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- 1. FunctionsUnit 4 - Homework 7 Homework Help
- 2. What is direct variation?
- 3. What is direct variation?• A linear equation that goes through the origin.
- 4. What is direct variation?• A linear equation that goes through the origin.• Remember a linear equation is in the form y = mx + b.
- 5. What is direct variation?• A linear equation that goes through the origin.• Remember a linear equation is in the form y = mx + b.• A direct variation would be y = mx + 0 or just y = mx.
- 6. What is direct variation?• A linear equation that goes through the origin.• Remember a linear equation is in the form y = mx + b.• A direct variation would be y = mx + 0 or just y = mx.• Typically the ‘m’ is replaced with ‘k’, which stands for constant of variation.
- 7. What is direct variation?• A linear equation that goes through the origin.• Remember a linear equation is in the form y = mx + b.• A direct variation would be y = mx + 0 or just y = mx.• Typically the ‘m’ is replaced with ‘k’, which stands for constant of variation.• General equation for direct variation is y = kx.
- 8. Which graphs are direct variations?
- 9. Which graphs are direct variations? Yes. Goes through origin.
- 10. Which graphs are direct variations? Yes. Goes through origin.No. Does NOT go through origin.Y-intercept something other than 0.
- 11. Which graphs are direct variations? Yes. Goes through origin. Yes. Goes through origin.No. Does NOT go through origin.Y-intercept something other than 0.
- 12. Which graphs are direct variations? Yes. Goes through origin. Yes. Goes through origin.No. Does NOT go through origin. No. Does NOT go through origin.Y-intercept something other than 0. Y-intercept something other than 0.
- 13. Steps to ﬁnd Direct VariationFind the direct variation equation ofthe graph through the points (0, 0) and(3, -5). Write in y=kx form.
- 14. Steps to ﬁnd Direct VariationFind the direct variation equation of • Need one point otherthe graph through the points (0, 0) and than (0, 0). Here we will(3, -5). Write in y=kx form. use (3, -5).
- 15. Steps to ﬁnd Direct VariationFind the direct variation equation of • Need one point otherthe graph through the points (0, 0) and than (0, 0). Here we will(3, -5). Write in y=kx form. use (3, -5). • Substitute the point into y = kx.
- 16. Steps to ﬁnd Direct VariationFind the direct variation equation of • Need one point otherthe graph through the points (0, 0) and than (0, 0). Here we will(3, -5). Write in y=kx form. use (3, -5). y = kx • Substitute the point into y = kx. −5 = k ⋅ 3
- 17. Steps to ﬁnd Direct VariationFind the direct variation equation of • Need one point otherthe graph through the points (0, 0) and than (0, 0). Here we will(3, -5). Write in y=kx form. use (3, -5). y = kx • Substitute the point into y = kx. −5 = k ⋅ 3 • Solve for k.
- 18. Steps to ﬁnd Direct VariationFind the direct variation equation of • Need one point otherthe graph through the points (0, 0) and than (0, 0). Here we will(3, -5). Write in y=kx form. use (3, -5). y = kx • Substitute the point into y = kx. −5 = k ⋅ 3 • Solve for k. 3 3
- 19. Steps to ﬁnd Direct VariationFind the direct variation equation of • Need one point otherthe graph through the points (0, 0) and than (0, 0). Here we will(3, -5). Write in y=kx form. use (3, -5). y = kx • Substitute the point into y = kx. −5 = k ⋅ 3 • Solve for k. 3 3 5 − =k 3
- 20. Steps to ﬁnd Direct VariationFind the direct variation equation of • Need one point otherthe graph through the points (0, 0) and than (0, 0). Here we will(3, -5). Write in y=kx form. use (3, -5). y = kx • Substitute the point into y = kx. −5 = k ⋅ 3 • Solve for k. 3 3 5 • Write direct variation − =k substituting value found 3 for k in y = kx.
- 21. Steps to ﬁnd Direct VariationFind the direct variation equation of • Need one point otherthe graph through the points (0, 0) and than (0, 0). Here we will(3, -5). Write in y=kx form. use (3, -5). y = kx • Substitute the point into y = kx. −5 = k ⋅ 3 • Solve for k. 3 3 5 • Write direct variation − =k substituting value found 3 for k in y = kx. 5 y=− x 3
- 22. You try...Find the direct variation equation ofthe graph through the points (0, 0) and(12, 2). Write in y=kx form.
- 23. You try...Find the direct variation equation of • Use (12, 2).the graph through the points (0, 0) and(12, 2). Write in y=kx form.
- 24. You try...Find the direct variation equation of • Use (12, 2).the graph through the points (0, 0) and(12, 2). Write in y=kx form. • Substitute the point into y = kx.
