1. Transformations Transforming Graphs

2. 7/9/2013 Transformations of Graphs 2 Basic Transformations Restructuring Graphs Vertical Translation f(x) to f(x) + k Horizontal Translation f(x) to f(x – h) y x x′ y′ x′ + h y = f(x) y = f(x) + k y = f(x – h) y′ + k Note:When x = x′ + h, y = f(x – h) = y′ = f(x′ + h – h) = f(x′) Vertical move k units

3. 7/9/2013 Transformations of Graphs 3 Basic Transformations Horizontal Translation f(x) to f(x – h) Example y x x′ y′ y = 2x y = 2(x – 1) x′ + 1 y = f(x) = 2x When x = x′, y = 2x′ When x = x′ + 1, y = 2(x – 1) = 2 ( (x′ + 1) – 1) = 2x′ y = f(x – 1) = 2(x – 1) = y′= f(x′)

4. 7/9/2013 Transformations of Graphs 4 Basic Transformations y x x y x′ y = f(x) y = f(x′) y x′x′ x′ = 0 when x = h So, x – h = 0 = x′ f(x′) = f(x – h) h x′ Horizontal Translation Assume h > 0 y = f(x – h) Replacing x with moves the graph h units to the right x – h

5. 7/9/2013 Transformations of Graphs 5 Basic Transformations Restructuring Graphs Vertical Stretching f(x) to a f(x), a > 1 stretch by factor of a Note that when y = f(x) = 0 then y = a f(x) = 0 y x y = f(x) y = a f(x) Stretch is away from x-axis Note:

6. 7/9/2013 Transformations of Graphs 6 Basic Transformations Restructuring Graphs Vertical Shrinking f(x) to a f(x), 0 < a < 1 shrink by factor of a Note that when y = f(x) = 0 then y = a f(x) = 0 y x y = f(x) y = a f(x) Stretch is toward x-axis Note:

7. 7/9/2013 Transformations of Graphs 7 Basic Transformations Restructuring Graphs Horizontal Stretching f(x) to f(cx), 0 < c < 1 stretch by factor of c Note that when y x y = f(x) y = f(cx) Stretch is away from y-axis Note: x = 0 then cx = 0 so y = f(x) = f(cx)

8. 7/9/2013 Transformations of Graphs 8 Basic Transformations Restructuring Graphs Horizontal Shrinking f(x) to f(cx), c > 1 shrink by factor of c Note that when y x y = f(x) y = f(cx) Shrink is toward y-axis Note: so y = f(x) = f(cx) x = 0 then cx = 0

9. 7/9/2013 Transformations of Graphs 9 Basic Transformations Restructuring Graphs: Reflections Vertical reflection through horizontal axis f(x) to -f(x) y x y = f(x) y = -f(x) Each point on the graph of f(x) projects to a corresponding point on the graph of -f(x) reflected through the x-axis

10. 7/9/2013 Transformations of Graphs 10 Basic Transformations Restructuring Graphs: Reflections Horizontal reflection through vertical axis f(x) to f(-x) y x y = f(x) y = f(-x) Each point on the graph of f(x) projects to a corresponding point on the graph of f(-x) … reflected through the y-axis Note:When x = 0, y = f(x) = f(-x)

11. 7/9/2013 Transformations of Graphs 11 Translations Vertical Translation y = f(x) Alters the value of y Horizontal Translation y = f(x) Alters the value of x x y y = f(x) + k k y = f(x – h) h

12. 7/9/2013 Transformations of Graphs 12 Examples y = -x 2 x y y = -x 2 + 4 y = -(x – 3) 2 y = -x 2 + 4 y = -(x – 3) 2 Moves graph up 4 units Moves graph right 3 units

13. 7/9/2013 Transformations of Graphs 13 Examples Vertical Stretching : y = f(x) Example: Vertical Shrinking : y = f(x) Example: x y y = a f(x) x y y = 2(x – 4) 2 – 6 a > 1 0 < a < 1 y = (x – 4) 2 – 3 = 2y= 2y y 1 2 3 2 – (x – 4) 2 = 1 2 = y f(x) = 0 = a f(x)

