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Basic Skills in Higher Mathematics

Basic Skills in Higher Mathematics. Mathematics 1(H) Outcome 2. Robert Glen Adviser in Mathematics. Mathematics 1(Higher). Outcome 2 Associate functions and graphs. y. Functions and graphs. x. Mathematics 1(Higher). Outcome 2 Associate functions and graphs. y. PC Index.

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Basic Skills in Higher Mathematics

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  1. Basic Skills in Higher Mathematics Mathematics 1(H)Outcome 2 Robert GlenAdviser in Mathematics

  2. Mathematics 1(Higher) Outcome 2 Associate functions and graphs y Functions and graphs x

  3. Mathematics 1(Higher) Outcome 2 Associate functions and graphs y PC Index Click on the one you want PC(a) Related functions and graphs x PC(b) Exponential and logarithmic graphs PC(c) Composite functions

  4. Mathematics 1(Higher) Outcome 2 Associate functions and graphs PC(a) Sketch and identify related functions and graphs Related functions and graphs

  5. Mathematics 1(Higher) Outcome 2 Associate functions and graphs PC(a) Sketch and identify related functions and graphs Index Click on the one you want 1 y = f (x)  a Slide up/down 2y = f (x a) Slide left/right 3y = -f (x) Reflect in x-axis 4y = f (-x) Reflect in y-axis 5y = kf (x) Stretch/compress vertically 6 y = f (kx) Stretch/compress horizontally

  6. Mathematics 1(Higher) Outcome 2 Associate functions and graphs PC(a) Sketch and identify related functions and graphs Section 1 y = f (x)  a Slide up/down

  7. Mathematics 1(Higher) Outcome 2 Associate functions and graphs PC(a) Sketch and identify related functions and graphs  In this graph every point on y = f(x) has moved up 5 units. y y = f (x) + 5  +5   y = f (x) 5 +5 +5 -2  In this graph every point on y = f(x) has moved down 2 units. y = f (x) - 2   -2 -2  0  x -2

  8. Mathematics 1(Higher) Outcome 2 Associate functions and graphs PC(a) Sketch and identify related functions and graphs In this graph every point on y = f(x) has moved up aunits. y y = f (x) + a y = f (x) In this graph every point on y = f(x) has moved down bunits. +a y = f (x) - b x -b

  9. Mathematics 1(Higher) Outcome 2 Associate functions and graphs PC(a) Sketch and identify related functions and graphs 1Write down an equation for each graph. y a y = x2 3 Here are the answers b 0 x a y = x2 + 3 -1 b y = x2 - 1

  10. Mathematics 1(Higher) Outcome 2 Associate functions and graphs PC(a) Sketch and identify related functions and graphs 2Write down an equation for each graph. y a 9 y = x3 - 5x 0 x Here are the answers a y = x3 - 5x +9 b -8 b y = x3 - 5x - 8

  11. Mathematics 1(Higher) Outcome 2 Associate functions and graphs PC(a) Sketch and identify related functions and graphs 3Write down an equation for each graph. y a 8 6 4 y = f (x) 2 0 x Here are the answers -2 b -4 a y = f (x) + 4 -6 b y = f (x) - 6

  12. Mathematics 1(Higher) Outcome 2 Associate functions and graphs PC(a) Sketch and identify related functions and graphs End of Section 1 Continue with Section 2

  13. Mathematics 1(Higher) Outcome 2 Associate functions and graphs PC(a) Sketch and identify related functions and graphs Section 2 y = f (x a)Slide left/right

  14. Mathematics 1(Higher) Outcome 2 Associate functions and graphs PC(a) Sketch and identify related functions and graphs y In this graph every point on y = f(x) has moved 5 units to the left. In this graph every point on y = f(x) has moved 3 units to the right. -5 +3    +3 -5    Notef (x+ 5) moves the graph -5. Notef (x- 3) moves the graph +3. -5 +3    x -5 3 0 y = f (x - 3) y = f (x + 5) y = f (x)

  15. Mathematics 1(Higher) Outcome 2 Associate functions and graphs PC(a) Sketch and identify related functions and graphs y In this graph every point on y = f(x) has moved aunits to the left. In this graph every point on y = f(x) has moved bunits to the right. Notef (x+a) moves the graph -a. Notef (x-b) moves the graph +b. x 0 -a +b y = f (x - b) y = f (x + a) y = f (x)

  16. Mathematics 1(Higher) Outcome 2 Associate functions and graphs PC(a) Sketch and identify related functions and graphs 1Write down an equation for each graph. Click the mouse to see the answers. y b a x 0 -5 3 y =(x + 5)2 y =(x - 3)2 y = x2

