TMJC H2 Chapter 2 Transformation of Curves Discussion Questions (Suggested Solutions)

Chapter 2 Transformations of Curves TMJC 2024 Page 1 of 14 H2 Mathematics (9758) Chapter 2 Transformations of Curves Discussion Questions (Suggested Solutions) Level 1 1 Describe a single transformation that will change ( )fyx= into each of the following functions in part (a) – (f). The diagram shows the graph of ( )fyx= . On separate diagrams, sketch the graphs of the following, showing clearly the coordinates of intersection with the axes. (a) ( )f1yx=− (b) ( )f2yx=− (c) ( )2fyx= (d) f 3 xy =  (e) ( )fyx=− (f) ( )fyx=− (g) ( )fyx= O y x 1 3 (–2, –1) (2, –1) 2

Chapter 2 Transformations of Curves TMJC 2024 Page 2 of 14 1 Solution (a) ( )f1yx=− Transformation: Translation of 1 unit in the negative y-direction. (b) ( )f2yx=− Transformation: Translation of 2 units in the positive x-direction. (c) ( )2fyx= Transformation: Stretch of factor 2 parallel to the y-axis. (d) f 3 xy =  Transformation: Stretch of factor 3 parallel to the x-axis. O y x (–2, –2) (2, –2) (0,1) (2, 2) O (0, –1) (4, –1) y x (3,0) (5,0) O y x (1,0) (3,0) (–2, –2) (2, –2) (0,4) O y x (3,0) (9,0) (–6, –1) (6, –1) (0,2) Replace x by x − 2 x-coordinates increases by 2, y-coordinates remain unchanged Replace y by 2 y x-coordinates remain unchanged, y-coordinates multiplied by 2 Replace y by y + 1 x-coordinates remain unchanged, y-coordinates decreases by 1 Replace x by 3 x x-coordinates multiplied by 3, y-coordinates remain unchanged

Chapter 2 Transformations of Curves TMJC 2024 Page 3 of 14 (e) ( )fyx=− Transformation: Reflection in the x-axis. (f) ( )fyx=− Transformation: Reflection in the y-axis. (f) ( )fyx= O y x (1,0) (3,0) (–2, –1) (2, –1) (0,2) Replace x by x− x-coordinates multiplied by 1− , y-coordinates remain unchanged Replace y by y− x-coordinates remain unchanged, y-coordinates multiplied by 1− y x (–3,0) (–1,0) (1,0) (3,0) O (–2, 1) (2, 1) (0,–2) y x (–3,0) (1,0) (3,0) O (–2, 1) (2, 1) (0,2) (–1,0)

Chapter 2 Transformations of Curves TMJC 2024 Page 4 of 14 Level 2 2 The diagram shows the graph of ( )fyx= with asymptotes 0x= and yx= . The points A, B, O have coordinates ( )1, 2−− , ( )1, 2 , ( )0, 0 respectively. Sketch on separate diagrams the graph of (i) 13f 2yx =  , (ii) ( )2f 4yx=+ , (iii ) ( )f 2 1yx=+ , showing in each case the coordinates of the points corresponding to A, B and O and the equations of the asymptotes where applicable. 2 Suggested Solutions (i) 3f 2 xy =  (ii) ( )2f 4yx=+ (iii) ( )f 2 1yx=+ Note: For (ii), y-intercept of asymptote y = 2x +8 is (0, 8) and should be higher than turning point (3, −4) in your sketch Replace x by 𝑥 2 Replace y by 𝑦 3 A2 (−2,−6) x = 0 y x B2 (2,6) O2 (0,0) y y = 2x + 8 x A3 (−5, −4) B3 (−3,4) O3 (−4,0) x y = 2x + 1 y y A O x B

Chapter 2 Transformations of Curves TMJC 2024 Page 5 of 14 3 The diagram shows the graph of f ( )yx= . On separate diagrams, sketch the graphs of (i) f (2 1) 1yx= + + , (ii) ( )f3 y x=− , labelling each graph clearly, showing the asymptotes (if any) and the coordinates of the points corresponding to A and B. 3 Suggested Solutions (i) Method 1: Translation followed by stretch for x-transformations Method 2: Stretch followed by translation for x-transformations x y y = 1 O 3) Replace y by y − 1 x = −3 x y y = 0 O 1) Replace x by x + 1 x = −1 x y y = 0 O 1) Replace x by 2x y = 0 y O B( 2, −1 ) A(0, 0) −2 x y = f (x) x y y = 0 O 2) Replace x by 2x x y y = 0 O 2) Replace x by