- 25. You try...Find the direct variation equation of • Use (12, 2).the graph through the points (0, 0) and(12, 2). Write in y=kx form. • Substitute the point into y = kx. y = kx 2 = k ⋅12
- 26. You try...Find the direct variation equation of • Use (12, 2).the graph through the points (0, 0) and(12, 2). Write in y=kx form. • Substitute the point into y = kx. y = kx • Solve for k. 2 = k ⋅12
- 27. You try...Find the direct variation equation of • Use (12, 2).the graph through the points (0, 0) and(12, 2). Write in y=kx form. • Substitute the point into y = kx. y = kx • Solve for k. 2 = k ⋅12 12 12
- 28. You try...Find the direct variation equation of • Use (12, 2).the graph through the points (0, 0) and(12, 2). Write in y=kx form. • Substitute the point into y = kx. y = kx • Solve for k. 2 = k ⋅12 12 12 1 =k 6
- 29. You try...Find the direct variation equation of • Use (12, 2).the graph through the points (0, 0) and(12, 2). Write in y=kx form. • Substitute the point into y = kx. y = kx • Solve for k. 2 = k ⋅12 • Write direct variation 12 12 substituting value found for k into y = kx. 1 =k 6
- 30. You try...Find the direct variation equation of • Use (12, 2).the graph through the points (0, 0) and(12, 2). Write in y=kx form. • Substitute the point into y = kx. y = kx • Solve for k. 2 = k ⋅12 • Write direct variation 12 12 substituting value found for k into y = kx. 1 =k 6 1 y= x 6
- 31. Find Direct Variation w/o pointWrite a direct variation equation thatrelates x to y. Then solve. Show boththe equation and the solution. Ify = 15 when x = 3, ﬁnd y when x = 4.
- 32. Find Direct Variation w/o pointWrite a direct variation equation that • The “if..when” has the piecesrelates x to y. Then solve. Show both found in an ordered pair. Usethe equation and the solution. If these values to ﬁnd k.y = 15 when x = 3, ﬁnd y when x = 4.
- 33. Find Direct Variation w/o pointWrite a direct variation equation that • The “if..when” has the piecesrelates x to y. Then solve. Show both found in an ordered pair. Usethe equation and the solution. If these values to ﬁnd k.y = 15 when x = 3, ﬁnd y when x = 4.
- 34. Find Direct Variation w/o pointWrite a direct variation equation that • The “if..when” has the piecesrelates x to y. Then solve. Show both found in an ordered pair. Usethe equation and the solution. If these values to ﬁnd k.y = 15 when x = 3, ﬁnd y when x = 4. • Substitute the “if...when” values into y = kx.
- 35. Find Direct Variation w/o pointWrite a direct variation equation that • The “if..when” has the piecesrelates x to y. Then solve. Show both found in an ordered pair. Usethe equation and the solution. If these values to ﬁnd k.y = 15 when x = 3, ﬁnd y when x = 4. y = kx • Substitute the “if...when” values into y = kx. 15 = k ⋅ 3
- 36. Find Direct Variation w/o pointWrite a direct variation equation that • The “if..when” has the piecesrelates x to y. Then solve. Show both found in an ordered pair. Usethe equation and the solution. If these values to ﬁnd k.y = 15 when x = 3, ﬁnd y when x = 4. y = kx • Substitute the “if...when” values into y = kx. 15 = k ⋅ 3 • Solve for k.
- 37. Find Direct Variation w/o pointWrite a direct variation equation that • The “if..when” has the piecesrelates x to y. Then solve. Show both found in an ordered pair. Usethe equation and the solution. If these values to ﬁnd k.y = 15 when x = 3, ﬁnd y when x = 4. y = kx • Substitute the “if...when” values into y = kx. 15 = k ⋅ 3 • Solve for k. 3 3
- 38. Find Direct Variation w/o pointWrite a direct variation equation that • The “if..when” has the piecesrelates x to y. Then solve. Show both found in an ordered pair. Usethe equation and the solution. If these values to ﬁnd k.y = 15 when x = 3, ﬁnd y when x = 4. y = kx • Substitute the “if...when” values into y = kx. 15 = k ⋅ 3 • Solve for k. 3 3 5=k
- 39. Find Direct Variation w/o pointWrite a direct variation equation that • The “if..when” has the piecesrelates x to y. Then solve. Show both found in an ordered pair. Usethe equation and the solution. If these values to ﬁnd k.y = 15 when x = 3, ﬁnd y when x = 4. y = kx • Substitute the “if...when” values into y = kx. 15 = k ⋅ 3 • Solve for k. 3 3 • Write direct variation 5=k substituting value found for k.
- 40. Find Direct Variation w/o pointWrite a direct variation equation that • The “if..when” has the piecesrelates x to y. Then solve. Show both found in an ordered pair. Usethe equation and the solution. If these values to ﬁnd k.y = 15 when x = 3, ﬁnd y when x = 4. y = kx • Substitute the “if...when” values into y = kx. 15 = k ⋅ 3 • Solve for k. 3 3 • Write direct variation 5=k substituting value found for k. y = 5x
- 41. Find Direct Variation w/o pointWrite a direct variation equation that • The “if..when” has the piecesrelates x to y. Then solve. Show both found in an ordered pair. Usethe equation and the solution. If these values to ﬁnd k.y = 15 when x = 3, ﬁnd y when x = 4. y = kx • Substitute the “if...when” values into y = kx. 15 = k ⋅ 3 • Solve for k. 3 3 • Write direct variation 5=k substituting value found for k. y = 5x • Now use the “ﬁnd...when” by substituting the given value.
- 42. Find Direct Variation w/o pointWrite a direct variation equation that • The “if..when” has the piecesrelates x to y. Then solve. Show both found in an ordered pair. Usethe equation and the solution. If these values to ﬁnd k.y = 15 when x = 3, ﬁnd y when x = 4. y = kx • Substitute the “if...when” values into y = kx. 15 = k ⋅ 3 • Solve for k. 3 3 • Write direct variation 5=k substituting value found for k. y = 5x • Now use the “ﬁnd...when” by substituting the given value. • In this case we are given x=4.
