2. f(x) + a 0 2 4 6 8 x y = f(x) -2-4 -6 2 4 6 -2 -4 -6 Graphs of Related Functions (1) f(x) = x 2 f(x) +2= x 2 + 2 f(x) - 5 = x 2 - 5 Vertical Translations In general f(x) + a gives a translation by the vector

3. 0 2 4 6 8 x y = f(x) -2-4 -6 2 4 6 -2 -4 -6 Graphs of Related Functions (1) Vertical Translations In general f(x) + a gives a translation by the vector f(x) f(x) + 3 f(x) - 2

4. 0 2 4 6 8 x y = f(x) -2-4 -6 2 4 6 -2 -4 -6 Graphs of Related Functions (2) Horizontal Translations f(x) f(x - 5) f(x + 2) In general f(x + a) gives a translation by the vector 52 In other words, ‘+’ inside the brackets means move to the LEFT

5. 0 2 4 6 8 x y = f(x) -2-4 -6 2 4 6 -2 -4 -6 Graphs of Related Functions (2) Horizontal Translations In general f(x + a) gives a translation by the vector f(x) f(?) f(x - 5) 5 f(?)f(x + 3) 3

6. Worksheet 1 f(x) x y = f(x) Grid 1: Sketch or trace (a) f(x) - 4 (b) f(x + 4) (c) f(x - 3) Grid 2: Sketch or trace (a) f(x + 4) - 2 (b) f(x - 3) + 1 (c) f(x - 3) - 5 y = f(x) x Grid 3: Sketch or trace (a) f(x) + 2 (b) f(x - 3) - 4 (c) f(x + 3) + 3 Grid 4: Sketch or trace (a) f(x) + 3 (b) f(x + 7) + 2 (c) f(x - 3) - 2 x x y = f(x) 3 4 1 2 f(x) Worksheet 1

7. Worksheet 1 Answers f(x) x y = f(x) Grid 1: Sketch or trace (a) f(x) - 4 (b) f(x + 4) (c) f(x - 3) Grid 2: Sketch or trace (a) f(x + 4) - 2 (b) f(x - 3) + 1 (c) f(x - 3) - 5 y = f(x) x Grid 3: Sketch or trace (a) f(x) + 2 (b) f(x - 3) - 4 (c) f(x + 3) + 3 Grid 4: Sketch or trace (a) f(x) + 3 (b) f(x + 7) + 2 (c) f(x - 3) - 2 x x y = f(x) 3 4 1 2 f(x) Worksheet 1

8. -f(x) 0 2 4 6 8 x y = f(x) -2-4 -6 2 4 6 -2 -4 -6 Graphs of Related Functions (4) f(x) = x 2 Reflections in the x axis -f(x) = -x 2 f(x) = x 2 - 10x + 25 -f(x) = -x 2 + 10x - 25 The graph of -f(x) is a reflection of f(x) in the x axis.

9. 0 2 4 6 8 x y = f(x) -2-4 -6 2 4 6 -2 -4 -6 Graphs of Related Functions (4) f(x) = x 2 + 1 Reflections in the x axis f(x) = x 2 - 10x + 23 -f(x) = -x 2 + 10x - 23 The graph of -f(x) is a reflection of f(x) in the x axis. -f(x) = -(x 2 + 1) = -x 2 - 1

10. 2 4 6 8 x y = f(x) -2-4 -6 10 20 30 -10 -20 -30 0 Reflections in the x axis The graph of -f(x) is a reflection of f(x) in the x axis. f(x) = x 3 - 3x 2 - 6x + 8 -f(x) = -x 3 + 3x 2 + 6x - 8 Graphs of Related Functions (4)

11. 2 4 6 8 x y = f(x) -2-4 -6 10 20 30 -10 -20 -30 0 Reflections in the x axis The graph of -f(x) is a reflection of f(x) in the x axis. f(x) -f(x) Graphs of Related Functions (4)

12. 90 180 x y = f(x) 0 270 360 -90 -180 -270 -360 1 2 -2 Reflections in the x axis The graph of -f(x) is a reflection of f(x) in the x axis. f(x) = Sinx -f(x) = -Sinx Graphs of Related Functions (4)

13. 90 180 x y = f(x) 0 270 360 -90 -180 -270 -360 1 2 -2 Reflections in the x axis The graph of -f(x) is a reflection of f(x) in the x axis. f(x) = 2Sinx -f(x) = -2Sinx Graphs of Related Functions (4)

14. Worksheet 2 f(x) x y = f(x) Draw the graph of -f(x) for each case on the grids below. y = f(x) x x x 3 4 1 2 f(x) Worksheet 2

15. Worksheet 2 Answers f(x) x y = f(x) Draw the graph of -f(x) for each case on the grids below. y = f(x) x x x 3 4 1 2 f(x) Worksheet 2

16. f(-x) 0 2 4 6 8 x y = f(x) -2-4 -6 2 4 6 -2 -4 -6 Graphs of Related Functions (5) f(x) = x 2 + 4x + 5 Reflections in the y axis f(x) = x 2 - 4x + 5 f(-x) f(-x) = (- x) 2 + 4(- x) + 5 = x 2 - 4x + 5 The graph of f(-x) is a reflection of f(x) in the y axis.