14. 7/9/2013 Transformations of Graphs 14 Consider y = f(x) = 2x + 1 y = f(x) Horizontal Shrinking & Stretching x y g(x) = f(2x) ● ● ● ● ● f(x) = 2x + 1 f(2x) = 4x + 1 ● ● ● f(x) f(2x) f(x/2) x 0 1 2 135 159 ??? ● = 4x + 1 What changes ? The graph shrinks toward the y-axis

15. 7/9/2013 Transformations of Graphs 15 Consider y = f(x) = 2x + 1 y = f(x) Horizontal Shrinking & Stretching x y h(x) = f ( (½ ) x) ● ●● ● f(x) = 2x + 1 f(2x) = 4x + 1 f((½)x) = x + 1 ● f(x) f(2x) f(x/2) x 0 1 2 135 159 123 ● ● ● = x + 1 What changes ? The graph stretches away from the y-axis

16. 7/9/2013 Transformations of Graphs 16 Consider y = f(x) = 2x + 1 y = f(x) Horizontal Shrinking & Stretching x y f(x) = 2x + 1 f(2x) = 4x + 1 f(( ½ )x) = x + 1 f(x) f(2x) f(x/2) x 0 4 8 16 11733 1 65 1917 For constant c, c ≠ 0, point (x, y) on the graph of y = f(x) corresponds to on the graph of f(cx) x c (, y ) 8 4 Let c = 2 x x c 16 (8,17) 17 x c 9 3 ½

17. 7/9/2013 Transformations of Graphs 17 How does it work ? y = f (x) Horizontal Shrinking x y y = g(x) = f(2x) b (b, f(b)) ● ● ● ● y = f(x) x1x1 ● Note: g(x/2) = f(2(x/2)) For some x 1 want g(x 1 ) = f(b) g(x 1 ) = f(2x 1 ) = f(b) 2x 1 = b and x 1 = b/2 g(b/2) = f(2(b/2)) = f(b) Let = f(x) f(b) g(b/2) * * Note: We have assumed f is 1-1 on interval [x 1, b]

18. 7/9/2013 Transformations of Graphs 18 How does it work ? y = f (x) Horizontal Shrinking - 2 x y y = g(x) = f(2x) b (b, f(b)) ● ● ● y = f(x) g(b/2) ● Note: g(x/2) = f(2(x/2)) For some x 2 want g(x 2 ) = f(c) g(x 2 ) = f(2x 2 ) = f(c) 2x 2 = c and x 2 = c/2 g(c/2) = f(2(c/2)) = f(c) Let = f(x) ● c ● ● (c, f(c)) ● x2x2 ● * f(c) g(c/2) * Note: We have assumed f is 1-1 on interval [x 2, c]

19. 7/9/2013 Transformations of Graphs 19 How does it work ? y = f (x) Horizontal Shrinking - 3 x y y = g(x) = f(2x) b (b, f(b)) ● ● ● y = f(x) g(b/2) ● Note: g(x/2) = f(2(x/2)) For some x 3 want g(x 3 ) = f(d) g(x 3 ) = f(2x 3 ) = f(d) 2x 3 = d and x 3 = d/2 g(d/2) = f(2(d/2)) = f(d) Let = f(x) ● c ● ● (c, f(c)) ● * f(c) g(c/2) ● ●● d (d, f(d)) f(d) x3x3 ● ● g(d/2) * Note: We have assumed f is 1-1 on interval [b, d]

20. 7/9/2013 Transformations of Graphs 20 How does it work ? y = f (x) Horizontal Shrinking - 4 x y y = g(x) = f(2x) b (b, f(b)) ● ● y = f(x) g(b/2) ● Note: g(x/2) = f(2(x/2)) Replacing x by 2x moves points (x,y) on the graph of f(x) to points (x/2, y) on the graph of f(2x) Let = f(x) ● c ● ● (c, f(c)) ● f(c) g(c/2) ● ●● d (d, f(d)) f(d) ● g(d/2) y = g(x) = f(2x) ● This shrinks the graph toward the y-axis

21. 7/9/2013 Transformations of Graphs 21 Horizontal Shrinking y = f 1 (x) Note: Example Examples x y y = f 2 (cx) y = | 2x |, c > 1 x (x, y) ● ● ● ● ● = f 1 (x) y f 2 (c( )) = f 1 (x) = y c x for x 1 = c x c x x1 =x1 = f 1 (x)f 2 (cx) f 2 (cx 1 ) = f 2 (c( )) c x y = | x |