  17. Mathematics 1(Higher) Outcome 2 Associate functions and graphs PC(a) Sketch and identify related functions and graphs 2Write down an equation for each graph. Click the mouse to see the answers. y a b x 0 -3 -5 4 y = f (x) y = f (x - 4) y = f (x + 3)

  18. Mathematics 1(Higher) Outcome 2 Associate functions and graphs PC(a) Sketch and identify related functions and graphs 3Write down an equation for each graph. Click the mouse to see the answers. y a b x -3 8 3 2 -5 -11 0 y =9 - (x - 5)2 y = 9 - x2 y =9 - (x + 8)2

  19. Mathematics 1(Higher) Outcome 2 Associate functions and graphs PC(a) Sketch and identify related functions and graphs End of Section 2 Continue with Section 3

  20. Mathematics 1(Higher) Outcome 2 Associate functions and graphs PC(a) Sketch and identify related functions and graphs Section 3 y = -f (x)Reflect in x-axis

  21. Mathematics 1(Higher) Outcome 2 Associate functions and graphs PC(a) Sketch and identify related functions and graphs y  y = f (x)   x In this graph every point on y = f(x) has beenreflected in the x - axis.  y = - f (x) 

  22. Mathematics 1(Higher) Outcome 2 Associate functions and graphs PC(a) Sketch and identify related functions and graphs 1Write down an equation for each graph. Click the mouse to see the answers. y y = x2 b y =(x - 5)2 x 5 a y = -x2

  23. Mathematics 1(Higher) Outcome 2 Associate functions and graphs PC(a) Sketch and identify related functions and graphs 2Write down an equation for each graph. Click the mouse to see the answers. y y = f (x) x -6 -2 b y = -f (x) a y = f (x + 6)

  24. Mathematics 1(Higher) Outcome 2 Associate functions and graphs PC(a) Sketch and identify related functions and graphs 3Write down an equation for each graph. Click the mouse to see the answers. y 1 0.5 0 x 90 360 180 270 -0.5 -1 y =sin x b a y =-sin x y =sin(x - 30)

  25. Mathematics 1(Higher) Outcome 2 Associate functions and graphs PC(a) Sketch and identify related functions and graphs End of Section 3 Continue with Section 4

  26. Mathematics 1(Higher) Outcome 2 Associate functions and graphs PC(a) Sketch and identify related functions and graphs Section 4 y = f (-x)Reflect in y-axis

  27. Mathematics 1(Higher) Outcome 2 Associate functions and graphs PC(a) Sketch and identify related functions and graphs y     x In this graph every point on y = f(x) has beenreflected in the y - axis.  y = f (x) y = f (-x)

  28. Mathematics 1(Higher) Outcome 2 Associate functions and graphs PC(a) Sketch and identify related functions and graphs 1Write down an equation for each graph. Click the mouse to see the answers. y y = f (x) y = f (-x) b x a y =-f(x)

  29. Mathematics 1(Higher) Outcome 2 Associate functions and graphs PC(a) Sketch and identify related functions and graphs 2Write down an equation for each graph. Click the mouse to see the answers. y 5 y = f (x) a y = f (-x) x 0 b y = f (x)- 5

  30. Mathematics 1(Higher) Outcome 2 Associate functions and graphs PC(a) Sketch and identify related functions and graphs 3Write down an equation for each graph. Click the mouse to see the answers. y a y = f (x) y = f (-x) x 0 y = - f (x) b

  31. Mathematics 1(Higher) Outcome 2 Associate functions and graphs PC(a) Sketch and identify related functions and graphs End of Section 4 Continue with Section 5

  32. Mathematics 1(Higher) Outcome 2 Associate functions and graphs PC(a) Sketch and identify related functions and graphs Section 5 y = kf (x)Stretch/compress vertically

  33. Mathematics 1(Higher) Outcome 2 Associate functions and graphs PC(a) Sketch and identify related functions and graphs In this graph everyy - value of y = sinx has been doubled. y =2 sin x y  2 1.5 y =sin x  1  0.5  0 x 90 270 360 180  -0.5 In this graph everyy - value of y = sinxhas been halved.  y =½sin x -1 -1.5  -2

  34. Mathematics 1(Higher) Outcome 2 Associate functions and graphs PC(a) Sketch and identify related functions and graphs y = k f (x) , k  1 y y = f (x) 0 x y = k f (x) , k 1