- 43. Find Direct Variation w/o pointWrite a direct variation equation that • The “if..when” has the piecesrelates x to y. Then solve. Show both found in an ordered pair. Usethe equation and the solution. If these values to ﬁnd k.y = 15 when x = 3, ﬁnd y when x = 4. y = kx • Substitute the “if...when” values into y = kx. 15 = k ⋅ 3 • Solve for k. 3 3 • Write direct variation 5=k substituting value found for k. y = 5x • Now use the “ﬁnd...when” by substituting the given value. y = 5⋅4 • In this case we are given x=4.
- 44. Find Direct Variation w/o pointWrite a direct variation equation that • The “if..when” has the piecesrelates x to y. Then solve. Show both found in an ordered pair. Usethe equation and the solution. If these values to ﬁnd k.y = 15 when x = 3, ﬁnd y when x = 4. y = kx • Substitute the “if...when” values into y = kx. 15 = k ⋅ 3 • Solve for k. 3 3 • Write direct variation 5=k substituting value found for k. y = 5x • Now use the “ﬁnd...when” by substituting the given value. y = 5⋅4 • In this case we are given x=4. • Simplify to ﬁnd y.
- 45. Find Direct Variation w/o pointWrite a direct variation equation that • The “if..when” has the piecesrelates x to y. Then solve. Show both found in an ordered pair. Usethe equation and the solution. If these values to ﬁnd k.y = 15 when x = 3, ﬁnd y when x = 4. y = kx • Substitute the “if...when” values into y = kx. 15 = k ⋅ 3 • Solve for k. 3 3 • Write direct variation 5=k substituting value found for k. y = 5x • Now use the “ﬁnd...when” by substituting the given value. y = 5⋅4 • In this case we are given x=4. y = 20 • Simplify to ﬁnd y.
- 46. Your turn...Write a direct variation equation thatrelates x to y. Then solve. Show boththe equation and the solution. Ify = 21 when x = 7, ﬁnd y when x = 6.
- 47. Your turn...Write a direct variation equation that • Identify “if..when” values torelates x to y. Then solve. Show both ﬁnd k.the equation and the solution. Ify = 21 when x = 7, ﬁnd y when x = 6.
- 48. Your turn...Write a direct variation equation that • Identify “if..when” values torelates x to y. Then solve. Show both ﬁnd k.the equation and the solution. Ify = 21 when x = 7, ﬁnd y when x = 6.
- 49. Your turn...Write a direct variation equation that • Identify “if..when” values torelates x to y. Then solve. Show both ﬁnd k.the equation and the solution. Ify = 21 when x = 7, ﬁnd y when x = 6. • Substitute the “if...when” values into y = kx.
- 50. Your turn...Write a direct variation equation that • Identify “if..when” values torelates x to y. Then solve. Show both ﬁnd k.the equation and the solution. Ify = 21 when x = 7, ﬁnd y when x = 6. • Substitute the “if...when” y = kx values into y = kx. 21 = k ⋅ 7
- 51. Your turn...Write a direct variation equation that • Identify “if..when” values torelates x to y. Then solve. Show both ﬁnd k.the equation and the solution. Ify = 21 when x = 7, ﬁnd y when x = 6. • Substitute the “if...when” y = kx values into y = kx. • Solve for k. 21 = k ⋅ 7
- 52. Your turn...Write a direct variation equation that • Identify “if..when” values torelates x to y. Then solve. Show both ﬁnd k.the equation and the solution. Ify = 21 when x = 7, ﬁnd y when x = 6. • Substitute the “if...when” y = kx values into y = kx. • Solve for k. 21 = k ⋅ 7 7 7
- 53. Your turn...Write a direct variation equation that • Identify “if..when” values torelates x to y. Then solve. Show both ﬁnd k.the equation and the solution. Ify = 21 when x = 7, ﬁnd y when x = 6. • Substitute the “if...when” y = kx values into y = kx. • Solve for k. 21 = k ⋅ 7 7 7 3= k
- 54. Your turn...Write a direct variation equation that • Identify “if..when” values torelates x to y. Then solve. Show both ﬁnd k.the equation and the solution. Ify = 21 when x = 7, ﬁnd y when x = 6. • Substitute the “if...when” y = kx values into y = kx. • Solve for k. 21 = k ⋅ 7 • Write direct variation 7 7 substituting value found for k. 3= k
- 55. Your turn...Write a direct variation equation that • Identify “if..when” values torelates x to y. Then solve. Show both ﬁnd k.the equation and the solution. Ify = 21 when x = 7, ﬁnd y when x = 6. • Substitute the “if...when” y = kx values into y = kx. • Solve for k. 21 = k ⋅ 7 • Write direct variation 7 7 substituting value found for k. 3= k y = 3x
- 56. Your turn...Write a direct variation equation that • Identify “if..when” values torelates x to y. Then solve. Show both ﬁnd k.the equation and the solution. Ify = 21 when x = 7, ﬁnd y when x = 6. • Substitute the “if...when” y = kx values into y = kx. • Solve for k. 21 = k ⋅ 7 • Write direct variation 7 7 substituting value found for k. 3= k • Now use the “ﬁnd...when” by y = 3x substituting the given value.
