Transformations of Functions and their Graphs

1. Transformations of Functions and their Graphs Mary Dwyer Wolfe, Ph.D. July 2009

2. Linear Transformations These are the common linear transformations used in high school algebra courses. • Translations (shifts) • Reflections • Dilations (stretches or shrinks) We examine the mathematics: • Graphically • Numerically • Symbolically • Verbally

3. Translations

4. Translations How do we get the flag figure in the left graph to move to the position in the right graph?

5. Translations This picture might help.

6. Translations How do we get the flag figure in the left graph to move to the position in the right graph? Here are the alternate numerical representations of the line graphs above.

7. Translations How do we get the flag figure in the left graph to move to the position in the right graph? This does it! + =

8. Translations Alternately, we could first add 1 to the y-coordinates and then 3 to the x-coordinates to arrive at the final image.

9. Translations What translation could be applied to the left graph to obtain the right graph? y = ???

10. Translations Graphic Representations: Following the vertex, it appears that the vertex, and hence all the points, have been shifted up 1 unit and right 3 units.

11. Translations Numeric Representations: Numerically, 3 has been added to each x-coordinate and 1 has been added to each y coordinate of the function on the left to produce the function on the right. Thus the graph is shifted up 1 unit and right 3 units.

12. Translations To find the symbolic formula for the graph that is seen above on the right, let’s separate our translation into one that shifts the function’s graph up by one unit, and then shift the graph to the right 3 units.

13. Translations To find the symbolic formula for the graph that is seen above on the right, let’s separate our translation into one that shifts the function’s graph up by one unit, and then shift the graph to the right 3 units. The graph on the left above has the equation y = x2. To translate 1 unit up, we must add 1 to every y-coordinate. We can alternately add 1 to x2 as y and x2 are equal. Thus we have y = x2 + 1

14. Translations We verify our results below: The above demonstrates a vertical shift up of 1. y = f(x) + 1 is a shift up of 1 unit that was applied to the graph y = f(x). How can we shift the graph of y = x2 down 2 units?

15. Translations Did you guess to subtract 2 units? We verify our results below: The above demonstrates a vertical shift down of 2. y = f(x) - 2 is a shift down 2 unit to the graph y = f(x) Vertical Shifts If k is a real number and y = f(x) is a function, we say that the graph of y = f(x) + k is the graph of f(x) shifted vertically by k units. If k > 0 then the shift is upward and if k < 0, the shift is downward.

16. x |x| 2 2 1 1 0 0 1 1 2 2 Vertical Translation Example Graphy = |x|

17. Aside: y = |x| on the TI83/84

18. x |x| |x|+2 2 2 4 1 1 3 0 0 2 1 1 3 2 2 4 Vertical Translation Example Graph y = |x| + 2

19. x |x| |x| -1 2 2 1 1 1 0 0 0 -1 1 1 0 2 2 1 Vertical Translation Example Graph y = |x| - 1

20. y = 3x2 Example Vertical Translations

21. y = 3x2 y = 3x2 – 3 y = 3x2 + 2 Example Vertical Translations

22. y = x3 Example Vertical Translations

23. y = x3 y = x3 – 3 y = x3 + 2 Example Vertical Translations

24. Translations Vertical Shift Animation: http://orion.math.iastate.edu/algebra/sp/xcurrent/applets/verticalshift.html

25. Translations Getting back to our unfinished task: The vertex has been shifted up 1 unit and right 3 units. Starting with y = x2 we know that adding 1 to x2, that is y = x2 +1 shifts the graph up 1 unit. Now, how to we also shift the graph 3 units to the right, that is a horizontal shift of 3 units?

26. Translations Starting with y = x2 we know that adding 1 to x2, that is y = x2 +1 shifts the graph up 1 unit. Now, how to we also shift the graph 3 units to the right, that is a horizontal shift of 3 units? We need to add 3 to all the x-coordinates without changing the y-coordinates, but how do we do that in the symbolic formula?

27. Translations We need to add 3 to all the x-coordinates without changing the y-coordinates, but how do we do that in the symbolic formula?

28. Translations We need to add 3 to all the x-coordinates without changing the y-coordinates, but how do we do that in the symbolic formula? So, let’s try y = (x + 3)2 + 1 ??? Oops!!!

29. Translations We need to add 3 to all the x-coordinates without changing the y-coordinates, but how do we do that in the symbolic formula? So, let’s try y = (x - 3)2 + 1 ??? Hurray!!!!!!

30. Translations Horizontal Shifts If h is a real number and y = f(x) is a function, we say that the graph of y = f(x - h) is the graph of f(x) shifted horizontally by h units. If h follows a minus sign, then the shift is right and if h follows a + sign, then the shift is left. Vertical Shifts If k is a real number and y = f(x) is a function, we say that the graph of y = f(x) + k is the graph of f(x) shifted vertically by k units. If k > 0 then the shift is upward and if k < 0, the shift is downward.

31. Graph g(x) = |x| x |x| 2 2 1 1 0 0 1 1 2 2 Example Horizontal Translation

32. Graph g(x) = |x + 1| x |x| |x + 1| 2 2 1 1 1 0 0 0 1 1 1 2 2 2 3 Example Horizontal Translation

33. Graph g(x) = |x - 2| x |x| |x - 2| 2 2 4 1 1 3 0 0 2 1 1 1 2 2 0 Example Horizontal Translation

34. y = 3x2 Horizontal Translation

35. y = 3x2 y = 3(x+2)2 y = 3(x-2)2 Horizontal Translation

36. Horizontal Shift Animation http://orion.math.iastate.edu/algebra/sp/xcurrent/applets/horizontalshift.html

37. Summary of Shift Transformations

38. Translations – Combining Shifts Investigate Vertex form of a Quadratic Function: y = x2 + bx + c y = x2 vertex: (0, 0) y = (x – 3)2 + 1 vertex: (3, 1) Vertex Form of a Quadratic Function (when a = 1): The quadratic function: y = (x – h)2 + k has vertex (h, k).

39. Translations Compare the following 2 graphs by explaining what to do to the graph of the first function to obtain the graph of the second function. f(x) = x4 g(x) = (x – 3)4 - 2

40. Reflections

41. Reflections How do we get the flag figure in the left graph to move to the position in the right graph?

42. Reflections How do we get the flag figure in the left graph to move to the position in the right graph? The numeric representations of the line graphs are:

43. Reflections So how should we change the equation of the function, y = x2 so that the result will be its reflection (across the x-axis)? Try y = - (x2) or simply y = - x2 (Note: - 22 = - 4 while (-2)2 = 4)

44. Reflection: Reflection: (across the x-axis) The graph of the function, y = - f(x) is the reflection of the graph of the function y = f(x).

45. f(x) = x2 Example Reflection over x-axis

46. f(x) = x2 f(x) = -x2 Example Reflection over x-axis

47. f(x) = x3 Example Reflection over x-axis

48. f(x) = x3 f(x) = -x3 Example Reflection over x-axis

49. f(x) = x + 1 Example Reflection over x-axis

50. f(x) = x + 1 f(x) = -(x + 1) = -x - 1 Example Reflection over x-axis