22. 7/9/2013 Transformations of Graphs 22 x y Horizontal Stretching y = f 1 (x) Examples y = f 2 (cx) for 0 < c < 1 y =  x  1 2 = f 1 (x) for x 1 = c x f 2 (cx 1 ) = f 2 (c( )) c x Note: x (x, y) ● ● ● ● ● y c x x1 =x1 = f 1 (x) f 2 (cx) f 2 (c( )) c x f 1 (x) = Example y = | x |

23. 7/9/2013 Transformations of Graphs 23 Examples Vertical Reflection y = f(x) Example y = |x| Horizontal Reflection y = f(x) Example y =  x + 1  x y y = –f(x) y = f(-x) y = –|x| y =  -x + 1  y x f(x) –f(x) f(x) f(-x)

24. 7/9/2013 Transformations of Graphs 24 Exercise What Transformations Are These ? x y f 1 (x) – 2b = ( a x + b) – 2b f 1 (x) = a x + b ● (0, b) ● (-b/ a, 0) f 2 (x) = a x – b ● (0, -b) ● (b/ a, 0) f 1 (x) f 2 (x) ? = a x – b = f 2 (x) f 1 (x – 2b/ a ) = a (x – 2b/ a ) + b = a x – 2b + b = f 2 (x) OR Vertical Translation Horizontal Translation ●

25. 7/9/2013 Transformations of Graphs 25 Exercise What Transformations Are These ? x y f 1 (x) = a x + b ● (0, b) ● (-b/ a, 0) ● (b/ a, 0) f 2 (x) = - a x + b f 1 (x) f 2 (x) ? f 1 (-x) = a (-x) + b = - a x + b = f 2 (x) Horizontal Reflection

26. 7/9/2013 Transformations of Graphs 26 Exercise What Transformations Are These ? x y x y f 1 (x) = a x + b ● (0, b) ● (-b/ a, 0) ● (0, -b) f 2 (x) = - a x – b f 1 (x) f 2 (x) ? -f 1 (x) = -( a x + b) = - a x – b = f 2 (x) Vertical Reflection f 1 (x) = a x + b ● (0, b) ● (-b/ a, 0) ● f 2 (x) = 2 a x + b (-b/(2 a), 0) f 1 (x) f 2 (x) ? f 1 (2x) = a (2x) + b = 2 a x + b = f 2 (x) Horizontal Shrink

27. 7/9/2013 Transformations of Graphs 27 Exercise What Transformations Are These ? x y x y f 1 (x) = a x + b ● (0, b) ● (-b/ a, 0) ● (0, -2b/ a ) f 1 (x) f 2 (x) ? = f 2 (x) Horizontal Stretch f 2 (x) = a x + b 1 2 f 1 (x) = a x + b ● ● (0, b) (-b/ a, 0) f 2 (x) = 2 a x + 2b ● (0, 2b) f 1 (x) f 2 (x) ? = f 2 (x) Vertical Stretch 2f 1 (x) = 2( a x + b) = 2 a x + 2b a x + b 1 2 = a ( x) + b 1 2 f 1 ( x) 1 2 =

28. 7/9/2013 Transformations of Graphs 28 Exercise What Transformation Is This ? x y f 1 (x) = b ● (0, b) f 1 (x) f 2 (x) ? = f 2 (x) Horizontal Stretch f 2 ( x ) = b 1 2... or Shrink... or Reflection f 1 (2x) = f 2 (x) = b f 1 (-x) = f 2 (x) = b f 1 ( x ) = b 1 2

29. 7/9/2013 Transformations of Graphs 29 Exercise What Transformation Is This ? x y f 1 (x) = b ● (0, b) f 2 (x) = -2b (0, -2b) f 1 (x) f 2 (x) ? = f 2 (x) Vertical Stretch and Reflection -2f 1 (x) = -2b ●... or Vertical Translation f 1 (x) – 3b = b – 3b = –2b = f 2 (x)

30. 7/9/2013 Transformations of Graphs 30 Think about it !