- 57. Your turn...Write a direct variation equation that • Identify “if..when” values torelates x to y. Then solve. Show both ﬁnd k.the equation and the solution. Ify = 21 when x = 7, ﬁnd y when x = 6. • Substitute the “if...when” y = kx values into y = kx. • Solve for k. 21 = k ⋅ 7 • Write direct variation 7 7 substituting value found for k. 3= k • Now use the “ﬁnd...when” by y = 3x substituting the given value. • In this case we are given x=6.
- 58. Your turn...Write a direct variation equation that • Identify “if..when” values torelates x to y. Then solve. Show both ﬁnd k.the equation and the solution. Ify = 21 when x = 7, ﬁnd y when x = 6. • Substitute the “if...when” y = kx values into y = kx. • Solve for k. 21 = k ⋅ 7 • Write direct variation 7 7 substituting value found for k. 3= k • Now use the “ﬁnd...when” by y = 3x substituting the given value. y = 3⋅ 6 • In this case we are given x=6.
- 59. Your turn...Write a direct variation equation that • Identify “if..when” values torelates x to y. Then solve. Show both ﬁnd k.the equation and the solution. Ify = 21 when x = 7, ﬁnd y when x = 6. • Substitute the “if...when” y = kx values into y = kx. • Solve for k. 21 = k ⋅ 7 • Write direct variation 7 7 substituting value found for k. 3= k • Now use the “ﬁnd...when” by y = 3x substituting the given value. y = 3⋅ 6 • In this case we are given x=6. • Simplify to ﬁnd y.
- 60. Your turn...Write a direct variation equation that • Identify “if..when” values torelates x to y. Then solve. Show both ﬁnd k.the equation and the solution. Ify = 21 when x = 7, ﬁnd y when x = 6. • Substitute the “if...when” y = kx values into y = kx. • Solve for k. 21 = k ⋅ 7 • Write direct variation 7 7 substituting value found for k. 3= k • Now use the “ﬁnd...when” by y = 3x substituting the given value. y = 3⋅ 6 • In this case we are given x=6. y = 18 • Simplify to ﬁnd y.
- 61. Write an equation in slope-intercept form from graph.
- 62. Write an equation in slope-intercept form from graph. • Identify 2 points on the graph. Use Integer coordinates only!
- 63. Write an equation in slope-intercept form from graph. • Identify 2 points on the graph. Use Integer coordinates only! ( 0, −4 )
- 64. Write an equation in slope-intercept form from graph. ( −8, 0 ) • Identify 2 points on the graph. Use Integer coordinates only! ( 0, −4 )
- 65. Write an equation in slope-intercept form from graph. ( −8, 0 ) • Identify 2 points on the graph. Use Integer coordinates only! ( 0, −4 ) • Find slope between 2 points.
- 66. Write an equation in slope-intercept form from graph. ( −8, 0 ) • Identify 2 points on the graph. Use Integer coordinates only! ( 0, −4 ) • Find slope between 2 points. y2 − y1m= x2 − x1
- 67. Write an equation in slope-intercept form from graph. ( −8, 0 ) • Identify 2 points on the graph. Use Integer coordinates only! ( 0, −4 ) • Find slope between 2 points. • Label points as 1’s and 2’s. y2 − y1m= x2 − x1
- 68. Write an equation in slope-intercept form from graph. ( x1, y1 ) ( −8, 0 ) • Identify 2 points on the graph. Use Integer coordinates only! ( x2 , y2 ) ( 0, −4 ) • Find slope between 2 points. • Label points as 1’s and 2’s. y2 − y1m= x2 − x1
- 69. Write an equation in slope-intercept form from graph. ( x1, y1 ) ( −8, 0 ) • Identify 2 points on the graph. Use Integer coordinates only! ( x2 , y2 ) ( 0, −4 ) • Find slope between 2 points. • Label points as 1’s and 2’s. y2 − y1m= x2 − x1 • Substitute.
- 70. Write an equation in slope-intercept form from graph. ( x1, y1 ) ( −8, 0 ) • Identify 2 points on the graph. Use Integer coordinates only! ( x2 , y2 ) ( 0, −4 ) • Find slope between 2 points. • Label points as 1’s and 2’s. y2 − y1 −4 − 0m= = x2 − x1 0 − ( −8 ) • Substitute.
- 71. Write an equation in slope-intercept form from graph. ( x1, y1 ) ( −8, 0 ) • Identify 2 points on the graph. Use Integer coordinates only! ( x2 , y2 ) ( 0, −4 ) • Find slope between 2 points. • Label points as 1’s and 2’s. y2 − y1 −4 − 0m= = x2 − x1 0 − ( −8 ) • Substitute. • Simplify.
- 72. Write an equation in slope-intercept form from graph. ( x1, y1 ) ( −8, 0 ) • Identify 2 points on the graph. Use Integer coordinates only! ( x2 , y2 ) ( 0, −4 ) • Find slope between 2 points. • Label points as 1’s and 2’s. y2 − y1 −4 − 0 −4 −1m= = = x2 − x1 0 − ( −8 ) 8 = 2 • Substitute. • Simplify.