17. 2 4 6 8 x y = f(x) -2-4 -6 10 20 30 -10 -20 -30 0 Reflections in the y axis f(x) = x 3 - 9x 2 + 18x f(-x) = (-x) 3 - 9(-x) 2 + 18(-x) Graphs of Related Functions (5) f(-x) = -x 3 - 9x 2 - 18x The graph of f(-x) is a reflection of f(x) in the y axis.

18. 2 4 6 8 x y = f(x) -2-4 -6 10 20 30 -10 -20 -30 0 Reflections in the y axis f(x) Graphs of Related Functions (5) The graph of f(-x) is a reflection of f(x) in the y axis. f(-x)

19. Page 511 Ex E17.3 A and A* questions Use a scale of 2 squares in your book = 1 square in the diagram Next lesson (Monday) Transformations of graphs part 2 – stretches. Transformations of sine, cosine Next Thursday & Friday Past paper practice #2 Calculator Paper (bring a calculator)

20. Worksheet 3 f(x) x y = f(x) Draw the graph of f(-x) for each case on the grids below. y = f(x) x x x 3 4 1 2 f(x) Worksheet 3

21. Worksheet 3 Answers f(x) x y = f(x) Draw the graph of f(-x) for each case on the grids below. y = f(x) x x x 3 4 1 2 f(x) Worksheet 3

22. kf(x) 2 4 6 8 x y = f(x) -2-4 -6 10 20 30 -10 -20 -30 0 Graphs of Related Functions (6) The graph of kf(x) gives a stretch of f(x) by scale factor k in the y direction. f(x) 2f(x) 3f(x) 0 Points located on the x axis remain fixed. Stretches in the y direction y co-ordinates doubled y co-ordinates tripled

23. 2 4 6 8 x y = f(x) -2-4 -6 10 20 30 -10 -20 -30 0 Graphs of Related Functions (6) The graph of kf(x) gives a stretch of f(x) by scale factor k in the y direction. f(x) ½f(x) 1/3f(x) 0 y co-ordinates halved y co-ordinates scaled by 1/3

24. 2 4 6 8 x y = f(x) -2-4 -6 10 20 30 -10 -20 -30 Stretches in y The graph of kf(x) gives a stretch of f(x) by scale factor k in the y direction. f(x) 2f(x) Graphs of Related Functions (6) 3f(x)

25. 90 180 x y = f(x) 0 270 360 -90 -180 -270 -360 1 2 -2 Sinx 2Sinx Graphs of Related Functions (6) 3 -3 3Sinx The graph of kf(x) gives a stretch of f(x) by scale factor k in the y direction. Stretches in y

26. Graphs of Related Functions (6) 90 180 x y = f(x) 0 270 360 -90 -180 -270 -360 1 2 -2 3 -3 Cosx ½Cosx 2Cosx 3Cosx The graph of kf(x) gives a stretch of f(x) by scale factor k in the y direction.

27. Worksheet 4 f(x) x y = f(x) Grid 1: Sketch or trace the graph of 2f(x) Grid 2: Sketch or trace the graph of 3f(x) y = f(x) x Grid 3: Sketch or trace the graph of ½f(x) Grid 4: Sketch or trace the graph of 2f(x) x x y = f(x) 3 4 1 2 f(x) Worksheet 4

28. Worksheet 4 Answers f(x) x y = f(x) Grid 1: Sketch or trace the graph of 2f(x) Grid 2: Sketch or trace the graph of 3f(x) y = f(x) x Grid 3: Sketch or trace the graph of ½f(x) Grid 4: Sketch or trace the graph of 2f(x) x x y = f(x) 3 4 1 2 f(x) Worksheet 4

29. f(kx) ½ the x co-ordinate 0 2 4 6 8 x y = f(x) -2-4 -6 2 4 6 -2 -4 -6 Graphs of Related Functions (7) f(x) f(2x) f(3x) The graph of f(kx) gives a stretch of f(x) by scale factor 1/k in the x direction. 1/3 the x co-ordinate Stretches in x

30. 0 2 4 6 8 x y = f(x) -2-4 -6 2 4 6 -2 -4 -6 Graphs of Related Functions (7) f(x) f(1/2x) f(1/3x) The graph of f(kx) gives a stretch of f(x) by scale factor 1/k in the x direction. All x co-ordinates x 3 All x co-ordinates x 2 Stretches in x

31. 0 2 4 6 8 x y = f(x) -2-4 -6 2 4 6 -2 -4 -6 Graphs of Related Functions (7) f(x) f(2x) f(1/2x) All x co-ordinates x 2 All x co-ordinates x 1/2 The graph of f(kx) gives a stretch of f(x) by scale factor 1/k in the x direction.

