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Absolute Value Functions

Absolute Value Functions. Algebra II Chapter 02 A BowerPoint Presentation. The graph of y = |x|. The graph of y = |x|. When x is 3, what is y ?. The graph of y = |x|. When x is 3, what is y ?. When x is -3, what is y?. The graph of y = |x|. When x is 3, what is y ?.

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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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