  35. Mathematics 1(Higher) Outcome 2 Associate functions and graphs PC(a) Sketch and identify related functions and graphs 1Write down an equation for each graph. Click the mouse to see the answers. y 4 a y =4sin x 3 2 y =2sin x b 1 0 x -1 y =sin x -2 -3 -4

  36. Mathematics 1(Higher) Outcome 2 Associate functions and graphs PC(a) Sketch and identify related functions and graphs 2Write down an equation for each graph. Click the mouse to see the answers. y 3 a y =4cos x 2 y =cos x 1 0 x -1 y =½cos x b -2 -3

  37. Mathematics 1(Higher) Outcome 2 Associate functions and graphs PC(a) Sketch and identify related functions and graphs 3Write down an equation for each graph. Click the mouse to see the answers. y a y =1.5sin x 1 0 x b y =sin x y =0.5sin x -1

  38. Mathematics 1(Higher) Outcome 2 Associate functions and graphs PC(a) Sketch and identify related functions and graphs End of Section 5 Continue with Section 6

  39. Mathematics 1(Higher) Outcome 2 Associate functions and graphs PC(a) Sketch and identify related functions and graphs Section 6 y = f (kx)Stretch/compress horizontally

  40. Mathematics 1(Higher) Outcome 2 Associate functions and graphs PC(a) Sketch and identify related functions and graphs In this graph everyy - value = sin (doublex) y = sin 2x y   1 y =sin x   0.5   0 x 270 360 90 180  y = sin 2x° runsthrough 2 cycles in the time y = sinx°runs through 1 cycle -0.5 -1

  41. Mathematics 1(Higher) Outcome 2 Associate functions and graphs PC(a) Sketch and identify related functions and graphs In this graph everyy - value = sin (halfx) y = sin ½x y   1   0.5 y =sin x   0 x 270 360 90 180  y = sin ½x° runsthrough ½ of acycle in the time y = sinx°runs through 1 cycle -0.5 -1

  42. Mathematics 1(Higher) Outcome 2 Associate functions and graphs PC(a) Sketch and identify related functions and graphs y = f (2x) y = f (x) y = f (½x) y 4 cycles 2 cycles 1 cycle 1 How manycycles? How manycycles? 0 x How manycycles? -1

  43. Mathematics 1(Higher) Outcome 2 Associate functions and graphs PC(a) Sketch and identify related functions and graphs 1Write down an equation for this graph. Click the mouse to see the answer. y 1 y =sin x x 0 90 180 360 270 y =sin 3x -1

  44. Mathematics 1(Higher) Outcome 2 Associate functions and graphs PC(a) Sketch and identify related functions and graphs 2Write down an equation for this graph. Click the mouse to see the answer. y 1 y =sin x x 0 900 180 540 720 1080 360 y =sin 1/3x -1

  45. Mathematics 1(Higher) Outcome 2 Associate functions and graphs PC(a) Sketch and identify related functions and graphs 3Write down an equation for this graph. Click the mouse to see the answer. y y =sin x 1 x 0 540 720 180 360 -1 y =sin ½x

  46. Mathematics 1(Higher) Outcome 2 Associate functions and graphs PC(a) Sketch and identify related functions and graphs 4Write down an equation for this graph. Click the mouse to see the answer. y y =sin x 1 0 x 90 270 180 360 -1 y =sin 2x

  47. Mathematics 1(Higher) Outcome 2 Associate functions and graphs PC(a) Sketch and identify related functions and graphs 5Write down an equation for this graph. Click the mouse to see the answer. y y =cos x 1 0 x 90 270 180 360 -1 y =cos 2x

  48. Mathematics 1(Higher) Outcome 2 Associate functions and graphs PC(a) Sketch and identify related functions and graphs 6Write down an equation for this graph. Click the mouse to see the answer. y y =cos x 1 0 x 90 450 540 720 360 270 180 630 -1 y =cos ½x

  49. Mathematics 1(Higher) Outcome 2 Associate functions and graphs PC(a) Sketch and identify related functions and graphs 7Write down an equation for each graph. Click the mouse to see the answers. y a y =3 sin x 3 y =-sin x b 2 1 0 x 90 450 540 720 360 270 180 630 -1 y =sin x -2 -3 y =sin x - 3 c

  50. Mathematics 1(Higher) Outcome 2 Associate functions and graphs PC(a) Sketch and identify related functions and graphs End of Section 6 and PC(a) Now do Sections A1 and A2 on pages15 - 17 of the Basic Skills booklet

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