32. 90 180 x y = f(x) 0 270 360-90 -180 -270 -360 1 2 -2 Graphs of Related Functions (7) f(x) = Sinx f(x) = Sin2x All x co-ordinates x 1/2 The graph of f(kx) gives a stretch of f(x) by scale factor 1/k in the x direction. Stretches in x

33. 90 180 x y = f(x) 0 270 360 -90 -180 -270 -360 1 2 -2 Graphs of Related Functions (7) f(x) = Sinx f(x) = Sin3x All x co-ordinates x 1/3 The graph of f(kx) gives a stretch of f(x) by scale factor 1/k in the x direction. Stretches in x

34. 90 180 x y = f(x) 0 270 360 -90 -180 -270 -360 1 2 -2 Graphs of Related Functions (7) f(x) = Cosx f(x) = Cos2x All x co-ordinates x 1/2 The graph of f(kx) gives a stretch of f(x) by scale factor 1/k in the x direction. f(x) = Cos ½ x All x co-ordinates x 2 Stretches in x

35. Worksheet 5 f(x) x y = f(x) Grid 1: Sketch or trace the graph of f(2x) Grid 2: Sketch or trace the graph of f(3x) y = f(x) x Grid 3: Sketch or trace the graph of (a) f(½x) (b) f((1/3)x) Grid 4: Sketch or trace the graph of f(½ x) x x y = f(x) 3 4 1 2 f(x) Worksheet 5

36. Worksheet 5 Answers f(x) x y = f(x) Grid 1: Sketch or trace the graph of f(2x) Grid 2: Sketch or trace the graph of f(3x) y = f(x) x Grid 3: Sketch or trace the graph of (a) f(½x) (b) f((1/3)x) Grid 4: Sketch or trace the graph of f(½ x) x x y = f(x) 3 4 1 2 f(x) Worksheet 5

37. GCSE Q’s Mark scheme 1.(a)Graph translated 2 units upwards through points (–4, 2), (–2, 4), (0, 2) and (3, 5) Sketch M1 for a vertical translation A1 curve through points (–4, 2), (–2, 4), (0, 2) and (3, 5) ± ½ square (b)Graph reflected in x-axis through points (–4, 0), (–2, –2), (0, 0) and (3, –3) Sketch2 M1 for reflection in x-axis or y-axis A1 curve through points (–4, 0), (–2, –2), (0, 0) and (3, –3) ± ½ square [4]

38. GCSE Q’s Mark scheme 2.(c)Reflection in the y axis1 mark 3.(a)(4, 3) 1 mark B1 for (4, 3) (b)(2, 6) 1 mark B1 for (2, 6)

39. 4.(a)y = f(x – 4)2 marks B2 cao (B1 for f(x – 4) or y = f(x + a), a ≠ –4, a ≠ 0) (b ) 2 B2 cao (B1 cosine curve with either correct amplitude or correct period, but not both)

40. 2 B2 parabola max (0,0), through (–2, –4) and (2, –4) To accuracy +/- ½sq (B1 parabola with single maximum point (0, 0) or through (–2, –4) and (2, –4),but not both or the given parabola translated along the y-axis by any other value than -4 – the translation must be such that the points (0, 4), (–2, 0), (2, 0) are translated by the same amount. To ½sq) Q5(a)

41. 2 B2 parabola max (0, 4), through (–4, 0) and (4, 0) To ½sq (B1 parabola with single maximum point (0, 4)) To ½sq Q5(b)

42. f(x) + a 0 2 4 6 8 x y = f(x) -2-4 -6 2 4 6 -2 -4 -6 Graphs of Related Functions (1) f(x) = x 2 f(x) = x 2 + 2 f(x) = x 2 - 5 Vertical Translations In general f(x) + a gives a translation by the vector

43. 0 2 4 6 8 x y = f(x) -2-4 -6 2 4 6 -2 -4 -6 Graphs of Related Functions (2) Horizontal Translations f(x) f(x + 2) In general f(x + a) gives a translation by the vector 2 Inside the brackets, “+” means move the curve _____