- 73. Write an equation in slope-intercept form from graph. ( x1, y1 ) ( −8, 0 ) • Identify 2 points on the graph. Use Integer coordinates only! ( x2 , y2 ) ( 0, −4 ) • Find slope between 2 points. • Label points as 1’s and 2’s. y2 − y1 −4 − 0 −4 −1m= = = x2 − x1 0 − ( −8 ) 8 = 2 • Substitute. • Simplify. • Use the slope and y- intercept to write equation.
- 74. Write an equation in slope-intercept form from graph. ( x1, y1 ) ( −8, 0 ) • Identify 2 points on the graph. Use Integer coordinates only! ( x2 , y2 ) ( 0, −4 ) • Find slope between 2 points. • Label points as 1’s and 2’s. y2 − y1 −4 − 0 −4 −1m= = = x2 − x1 0 − ( −8 ) 8 = 2 • Substitute. • Simplify. y = mx + b • Use the slope and y- intercept to write equation.
- 75. Write an equation in slope-intercept form from graph. ( x1, y1 ) ( −8, 0 ) • Identify 2 points on the graph. Use Integer coordinates only! ( x2 , y2 ) ( 0, −4 ) • Find slope between 2 points. • Label points as 1’s and 2’s. y2 − y1 −4 − 0 −4 −1m= = = x2 − x1 0 − ( −8 ) 8 = 2 • Substitute. • Simplify. y = mx + b • Use the slope and y- intercept to write equation.
- 76. Write an equation in slope-intercept form from graph. ( x1, y1 ) ( −8, 0 ) • Identify 2 points on the graph. Use Integer coordinates only! ( x2 , y2 ) ( 0, −4 ) • Find slope between 2 points. • Label points as 1’s and 2’s. y2 − y1 −4 − 0 −4 −1m= = = x2 − x1 0 − ( −8 ) 8 = 2 • Substitute. • Simplify. y = mx + b • Use the slope and y- 1 intercept to write y=− x−4 equation. 2
- 77. Your turn to write the equation...
- 78. Your turn to write the equation... • Identify 2 points on the graph. Use Integer coordinates only!
- 79. Your turn to write the equation... • Identify 2 points on the graph. Use Integer ( 0, 4 ) coordinates only!
- 80. Your turn to write the equation... ( −5, 0 ) • Identify 2 points on the graph. Use Integer ( 0, 4 ) coordinates only!
- 81. Your turn to write the equation... ( −5, 0 ) • Identify 2 points on the graph. Use Integer ( 0, 4 ) coordinates only! • Find slope between 2 points.
- 82. Your turn to write the equation... ( −5, 0 ) • Identify 2 points on the graph. Use Integer ( 0, 4 ) coordinates only! • Find slope between 2 points. y2 − y1m= x2 − x1
- 83. Your turn to write the equation... ( −5, 0 ) • Identify 2 points on the graph. Use Integer ( 0, 4 ) coordinates only! • Find slope between 2 points. • Label points as 1’s and 2’s. y2 − y1m= x2 − x1
- 84. Youry )turn to write the equation... (x ,1 1 ( −5, 0 ) • Identify 2 points on the ( x2 , y2 ) graph. Use Integer ( 0, 4 ) coordinates only! • Find slope between 2 points. • Label points as 1’s and 2’s. y2 − y1m= x2 − x1
- 85. Youry )turn to write the equation... (x ,1 1 ( −5, 0 ) • Identify 2 points on the ( x2 , y2 ) graph. Use Integer ( 0, 4 ) coordinates only! • Find slope between 2 points. • Label points as 1’s and 2’s. y2 − y1m= x2 − x1 • Substitute.
- 86. Youry )turn to write the equation... (x , 1 1 ( −5, 0 ) • Identify 2 points on the ( x2 , y2 ) graph. Use Integer ( 0, 4 ) coordinates only! • Find slope between 2 points. • Label points as 1’s and 2’s. y2 − y1 4−0m= = x2 − x1 0 − ( −5 ) • Substitute.
- 87. Youry )turn to write the equation... (x , 1 1 ( −5, 0 ) • Identify 2 points on the ( x2 , y2 ) graph. Use Integer ( 0, 4 ) coordinates only! • Find slope between 2 points. • Label points as 1’s and 2’s. y2 − y1 4−0m= = x2 − x1 0 − ( −5 ) • Substitute. • Simplify.
- 88. Youry )turn to write the equation... (x ,1 1 ( −5, 0 ) • Identify 2 points on the ( x2 , y2 ) graph. Use Integer ( 0, 4 ) coordinates only! • Find slope between 2 points. • Label points as 1’s and 2’s. y2 − y1 4−0 4m= = = x2 − x1 0 − ( −5 ) 5 • Substitute. • Simplify.
- 89. Youry )turn to write the equation... (x ,1 1 ( −5, 0 ) • Identify 2 points on the ( x2 , y2 ) graph. Use Integer ( 0, 4 ) coordinates only! • Find slope between 2 points. • Label points as 1’s and 2’s. y2 − y1 4−0 4m= = = x2 − x1 0 − ( −5 ) 5 • Substitute. • Simplify. • Use the slope and y- intercept to write equation.
