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Higher Maths

Higher Maths. Graph Transformations. Trig Graphs. Derivative Graphs f’(x). Exponential and Log Graphs. Completing the Square. Solving equations / /Inequations. Composite Functions. Inverse function. Mindmap. Exam Question Type. Graph Transformations.

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Higher Maths

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  1. Higher Maths Graph Transformations Trig Graphs Derivative Graphs f’(x) Exponential and Log Graphs Completing the Square Solving equations / /Inequations Composite Functions Inverse function Mindmap Exam Question Type www.mathsrevision.com

  2. Graph Transformations • We will investigate f(x) graphs of the form • 1. f(x) ± k • f(x ±k) • -f(x) • f(-x) • kf(x) • f(kx) Each affect the Graph of f(x) in a certain way !

  3. 6 4 2 x 4 -6 2 8 6 -4 -2 -2 -4 -6 f(x) - 3 Transformation f(x) ± k f(x) + 5 Mapping (x , y)  (x , y ± k) 0 f(x) f(x)

  4. Transformation f(x) ± k • Keypoints • y = f(x) ± k • moves original f(x) graph vertically up or down • + k move up • - k  move down • Only y-coordinate changes • NOTE: Always state any coordinates given on f(x) • on f(x) ± k graph Demo

  5. f(x) - 2 B(1,-2) A(-1,-2) C(0,-3)

  6. f(x) + 1 A(45o,1.5) B(90o,1) C(135o,0.5) A(45o,0.5) B(90o,0) C(135o,-0.5)

  7. Extra Practice HHM Ex 3C

  8. 6 4 2 x 4 -6 2 8 6 -4 -2 -2 -4 -6 f(x + 4) Transformation f(x ± k) f(x - 2) Mapping (x, y)  (x ± k , y) 0 f(x) f(x)

  9. Transformation f(x ± k) • Keypoints • y = f(x ± k) • moves original f(x) graph horizontally left or right • + k move left • - k  move right • Only x-coordinate changes • NOTE: Always state any coordinates given on f(x) • on f(x ± k) graph Demo

  10. Extra Practice HHM Ex 3E

  11. 6 4 2 x 4 -6 2 8 6 -4 -2 -2 -4 -6 Transformation -f(x) Mapping (x, y)  (x , -y) Flip in x-axis Flip in x-axis 0 f(x) f(x)

  12. Transformation -f(x) • Keypoints • y = -f(x) • Flips original f(x) graph in the x-axis • y-coordinate changes sign • NOTE: Always state any coordinates given on f(x) • on -f(x) graph Demo

  13. - f(x) C(0,1) A(-1,0) B(1,0)

  14. - f(x) C(135o,0.5) A(45o,0.5) B(90o,0) A(45o,-0.5) C(135o,-0.5)

  15. Extra Practice HHM Ex 3G

  16. 6 4 2 x 4 -6 2 8 6 -4 -2 -2 -4 -6 Transformation f(-x) Flip in y-axis Mapping (x, y)  (-x , y) 0 f(x) f(x) Flip in y-axis

  17. Transformation f(-x) • Keypoints • y = f(-x) • Flips original f(x) graph in the y-axis • x-coordinate changes sign • NOTE: Always state any coordinates given on f(x) • on f(-x) graph Demo

  18. f(-x) C (1,1) C’(-1,1) B(0,0) A’(1,-1) A(-1,-1)

  19. Extra Practice HHM Ex 3I

  20. 6 4 2 x 4 -6 2 8 6 -4 -2 -2 -4 -6 0.5f(x) Transformation kf(x) 2f(x) Stretch in y-axis Compress in y-axis Mapping (x, y)  (x , ky) 0 f(x) f(x)

  21. Transformation kf(x) Keypoints y = kf(x) Stretch / Compress original f(x) graph in the y-axis direction y-coordinate changes by a factor of k NOTE: Always state any coordinates given on f(x) on kf(x) graph Demo

  22. 6 4 2 x 4 -6 2 8 6 -4 -2 -2 -4 -6 f(0.5x) Transformation f(kx) f(2x) Compress in x-axis Mapping (x, y)  (1/kx , y) 0 f(x) Stretch in x-axis f(x)

  23. Transformation f(kx) Keypoints y = f(kx) Stretch / Compress original f(x) graph in the x-axis direction x-coordinate changes by a factor of 1/k NOTE: Always state any coordinates given on f(x) on f(kx) graph Demo

  24. Extra Practice HHM Ex 3K & 3M

  25. Combining Transformations You need to be able to work with combinations Drill

  26. Explain the effect the following have • -f(x) • f(-x) • f(x) ± k (1,3) (1,3) (-1,-3) (-1,-3) 2f(x) + 1 f(x + 1) + 2 (1,3) (1,3) Name : f(-x) + 1 (-1,-3) -f(x) - 2 (-1,-3) (1,3) • Explain the effect the following have • f(x ± k) • kf(x) • f(kx) (1,3) f(0.5x) - 1 -f(x + 1) - 3 (-1,-3) (-1,-3)

  27. Explain the effect the following have • -f(x) flip in x-axis • f(-x) flip in y-axis • f(x) ± k move up or down (1,7) (0,5) (1,3) (1,3) (-2,-1) (-1,-3) (-1,-3) 2f(x) + 1 (-1,-5) f(x + 1) + 2 (1,3) (-1,4) (1,3) Name : (-1,1) f(-x) + 1 (-1,-3) -f(x) - 2 (1,-2) (1,-5) (-1,-3) (1,3) • Explain the effect the following have • f(x ± k) move left or right • kf(x) stretch / compress • in y direction • f(kx) stretch / compress • in x direction (1,3) f(0.5x) - 1 (-2,0) (2,2) -f(x + 1) - 3 (-1,-3) (-1,-3) (-2,-4) (0,-6)

