1 / 42

f(x) + a

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

ksena
Télécharger la présentation

f(x) + a

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. f(x) + a 6 y = f(x) 4 In general f(x) + a gives a translation by the vector 2 x 4 -6 2 8 6 -4 -2 -2 -4 -6 Graphs of Related Functions (1) Vertical Translations f(x) +2= x2 + 2 f(x) = x2 0 f(x) - 5 = x2 - 5

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

  3. 6 y = f(x) 4 2 5 2 f(x - 5) x 4 -6 2 8 6 -4 -2 f(x + 2) -2 In general f(x + a) gives a translation by the vector -4 -6 Graphs of Related Functions (2) Horizontal Translations f(x) 0 In other words, ‘+’ inside the brackets means move to the LEFT

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

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

  6. Worksheet 1 Worksheet 1 Answers 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) 2 1 y = f(x) f(x) f(x) 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 y = f(x) 4 y = f(x) 3 f(x) f(x) x x

  7. -f(x) 6 y = f(x) 4 2 x 4 -6 2 8 6 -4 -2 -2 -4 -6 Graphs of Related Functions (4) Reflections in the x axis f(x) = x2 - 10x + 25 f(x) = x2 The graph of -f(x) is a reflection of f(x) in the x axis. 0 -f(x) = -x2 -f(x) = -x2 + 10x - 25

  8. 6 y = f(x) 4 2 x 4 -6 2 8 6 -4 -2 -2 -4 -6 Graphs of Related Functions (4) Reflections in the x axis f(x) = x2+ 1 f(x) = x2 - 10x + 23 The graph of -f(x) is a reflection of f(x) in the x axis. 0 -f(x) = -(x2 + 1) = -x2 - 1 -f(x) = -x2 + 10x - 23 = -x2 - 1

  9. Graphs of Related Functions (4) 30 y = f(x) Reflections in the x axis 20 f(x) = x3 - 3x2 - 6x + 8 The graph of -f(x) is a reflection of f(x) in the x axis. 10 x 4 -6 2 8 0 6 -4 -2 -10 -f(x) = -x3 + 3x2 + 6x - 8 -20 -30

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

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

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

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

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

  15. 6 y = f(x) 4 2 x 4 -6 2 8 6 -4 -2 -2 -4 -6 Graphs of Related Functions (5) f(-x) Reflections in the y axis f(x) = x2 - 4x + 5 f(x) = x2 + 4x + 5 0 f(-x) The graph of f(-x) is a reflection of f(x) in the y axis. f(-x) = (- x)2 + 4(- x) + 5 = x2 - 4x + 5

  16. Graphs of Related Functions (5) 30 y = f(x) Reflections in the y axis 20 f(x) = x3 - 9x2 + 18x 10 x 4 -6 2 8 0 6 -4 -2 -10 f(-x) = (-x)3 - 9(-x)2 + 18(-x) The graph of f(-x) is a reflection of f(x) in the y axis. -20 f(-x) = -x3 - 9x2 - 18x -30

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

  18. 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)

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

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

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

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

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

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

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

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

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

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

  29. 6 y = f(x) 4 2 x 4 -6 2 8 6 -4 -2 -2 -4 -6 Graphs of Related Functions (7) Stretches in x f(1/3x) f(x) f(1/2x) 0 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 2 All x co-ordinates x 3

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

  31. Graphs of Related Functions (7) y = f(x) Stretches in x 2 f(x) = Sinx f(x) = Sin2x 1 x -360 90 -90 360 -180 0 270 180 -270 -1 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 1/2 -2

  32. Graphs of Related Functions (7) y = f(x) Stretches in x 2 f(x) = Sinx f(x) = Sin3x 1 x -360 90 -90 -180 0 270 180 -270 360 -1 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 1/3 -2

  33. Graphs of Related Functions (7) y = f(x) Stretches in x 2 f(x) = Cos ½ x f(x)=Cos2x f(x) = Cosx 1 x -360 90 -90 360 -180 0 270 180 -270 -1 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 1/2 -2 All x co-ordinates x 2

  34. Grid 1: Sketch or trace the graph of f(2x) Grid 2: Sketch or trace the graph of f(3x) y = f(x) 2 1 y = f(x) f(x) f(x) 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) f(x) y = f(x) 4 y = f(x) 3 f(x) x x Worksheet 5 Worksheet 5

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

  36. 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)Sketch 2 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]

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

  38. 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)

  39. Q5(a) 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)

  40. 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)

  41. f(x) + a 6 y = f(x) 4 In general f(x) + a gives a translation by the vector 2 x 4 -6 2 8 6 -4 -2 -2 -4 -6 Graphs of Related Functions (1) Vertical Translations f(x) = x2 + 2 f(x) = x2 0 f(x) = x2 - 5

  42. 6 y = f(x) 4 2 2 x 4 -6 2 8 6 -4 -2 f(x + 2) -2 In general f(x + a) gives a translation by the vector -4 -6 Graphs of Related Functions (2) Horizontal Translations f(x) 0 Inside the brackets, “+” means move the curve _____

More Related