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Transforming Graphs

Transforming Graphs. Learning Objectives: Understand how graphs of functions are transformed. Assumed knowledge. Example. Graphs:. translation. y = x 2 ± c. Plot using a graphic calculator and then sketch y=x 2 , y=x 2 +3 and y=x 2 - 2

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Transforming Graphs

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  1. Transforming Graphs Learning Objectives: Understand how graphs of functions are transformed.

  2. Assumed knowledge

  3. Example

  4. Graphs: translation y = x2±c • Plot using a graphic calculator and then sketch y=x2, y=x2 +3 and y=x2 - 2 • How is the graph of y=x2 transformed to make these two graphs? • For y=x2 +3 there is a translation of [ ] 0 3 [ ] 0 -2 For y=x2 - 2 there is a translation of

  5. Graphs: translation y = (x ±k)2 • Plot using a graphic calculator and then sketch y=x2, y=(x- 3)2 and y=(x + 1)2 • How is the graph of y=x2 transformed to make these two graphs? • For y=(x- 3)2 there is a translation of • For y =(x + 1)2 there is a translation of [ ] 3 0 [ ] -1 0

  6. Have a go • New Try These • What transformations are occurring? y = x2 + 3x + 1 y = (x-3)2 + 3(x-3) + 1 y + 4 = x2 + 3x + 1 y + 4 = (x - 3)2 + 3(x - 3) + 1

  7. Example y = x2 + 3x + 1

  8. [ ] 3 0 is translated by Example y = (x-3)2 + 3(x-3) + 1 y = x2 + 3x + 1

  9. [ ] 0 -4 is translated by Example y + 4 = x2 + 3x + 1 y = x2 + 3x + 1 y = x2 + 3x + 1 y = x2 + 3x + 1 - 4

  10. [ ] 3 -4 is translated by Example y = (x - 3)2 + 3(x - 3) + 1 - 4 y = x2 + 3x + 1

  11. [ ] a 0 Translation Rules - in a nutshell (1) For any graph y = f(x) The translation y = f(x - a) moves it ‘a’ units to the right The translation y = f(x + a) moves it ‘a’ units to the left i.e. for y = f(x - a) is translated by Click to see

  12. [ ] 0 b Translation Rules - in a nutshell (2) For any graph y = f(x) The translation y = f(x) + b moves it ‘b’ units up … this can be considered as y - b = f(x) i.e. for y - b = f(x) is translated by Same as ‘y = f(x) + b’ Click to see

  13. [ ] a b Translation Rules - in a nutshell (3) The previous 2 rules can be combined…. For any graph y = f(x) i.e. for y - b = f(x - a) is translated by Same as y = f(x - a) + b i.e. ‘a’ units right …. and ‘b’ up

  14. Graphs: translation • Plot using a graphic calculator and then sketch y=x3, y=x3 +1 y=x3– 3 • How is the y=x3 transformed to make the other two graphs? • For y=x3 +1 there is a translation of 1 unit up. • For y=x3– 3 there is a translation of -3 unit up. • What about y=x4 + 2 or y= x3 + x2

  15. Graphs: translation • Plot using a graphic calculator and then sketch y=sin (x), y=sin (x) +1 y=sin (x)– 3 work in degrees. • How is the y=sin x transformed to make the other two graphs? • For y=sin (x) + 1 there is a translation of 1 unit up. • For y=sin (x) - 3 there is a translation of -3 unit up. • What about y=cos (x) + 2 or y=tan (x) – 4?

  16. Graphs: more translation • Plot using a graphic calculator and then sketch y=sin x, y=sin( x+ 90) and y=sin( x - 45). • How is the y=sin x transformed to make these two graphs? • For y=sin( x+ 90) there is a translation of -90 units in the x direction. • For y=sin( x - 45). there is a translation of 45 units in the x direction.

  17. Vertex must be at (-5,-9) Our old friend : completing the square Find translation from y=x2 by writing in completed square form.

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