2024 JPJC Chapter 5 Graphing Techniques II

Jurong Pioneer Junior College Mathematics (9758) JC 1 - 2023 Chapter 5 Graphing Techniques II (Students’ Version) / Pg 1 Content outline :  Effect of transformations on the graph of y = f(x) as represented by y = af(x), y = f(x)  a,  fy x a  and y = f(ax), and combinations of these transformations  Relating the graphs of  fy x ,  fy x , and   1 fy x to the graph of  fy x References  http://www.h2maths.site [This website offers Applets which allow self-exploration of numerous types of graphs covered in this topic. Refer to "Transformation of Graphs" under the link "Graphs and Functions" in this website]  H2 Mathematics For 'A' Level, Federick Ho, David Khor, Yui-P'ng Lam, B.S. Ong Volume 1, Call No 510.76 HO  Mathematics The Core Course for A-level, L Bostock and Chandler, Call No 510 BOS  MEI Structured Mathematics (2nd Edition), Pure Mathematics 4, Terry Heard and David Martin, Call No 510 HEA  Pure Mathematics 4, Hugh Neill and Douglas Quadling, Call No 510 NEI Prerequisites  Coordinate geometry from Add Maths  Using GC to draw simple curves from Chapter 2 Graphing Techniques I  Functions learnt in Chapter 4  O Level Trigonometry Chapter 5: Graphing Techniques II

Jurong Pioneer Junior College Mathematics (9758) JC 1 - 2023 Chapter 5 Graphing Techniques II (Students’ Version) / Pg 2 1.1 Translations (I) Graphs of the form y = f(x) + a Example 1 The diagram below shows the graphs of 2y x , 2 5y x  and 2 3y x  on the same axes. How are the graphs of 2 5y x  and 2 3y x  related to the graph of 2y x ? Solution: The graph of 2 5y x  is obtained by translating the graph of 2y x ____________ ______________________________________ The graph of 2 3y x  is obtained by translating the graph of 2y x ____________ ______________________________________ This can be generalized as follows: For a > 0, the graph of y = f( x) + a is a translation of the graph of y = f( x) by a units in the positive y direction, the graph of y = f(x)  a is a translation of the graph of y = f(x) by a units in the negative y direction. Note: This transformation affects the y-coordinates only. The x-coordinates remain unchanged. Example 2: Given the graph of y = f(x) below, where f(x) = 2 4 2 x , sketch, on separate diagrams, the graphs of (i) y = f(x) + 1, (ii) y = f(x) – 2. y x O 2y x 2 5y x  2 3y x  5 –3 2 O x y Students may use the GC to check the accuracy of their answers. However, this may take a longer time.

Jurong Pioneer Junior College Mathematics (9758) JC 1 - 2023 Chapter 5 Graphing Techniques II (Students’ Version) / Pg 3 Solution: (II) Graphs of the form y = f(x  a) Example 3 The diagram below shows the graphs of 2y x , 2( 3)y x  and 2( 4)y x  on the same axes. How are the graphs of 2( 3)y x  and 2( 4)y x  related to the graph of 2y x ? Solution: This can be generalized as follows: For a > 0, the graph of y = f(x  a) is obtained by a translation of the graph of y = f(x) by a units in the positive x direction. the graph of y = f(x + a) is obtained by a translation of the graph of y = f(x) by a units in the negative x direction. Note: (I) The formal descriptive word to use is “translation”. Avoid using other vague descriptive words such as “shift” or “move”. (II) This transformation affects the x-coordinates only. The y-coordinates remain unchanged. The graph of 2( 3)y x  is obtained by translating the curve of 2y x by _________________________________ The graph of 2( 4)y x  is obtained by translating the curve of 2y x by _________________________________ x O 2y x 2( 4)y x  2( 3)y x  y –3 4 3 O x y y = f(x) + 1 1y (i) O x y y = f(x) 2 2y (ii)

Jurong Pioneer Junior College Mathematics (9758) JC 1 - 2023 Chapter 5 Graphing Techniques II (Students’ Version) / Pg 4 Example 4: The diagram shows the graph of f ( )y x . On separate diagrams, sketch the graphs of (i) f ( 2)y x  , (ii) f ( 1)y x  . Solution: (i) (ii) Check: Have you ensured that key features such as the given points and the asymptotes in the ( ) fy x graph are indicated in your answers? 2 x f ( 1)y x   0,1 y 2y 1x 1 x f ( 2)y x   3,1 y 2y 2x 1 x f ( )y x  1,1 y 2y 0x O

Jurong Pioneer Junior College Mathematics (9758) JC 1 - 2023 Chapter 5 Graphing Techniques II (Students’ Version) / Pg 5 1.2 Scaling (I) Graphs of the form y = a f(x), a > 0 (Scaling parallel to the y-axis) If f(x) is multiplied by a positive constant a where a > 1, the geometric effect is to “stretch" the graph of y = f(x) parallel to the y-axis. If f( x) is multiplied by a positive constant a where 0 1 a  , the geometric effect is to “compress” the graph of y = f(x) parallel to the y-axis. Example 5: The diagram below shows the graphs of siny x , 3siny x and 1 2 siny x , for 0 2 x   , on the same axes. How are the graphs of 3siny x and 1 2 siny x related to the graph of siny x ? Solution: In general, For a > 0, the graph of y = a f(x) is obtained from the graph of y = f(x) by a scaling parallel to the y-axis by a scale factor of a. i.e. “stretch” the original curve ( fy x ) away from the x-axis in both directions (a > 1) or “compress” the original curve ( fy x ) towards the x-axis in both directions. (0 < a < 1) Note: (1) Scale factor has no units. Do not write “scale factor of k units”. (2) The x-coordinates of all the points remain unchanged. The graph of 3siny x is obtained by scaling the curve siny x _____________ ___________________________________ The graph of 1 2 siny x is obtained by scaling the curve siny x ______________ ____________________________________ __ x O siny x 3siny x 1 sin2y x 3 - 1 - 1 2 - –1 –1 2 - - y –3 -

Jurong Pioneer Junior College Mathematics (9758)