1. Absolute Value Functions and Graphs Lesson 2-5

2. Important Terms Parent function: the simplest function with these characteristics. The equations of the function in a family resemble each other, and so do the graphs. Offspring of parent functions include translations, stretches, and shrinks. Translation: it shifts a graph horizontally, vertically, or both. It results in a graph of the same shape and size but possibly in a different position Stretch: a vertical stretch multiplies all y-values by the same factor greater than 1, thereby stretching the graph vertically Shrink: a vertical shrink reduces y-values by a factor between 0 and 1, thereby compressing the graph vertically Reflection: in the x-axis changes y-values to their opposites. When you change the y-value of a graph to their opposites, the graph reflects across the x-axis (creates a mirror image)

3. The Family of Absolute Value Functions Vertical Translation Parent functionY=|x|Y=f(x) Translation up k units, k>0Y=|x|+kY=f(x)+k Translation down k units, k<0Y=|x|-kY=f(x)-k Horizontal Translation Parent FunctionY=|x|Y=f(x) Translation right h units, h>0Y=|x-h|Y=f(x-h) Translation left h units, h<0Y=|x+h|Y=f(x+k) Combined Translation (right h units, up k units)Y=|x-h|+kY=f(x-h)+k

4. Families of Functions: Absolute Value Functions Vertical Stretch or Shrink, and Reflection in x-axis Parent functionY=|x|Y=f(x) Reflection in x-axisY=-|x|Y= -f(x) Stretch (a>1)Y=a|x|Y=af(x) Shrink (0<a<1) Reflection in x-axis Y=-a|x|Y=-af(x) Combined Translation Y=a|x-h|+kY=af(x-h)+k

5. Absolute Value An Absolute Value graph is always in a “V” shape.

6. Given the following function, If: a > 0, then shift the graph “a” units up If: a < 0, then shift the graph “a” units down

7. Given the following function, Since a > 0, then shift the graph “3” units up

8. Let’s Graph

9. How will the graph look?

10. Let’s Graph

11. How will the graph look?

12. Let’s Graph

13. How will the graph look?

14. Let’s Graph

15. Given the following function, We get the expression (x - b) and equal it to zero x - b = 0 x = b If: b > 0, then shift the graph “b” units to the right If: b < 0, then shift the graph “b” units to the left

16. Given the following function, x – 1 = 0 x = 1 Since 1 > 0, then shift the graph “1” unit right

17. Let’s Graph

18. How will the graph look?

19. Let’s Graph

20. How will the graph look?

21. Let’s Graph

22. How will the graph look?

23. Let’s Graph

24. Graphing Recall: Shift “3” units up since 3 > 0 then we use the expression x + 1, and equal it to zero x + 1 = 0 x = -1 Since –1 < 0, then we shift “1” unit to the left

25. Let’s Graph

26. How will the graph look?

27. Let’s Graph

28. How will the graph look?

29. Let’s Graph

30. How will the graph look?

31. Let’s Graph

32. Given the following function, For this equation, c determines how wide or thin it will be. if: |c|>1, then the graph is closer to the y-axis if: |c|=1, then the graph remains the same if: 0<|c|<1, then the graph is further from the y-axis if c is a negative number, then the graph will reflect on the x-axis

33. Given the following function, Since |5| > 0, then the graph is closer to the y-axis

34. Let’s Graph

35. How will the graph look?

36. Let’s Graph

37. How will the graph look?

38. Let’s Graph

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42. Let’s Graph

43. Given the following function, Since 4 > 0, shift the graph “4” units up x – 1 = 0 x = 1 Since 1 > 0, then shift the graph “1” unit to the right Since |5| > 0 shift the graph closer to the y-axis.

44. Let’s Graph

45. How will the graph look?

46. Let’s Graph

47. How will the graph look?

48. Let’s Graph

49. How will the graph look?

50. Let’s Graph

51. How will the graph look?

52. Let’s Graph

53. How will the graph look?

54. Let’s Graph

55. How will the graph look?

56. Let’s Graph

57. How will the graph look?

58. Let’s Graph

59. Congratulations!! You just completed the transformation of