- 90. Youry )turn to write the equation... (x ,1 1 ( −5, 0 ) • Identify 2 points on the ( x2 , y2 ) graph. Use Integer ( 0, 4 ) coordinates only! • Find slope between 2 points. • Label points as 1’s and 2’s. y2 − y1 4−0 4m= = = x2 − x1 0 − ( −5 ) 5 • Substitute. • Simplify. y = mx + b • Use the slope and y- 4 intercept to write y= x+4 equation. 5
- 91. A couple comments aboutpicking points on a Graph...
- 92. A couple comments about picking points on a Graph...• Only use Integer coordinates. (No fractions or decimals.)
- 93. A couple comments about picking points on a Graph...• Only use Integer coordinates. (No fractions or decimals.)• Never estimate coordinates. You may get lucky but more often your equation is slightly off and harder to ﬁnd.
- 94. A couple comments about picking points on a Graph...• Only use Integer coordinates. (No fractions or decimals.)• Never estimate coordinates. You may get lucky but more often your equation is slightly off and harder to ﬁnd.• Try to use the x- and y-intercepts as your points.
- 95. Write the equation given 2 pointsWrite an equation in slope interceptform of the line that passes through(1, 2) and (4, -5).
- 96. Write the equation given 2 pointsWrite an equation in slope intercept • Find the slope.form of the line that passes through(1, 2) and (4, -5).
- 97. Write the equation given 2 pointsWrite an equation in slope intercept • Find the slope.form of the line that passes through(1, 2) and (4, -5). y2 − y1m= x2 − x1
- 98. Write the equation given 2 pointsWrite an equation in slope intercept • Find the slope.form of the line that passes through(1, 2) and (4, -5). • Label points as 1’s and 2’s. Doesn’t matter which is which. y2 − y1 Will get the same slope.m= x2 − x1
- 99. Write the equation given 2 points Write an equation in slope intercept • Find the slope. form of the line that passes through (1, 2) and (4, -5). • Label points as 1’s and 2’s.( x1, y1 ) Doesn’t matter which is which. y2 − y1 Will get the same slope. m= x2 − x1
- 100. Write the equation given 2 points Write an equation in slope intercept • Find the slope. form of the line that passes through (1, 2) and (4, -5). • Label points as 1’s and 2’s.( x1, y1 ) ( x2 , y2 ) Doesn’t matter which is which. y2 − y1 Will get the same slope. m= x2 − x1
- 101. Write the equation given 2 points Write an equation in slope intercept • Find the slope. form of the line that passes through (1, 2) and (4, -5). • Label points as 1’s and 2’s.( x1, y1 ) ( x2 , y2 ) Doesn’t matter which is which. y2 − y1 Will get the same slope. m= x2 − x1 • Substitute and simplify.
- 102. Write the equation given 2 points Write an equation in slope intercept • Find the slope. form of the line that passes through (1, 2) and (4, -5). • Label points as 1’s and 2’s.( x1, y1 ) ( x2 , y2 ) Doesn’t matter which is which. y2 − y1 −5 − 2 Will get the same slope. m= = x2 − x1 4 −1 • Substitute and simplify.
- 103. Write the equation given 2 points Write an equation in slope intercept • Find the slope. form of the line that passes through (1, 2) and (4, -5). • Label points as 1’s and 2’s.( x1, y1 ) ( x2 , y2 ) Doesn’t matter which is which. y2 − y1 −5 − 2 −7 Will get the same slope. m= = = x2 − x1 4 −1 3 • Substitute and simplify.
- 104. Write the equation given 2 points Write an equation in slope intercept • Find the slope. form of the line that passes through (1, 2) and (4, -5). • Label points as 1’s and 2’s.( x1, y1 ) ( x2 , y2 ) Doesn’t matter which is which. y2 − y1 −5 − 2 −7 Will get the same slope. m= = = x2 − x1 4 −1 3 • Substitute and simplify. • Use slope and one point to ﬁnd y-intercept. Choose the “easier” point to work with.
- 105. Write the equation given 2 points Write an equation in slope intercept • Find the slope. form of the line that passes through (1, 2) and (4, -5). • Label points as 1’s and 2’s.( x1, y1 ) ( x2 , y2 ) Doesn’t matter which is which. y2 − y1 −5 − 2 −7 Will get the same slope. m= = = x2 − x1 4 −1 3 • Substitute and simplify. • Use slope and one point to ﬁnd y-intercept. Choose the “easier” point to work with.
- 106. Write the equation given 2 points Write an equation in slope intercept • Find the slope. form of the line that passes through (1, 2) and (4, -5). • Label points as 1’s and 2’s.( x1, y1 ) ( x2 , y2 ) Doesn’t matter which is which. y2 − y1 −5 − 2 −7 Will get the same slope. m= = = x2 − x1 4 −1 3 • Substitute and simplify. y = mx + b • Use slope and one point to ﬁnd y-intercept. Choose the −7 2= ⋅1 + b “easier” point to work with. 3
- 107. Write the equation given 2 points Write an equation in slope intercept • Find the slope. form of the line that passes through (1, 2) and (4, -5). • Label points as 1’s and 2’s.( x1, y1 ) ( x2 , y2 ) Doesn’t matter which is which. y2 − y1 −5 − 2 −7 Will get the same slope. m= = = x2 − x1 4 −1 3 • Substitute and simplify. y = mx + b • Use slope and one point to ﬁnd y-intercept. Choose the −7 2= ⋅1 + b “easier” point to work with. 3 • Solve for b.