  28. Graphs & Functions Higher The diagram shows the graph of a function f. f has a minimum turning point at (0, -3) and a point of inflexion at (-4, 2). a) sketch the graph of y = f(-x). b) On the same diagram, sketch the graph of y = 2f(-x) a) Reflect across the y axis b) Now scale by 2 in the y direction

  29. Graphs & Functions Higher Part of the graph of is shown in the diagram. On separate diagrams sketch the graph of a) b) Indicate on each graph the images of O, A, B, C, and D. a) graph moves to the left 1 unit graph is reflected in the x axis b) graph is then scaled 2 units in the y direction

  30. Graphs & Functions Higher = a) On the same diagram sketch i) the graph of ii) the graph of b) Find the range of values of x for which is positive a) b) Solve: 10 - f(x) is positive for -1 < x < 5

  31. A sketch of the graph of y = f(x) where is shown. The graph has a maximum at A (1,4) and a minimum at B(3, 0) . Sketch the graph of Indicate the co-ordinates of the turning points. There is no need to calculate the co-ordinates of the points of intersection with the axes. (-1,8) Graphs & Functions Higher (1,4) moved 2 units to the left, and 4 units up Graph is t.p.’s are:

  32. Trig Graphs The same transformation rules apply to the basic trig graphs. NB: If f(x) =sinx then 3f(x) = 3sinx and f(5x) = sin5x Think about sin replacing f ! Also if g(x) = cosx then g(x) – 4 = cosx – 4 and g(x + 90) = cos(x + 90)  Think about cos replacing g !

  33. Trig Graphs Demo Sketch the graph of y = sinx - 2 If sinx = f(x) then sinx - 2 = f(x) - 2 So move the sinx graph 2 units down. 1 0 90o 180o 270o 360o -1 -2 y = sinx - 2 -3

  34. Trig Graphs Demo Sketch the graph of y = cos(x - 50) If cosx = f(x) then cos(x - 50) = f(x - 50) So move the cosx graph 50 units right. 50o 1 0 90o 180o 270o 360o -1 -2 y = cos(x - 50)o -3

  35. Trig Graphs Demo Sketch the graph of y = 3sinx If sinx = f(x) then 3sinx = 3f(x) So stretch the sinx graph 3 times vertically. 3 2 1 0 90o 180o 270o 360o -1 -2 y = 3sinx -3

  36. Trig Graphs Demo Sketch the graph of y = cos4x If cosx = f(x) then cos4x = f(4x) So squash the cosx graph to 1/4 size horizontally 1 0 90o 180o 270o 360o -1 y = cos4x

  37. Trig Graphs Demo Sketch the graph of y = 2sin3x If sinx = f(x) then 2sin3x = 2f(3x) So squash the sinx graph to 1/3 size horizontally and also double its height. 3 2 1 0 90o 180o 270o 360o -1 -2 y = 2sin3x -3

  38. Write down equations for graphs shown ? y = 0.5sin2xo + 0.5 y = 2sin4xo- 1 Write down the equations in the form f(x) for the graphs shown? y = 0.5f(2x) + 0.5 y = 2f(4x) - 1 Trig Graph DEMO Combinations Higher 3 Demo 2 1 0 www.mathsrevision.com 90o 180o 270o 360o -1 -2 -3 created by Mr. Lafferty

  39. Write down the equations for the graphs shown? Write down the equations in the form f(x) for the graphs shown? Trig Graphs y = cos2xo + 1 y = -2cos2xo - 1 y = f(2x) + 1 y = -2f(2x) - 1 DEMO Combinations Higher 3 Demo 2 1 0 www.mathsrevision.com 90o 180o 270o 360o -1 -2 -3 created by Mr. Lafferty

  40. Extra Practice Demo HHM Ex 4A & 4B Show-me boards

  41. Exponential (to the power of) Graphs Exponential Functions A function in the form f(x) = ax where a > 0, a ≠ 1 is called an exponential function to base a . Consider f(x) = 2x x -3 -2 -1 0 1 2 3 f(x) 1 1/8 ¼ ½ 1 2 4 8

  42. Graph The graph of y = 2x (1,2) (0,1) Major Points (i) y = 2x passes through the points (0,1) & (1,2) (ii) As x ∞ y ∞ however as x -∞ y 0 . (iii) The graph shows a GROWTH function.

  43. Log Graphs ie x 1/8 ¼ ½ 1 2 4 8 y -3 -2 -1 0 1 2 3 To obtain y from x we must ask the question “What power of 2 gives us…?” This is not practical to write in a formula so we say “the logarithm to base 2 of x” y = log2x or “log base 2 of x”

  44. Graph The graph of (2,1) y = log2x (1,0) NB: x > 0 Major Points (i) y = log2x passes through the points (1,0) & (2,1) . • As x ∞ y ∞but at a very slow rate and as x  0 y  -∞ .

  45. Exponential (to the power of) Graphs The graph of y = axalways passes through (0,1) & (1,a) It looks like .. Y y = ax (1,a) (0,1) x

  46. Log Graphs The graph of y = logaxalways passes through (1,0) & (a,1) Y It looks like .. (a,1) (1,0) x y = logax

  47. Connection f(x) = ax (1,a) Y (0,1) (a,1) x (1,0) f-1(x) = logax

  48. Extra Practice HHM Ex 2H HHM Ex 3N , 3O and 15K HHM Ex 3P

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