Chapter 2 Transformations of Curves TMJC 2024 Page 6 of 14 (ii) ( ) ( )ff33 yy xx=−  − = x y y = 1 O 3) Replace y by y − 1 x y 2) Replace y by –y 3) Replace y by x y 1) Replace x by |x| f (2 1) 1yx= + +

Chapter 2 Transformations of Curves TMJC 2024 Page 7 of 14 4 The diagram shows the graph of ( )fyx= . The curve crosses the x-axis at the origin O and the point ( )2,0A , and has a maximum point at ( )1, 2B . Sketch, on separate diagrams, the graphs of (i) ( )fyx= , (ii) ( )fyx=− , (iii) ( ) 1 fy x= , indicating in each case the coordinates of the axial intercepts and turning points, and the equation of the asymptotes where applicable. 4 Suggested Solutions (i) ( )fyx= (ii) ( )fyx=− (iii) ( ) 1 fy x= Note for (ii) ( )fyx=− , we have to reflect in the y-axis first before we apply the transformation for |x|: ( )fyx= → ( )fyx=− → Incorrect order if you apply the transformation for |x| followed by a reflection in the y-axis: ( )fyx= → ( )fyx= → ( ) ( )ffy x x= − = y = 3 x O y B1(1,2) A1(2,0) O1(0,0) O x y y = −3 O2(0,0) y x y = − B3(1, ½ ) x = 2 x = 0 y = 0 x y B(1,2) y = −3 A(2,0) O O

Chapter 2 Transformations of Curves TMJC 2024 Page 8 of 14 5 2016/Specimen Paper/I/2 The curve C with equation 3yx= is transformed onto the curve with equation ( )fyx= by a translation of 2 units in the negative x-direction, followed by a stretch of factor 1 2 parallel to the y-axis, followed by a translation of 1 unit in the positive y-direction. (i) Write down the equation of the new curve. [1] (ii) Sketch C and the curve with equation ( )fyx= on the same diagram, stating the exact values of the coordinates of the points where ( )fyx= crosses the x- and y- axes. Find the x-coordinate(s) of the point(s) where the two curves intersect, giving your answer(s) correct to 3 decimal places. [4] 5 Suggested Solutions (i) 3yx= Replace x by 2x+ ( ) 3 2yx=+ Replace y by 21 2 y y= ( ) ( ) 33 12 2 2 2y x y x= +  = + Replace y by 1y− ( ) ( ) 33111 2 2 122y x y x− = +  = + +  The equation of the curve is ( ) 31 212yx= + + (ii) x-coordinate = 7.722 (3 d.p.) x y (0,0) Have to show that the 2 graphs intersect at a point Read the question ‘exact coordinates’ so you cannot label as (−3.26, 0)

Chapter 2 Transformations of Curves TMJC 2024 Page 9 of 14 6 A curve ( )fyx= undergoes, in succession, the following transformations. A: A translation of magnitude 2 units in the positive x-direction. B: A stretching parallel to the x-axis by a factor 1 2 . C: A reflection in the y-axis. D: A translation of magnitude 1 unit in the positive y-direction. The resulting curve has equation 2( 1)1e xy +=+ . Determine the equation of the curve before the four transformations were effected. 6 Suggested Solutions 2( 1)1e xy +=+ ':D Replace y by y + 1 ( ) ( )2 1 2 1 1 1 e e xx yy ++ + = +  = ':C Replace x by x− ( )21 e x y −+ = ':B Replace x by 2 x 21 22ee x xyy −+ −+=  = ':A Replace x by 2x+ e xy −=  The equation of the curve is e xy −= . Learning points 1. Question asks to determine the equation of the curve before the transformations, thus start from the last transformation and undo each in order (i.e. D′→C ′→B′→A′) 2. If D: Replace y by 1y− To get ':D Replace y by y + 1 3. Check your final answer by working forward using the transformations given to see if you are able to get the resulting curve A: Replace x by 2x− B: Replace x by 1 2 x  C: Replace x by x− D: Replace y by 1y−

Chapter 2 Transformations of Curves TMJC 2024 Page 10 of 14 7 2013/SRJC Prelim/I/3 (M