- 108. Write the equation given 2 points Write an equation in slope intercept • Find the slope. form of the line that passes through (1, 2) and (4, -5). • Label points as 1’s and 2’s.( x1, y1 ) ( x2 , y2 ) Doesn’t matter which is which. y2 − y1 −5 − 2 −7 Will get the same slope. m= = = x2 − x1 4 −1 3 • Substitute and simplify. y = mx + b • Use slope and one point to ﬁnd y-intercept. Choose the −7 2= ⋅1 + b “easier” point to work with. 3 7 −7 7 • Solve for b. 2+ = +b+ 3 3 3
- 109. Write the equation given 2 points Write an equation in slope intercept • Find the slope. form of the line that passes through (1, 2) and (4, -5). • Label points as 1’s and 2’s.( x1, y1 ) ( x2 , y2 ) Doesn’t matter which is which. y2 − y1 −5 − 2 −7 Will get the same slope. m= = = x2 − x1 4 −1 3 • Substitute and simplify. y = mx + b • Use slope and one point to ﬁnd y-intercept. Choose the −7 2= ⋅1 + b “easier” point to work with. 3 7 −7 7 • Solve for b. 2+ = +b+ 3 3 3 13 =b 3
- 110. Write the equation given 2 points Write an equation in slope intercept • Find the slope. form of the line that passes through (1, 2) and (4, -5). • Label points as 1’s and 2’s.( x1, y1 ) ( x2 , y2 ) Doesn’t matter which is which. y2 − y1 −5 − 2 −7 Will get the same slope. m= = = x2 − x1 4 −1 3 • Substitute and simplify. y = mx + b • Use slope and one point to ﬁnd y-intercept. Choose the −7 2= ⋅1 + b “easier” point to work with. 3 7 −7 7 • Solve for b. 2+ = +b+ 3 3 3 • Use the slope and y-intercept 13 =b to write the equation in slope- 3 intercept form.
- 111. Write the equation given 2 points Write an equation in slope intercept • Find the slope. form of the line that passes through (1, 2) and (4, -5). • Label points as 1’s and 2’s.( x1, y1 ) ( x2 , y2 ) Doesn’t matter which is which. y2 − y1 −5 − 2 −7 Will get the same slope. m= = = x2 − x1 4 −1 3 • Substitute and simplify. y = mx + b • Use slope and one point to ﬁnd y-intercept. Choose the −7 2= ⋅1 + b “easier” point to work with. 3 7 −7 7 • Solve for b. 2+ = +b+ 3 3 3 • Use the slope and y-intercept 13 =b to write the equation in slope- 3 intercept form.
- 112. Write the equation given 2 points Write an equation in slope intercept • Find the slope. form of the line that passes through (1, 2) and (4, -5). • Label points as 1’s and 2’s.( x1, y1 ) ( x2 , y2 ) Doesn’t matter which is which. y2 − y1 −5 − 2 −7 Will get the same slope. m= = = x2 − x1 4 −1 3 • Substitute and simplify. y = mx + b • Use slope and one point to ﬁnd y-intercept. Choose the −7 2= ⋅1 + b “easier” point to work with. 3 7 −7 7 • Solve for b. 2+ = +b+ 3 3 3 • Use the slope and y-intercept 13 =b to write the equation in slope- 3 intercept form.
- 113. Write the equation given 2 points Write an equation in slope intercept • Find the slope. form of the line that passes through (1, 2) and (4, -5). • Label points as 1’s and 2’s.( x1, y1 ) ( x2 , y2 ) Doesn’t matter which is which. y2 − y1 −5 − 2 −7 Will get the same slope. m= = = x2 − x1 4 −1 3 • Substitute and simplify. y = mx + b • Use slope and one point to ﬁnd y-intercept. Choose the −7 2= ⋅1 + b “easier” point to work with. 3 7 −7 7 • Solve for b. 2+ = +b+ 3 3 3 • Use the slope and y-intercept 13 =b 7 13 to write the equation in slope- 3 y=− x+ intercept form. 3 3
- 114. You try...Write an equation in slope interceptform of the line that passes through(-3, 7) and (2, 4).
- 115. You try...Write an equation in slope intercept • Find the slope.form of the line that passes through(-3, 7) and (2, 4).
- 116. You try...Write an equation in slope intercept • Find the slope.form of the line that passes through(-3, 7) and (2, 4). y2 − y1m= x2 − x1
- 117. You try...Write an equation in slope intercept • Find the slope.form of the line that passes through(-3, 7) and (2, 4). • Label points as 1’s and 2’s. Doesn’t matter which is which. y2 − y1 Will get the same slope.m= x2 − x1
- 118. You try...Write an equation in slope intercept • Find the slope.form of the line that passes through(-3, 7) and (2, 4). • Label points as 1’s and 2’s. ( x1, y1 ) Doesn’t matter which is which. y2 − y1 Will get the same slope.m= x2 − x1
- 119. You try...Write an equation in slope intercept • Find the slope.form of the line that passes through(-3, 7) and (2, 4). • Label points as 1’s and 2’s.( x2 , y2 ) ( x1, y1 ) Doesn’t matter which is which. y2 − y1 Will get the same slope.m= x2 − x1
- 120. You try...Write an equation in slope intercept • Find the slope.form of the line that passes through(-3, 7) and (2, 4). • Label points as 1’s and 2’s.( x2 , y2 ) ( x1, y1 ) Doesn’t matter which is which. y2 − y1 Will get the same slope.m= x2 − x1 • Substitute and simplify.
