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Absolute Value Functions

This presentation provides an insightful exploration of absolute value functions, focusing on their graphs. It covers essential concepts, such as identifying the vertex, determining slopes of the rays, and understanding how transformations impact the graph’s shape and position. Through examples like y = |x| and y = 2|x - 1| + 3, the presentation demonstrates how to create tables of points and graph these functions step-by-step. Ideal for Algebra II students, it emphasizes key features and properties of absolute value equations.

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Absolute Value Functions

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  1. Absolute Value Functions Algebra II Chapter 02 A BowerPoint Presentation

  2. The graph of y = |x|

  3. The graph of y = |x| When x is 3, what is y?

  4. The graph of y = |x| When x is 3, what is y? When x is -3, what is y?

  5. The graph of y = |x| When x is 3, what is y? When x is -3, what is y? What point is the VERTEX of this function?

  6. The graph of y = |x| When x is 3, what is y? When x is -3, what is y? What point is the VERTEX of this function? What is the slope of the right-side ray?

  7. The graph of y = 2|x – 1| + 3 Let’s make a table of points

  8. The graph of y = 2|x – 1| + 3 Let’s make a table of points X Y

  9. The graph of y = 2|x – 1| + 3 Let’s make a table of points X Y -1 0 1 2 3 Find the corresponding y values

  10. The graph of y = 2|x – 1| + 3 Let’s make a table of points X Y -1 0 1 2 3 7 5 3 5 7 Do you notice anything?

  11. The graph of y = 2|x – 1| + 3 Let’s make a graph using those points

  12. The graph of y = 2|x – 1| + 3

  13. The graph of y = 2|x – 1| + 3 What is the VERTEX of this graph?

  14. The graph of y = 2|x – 1| + 3 What is the VERTEX of this graph? What is the slope of the right-side ray?

  15. The graph of y = 2|x – 1| + 3 What is the VERTEX of this graph? What is the slope of the right-side ray? Does this graph open UP or DOWN?

  16. The graph of y = 2|x – 1| + 3 What is the VERTEX of this graph? What is the slope of the right-side ray? Does this graph open UP or DOWN? Is this graph WIDER, NARROWER, or THE SAME as the graph of y = |x|?

  17. What’s up w/absolute value functions y = a | x – h | + k Do you see how this looks like y – y1 = m (x – x1) ? [Maybe not yet – let’s move y1…]

  18. What’s up w/absolute value functions y = a | x – h | + k Do you see how this looks like y – y1 = m (x – x1) ?

  19. What’s up w/absolute value functions y = a | x – h | + k Do you see how this looks like y= m (x – x1) + y1 ?

  20. What’s up w/absolute value functions y = a | x – h | + k Do you see how this looks like y= m (x – x1) + y1 ?

  21. What’s up w/absolute value functions y = a | x – h| + k The vertex of this graph will be the point (h, k)

  22. What’s up w/absolute value functions y = a | x – h | + k The slope of the right-side ray will be a

  23. What’s up w/absolute value functions y = a | x – h | + k The slope of the right-side ray will be a The slope of the left-side ray will be -a

  24. What’s up w/absolute value functions y = a | x – h | + k If a is POSITIVE If a is NEGATIVE Graph opens Graph opens UP DOWN

  25. What’s up w/absolute value functions y = a | x – h | + k If |a| > 1 If |a| = 1 If |a| < 1 Narrower Same width Wider than y =|x| than y =|x| than y =|x|

  26. Let’s graph! Graph this absolute value function:y = – |x + 2| – 3

  27. Let’s graph! Graph this absolute value function:y = – |x + 2| – 3 • Will this graph open UP or DOWN?

  28. Let’s graph! Graph this absolute value function:y = – |x + 2| – 3 • Will this graph open UP or DOWN? • What will be the VERTEX of this graph?

  29. Let’s graph! Graph this absolute value function:y = – |x + 2| – 3 • Will this graph open UP or DOWN? • What will be the VERTEX of this graph? • What is the slope of the right-side ray?

  30. Let’s graph! Graph this absolute value function:y = – |x + 2| – 3 • Will this graph open UP or DOWN? • What will be the VERTEX of this graph? • What is the slope of the right-side ray? • Use symmetry to draw the left-side ray.

  31. Let’s graph! Graph this absolute value function:y = – |x + 2| – 3 • Will this graph open UP or DOWN? • What will be the VERTEX of this graph? • What is the slope of the right-side ray? • Use symmetry to draw the left-side ray. • Is this NARROWER, WIDER, or THE SAME as y = |x| ?

  32. Let’s graph!

  33. Let’s graph again! Graph this absolute value function:y = 2/3 |x – 4| + 2

  34. Let’s graph again! Graph this absolute value function: y = 2/3 |x – 4| + 2 • Will this graph open UP or DOWN?

  35. Let’s graph! Graph this absolute value function: y = 2/3 |x – 4| + 2 • Will this graph open UP or DOWN? • What will be the VERTEX of this graph?

  36. Let’s graph again! Graph this absolute value function: y = 2/3 |x – 4| + 2 • Will this graph open UP or DOWN? • What will be the VERTEX of this graph? • What is the slope of the right-side ray?

  37. Let’s graph again! Graph this absolute value function: y = 2/3 |x – 4| + 2 • Will this graph open UP or DOWN? • What will be the VERTEX of this graph? • What is the slope of the right-side ray? • Use symmetry to draw the left-side ray.

  38. Let’s graph again! Graph this absolute value function: y = 2/3 |x – 4| + 2 • Will this graph open UP or DOWN? • What will be the VERTEX of this graph? • What is the slope of the right-side ray? • Use symmetry to draw the left-side ray. • Is this NARROWER, WIDER, or THE SAME as y = |x| ?

  39. Let’s graph again!

  40. Turning graph into a function We will follow these steps to turn a graph into an absolute value function… • Find the vertex – this gives us h and k. • Find the slope of the right side ray – this gives us a. • Put our h, k, and a into y = a | x – h| + k

  41. Turning graph into a function • Let’s turn the following graph into a function!

  42. Turning graph into a function Step 1- Find the vertex (to get h & k)

  43. Turning graph into a function Step 1- Find the vertex (to get h & k) What is the vertex?

  44. Turning graph into a function Step 1- Find the vertex (to get h & k) What is the vertex? (– 4, –1)

  45. Turning graph into a function Step 1- Find the vertex (to get h & k) What is the vertex? (– 4, –1) h = –4 & k = –1

  46. Turning graph into a function Step 2- Find the slope of the right-side ray (to get a)

  47. Turning graph into a function Step 2- Find the slope of the right-side ray (to get a) What is the slope (go from vertex to P1)?

  48. Turning graph into a function Step 2- Find the slope of the right-side ray (to get a) What is the slope (go from vertex to P1)? Slope is –3/2 , so a = –3/2.

  49. Turning graph into a function Step 3- Put our h, k, and a into y = a | x – h| + k

  50. Turning graph into a function Step 3- Put our h, k, and a into y = a | x – h| + k a = –3/2 h = –4 k = –1

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