f ( ) siny x x== change with different values of a ? f ( )yx= Transformed equation ( )f , 0y a x a= Replacement Geometrical description of transformation sinyx= 3sinyx= Replace y by 3 y Stretch the graph of sinyx= by factor 3 parallel to the y-axis. i.e. replace y by 3 y , thus we multiply  to all the original y-coordinates. 1 sin2yx= Replace y by 1 2 y  Stretch the graph sinyx= by factor 1 2 parallel to the y-axis. i.e. replace y by 1 2 y  , thus we multiply 1 2 to all the original y-coordinates. To get ( )fyax= from the graph of ( )fyx= , where 0a , • Replace y by y a • ( ) ( ) ( )f f f yy x x y a x a= ⎯⎯ → =  = • The graph of ( )fyax= is a stretch of the graph ( )fyx= by factor a parallel to the y-axis. • ( )( ) ( )( ), f , fi i i ix x x a x ⎯⎯ → 3𝜋 2ൗ 𝜋 2𝜋 y O x 𝜋 2ൗ 1 –1 y O x 3 –3 y O x

Chapter 2 Transformation of Curves TMJC 2024 Page 7 2.2 Stretch parallel to the x-axis Let f ( ) siny x x== . Using GC, sketch the graphs of f xy a =  for 1 2a= and 3a= . How does the graph of f ( ) siny x x== change with different values of a? f ( )yx= Transformed equation f , 0xya a = Replacement Geometrical description of transformation sinyx= sin 2yx= Replace x by 1 2 x  Stretch the graph sinyx= by factor 1 2 parallel to the x-axis. i.e replace x by (