- 121. You try...Write an equation in slope intercept • Find the slope.form of the line that passes through(-3, 7) and (2, 4). • Label points as 1’s and 2’s.( x2 , y2 ) ( x1, y1 ) Doesn’t matter which is which. y2 − y1 7−4 Will get the same slope.m= = x2 − x1 −3 − 2 • Substitute and simplify.
- 122. You try...Write an equation in slope intercept • Find the slope.form of the line that passes through(-3, 7) and (2, 4). • Label points as 1’s and 2’s.( x2 , y2 ) ( x1, y1 ) Doesn’t matter which is which. y2 − y1 7−4 3 Will get the same slope.m= = = x2 − x1 −3 − 2 −5 • Substitute and simplify.
- 123. You try...Write an equation in slope intercept • Find the slope.form of the line that passes through(-3, 7) and (2, 4). • Label points as 1’s and 2’s.( x2 , y2 ) ( x1, y1 ) Doesn’t matter which is which. y2 − y1 7−4 3 Will get the same slope.m= = = x2 − x1 −3 − 2 −5 • Substitute and simplify. • Use slope and one point to ﬁnd y-intercept. Choose the “easier” point to work with.
- 124. You try...Write an equation in slope intercept • Find the slope.form of the line that passes through(-3, 7) and (2, 4). • Label points as 1’s and 2’s.( x2 , y2 ) ( x1, y1 ) Doesn’t matter which is which. y2 − y1 7−4 3 Will get the same slope.m= = = x2 − x1 −3 − 2 −5 • Substitute and simplify. • Use slope and one point to ﬁnd y-intercept. Choose the “easier” point to work with.
- 125. You try...Write an equation in slope intercept • Find the slope.form of the line that passes through(-3, 7) and (2, 4). • Label points as 1’s and 2’s.( x2 , y2 ) ( x1, y1 ) Doesn’t matter which is which. y2 − y1 7−4 3 Will get the same slope.m= = = x2 − x1 −3 − 2 −5 • Substitute and simplify. y = mx + b • Use slope and one point to 3 ﬁnd y-intercept. Choose the 4 = − ⋅2 + b “easier” point to work with. 5
- 126. You try...Write an equation in slope intercept • Find the slope.form of the line that passes through(-3, 7) and (2, 4). • Label points as 1’s and 2’s.( x2 , y2 ) ( x1, y1 ) Doesn’t matter which is which. y2 − y1 7−4 3 Will get the same slope.m= = = x2 − x1 −3 − 2 −5 • Substitute and simplify. y = mx + b • Use slope and one point to 3 ﬁnd y-intercept. Choose the 4 = − ⋅2 + b “easier” point to work with. 5 • Solve for b.
- 127. You try...Write an equation in slope intercept • Find the slope.form of the line that passes through(-3, 7) and (2, 4). • Label points as 1’s and 2’s.( x2 , y2 ) ( x1, y1 ) Doesn’t matter which is which. y2 − y1 7−4 3 Will get the same slope.m= = = x2 − x1 −3 − 2 −5 • Substitute and simplify. y = mx + b • Use slope and one point to 3 ﬁnd y-intercept. Choose the 4 = − ⋅2 + b “easier” point to work with. 5 6 4 = − +b • Solve for b. 5
- 128. You try...Write an equation in slope intercept • Find the slope.form of the line that passes through(-3, 7) and (2, 4). • Label points as 1’s and 2’s.( x2 , y2 ) ( x1, y1 ) Doesn’t matter which is which. y2 − y1 7−4 3 Will get the same slope.m= = = x2 − x1 −3 − 2 −5 • Substitute and simplify. y = mx + b • Use slope and one point to 3 ﬁnd y-intercept. Choose the 4 = − ⋅2 + b “easier” point to work with. 5 6 4 = − +b • Solve for b. 5 6 6 64+ = − +b+ 5 5 5
- 129. You try...Write an equation in slope intercept • Find the slope.form of the line that passes through(-3, 7) and (2, 4). • Label points as 1’s and 2’s.( x2 , y2 ) ( x1, y1 ) Doesn’t matter which is which. y2 − y1 7−4 3 Will get the same slope.m= = = x2 − x1 −3 − 2 −5 • Substitute and simplify. y = mx + b • Use slope and one point to 3 ﬁnd y-intercept. Choose the 4 = − ⋅2 + b “easier” point to work with. 5 6 4 = − +b • Solve for b. 5 6 6 64+ = − +b+ 5 5 5 26 =b 5
- 130. You try...Write an equation in slope intercept • Find the slope.form of the line that passes through(-3, 7) and (2, 4). • Label points as 1’s and 2’s.( x2 , y2 ) ( x1, y1 ) Doesn’t matter which is which. y2 − y1 7−4 3 Will get the same slope.m= = = x2 − x1 −3 − 2 −5 • Substitute and simplify. y = mx + b • Use slope and one point to 3 ﬁnd y-intercept. Choose the 4 = − ⋅2 + b “easier” point to work with. 5 6 4 = − +b • Solve for b. 5 6 6 6 • Use the slope and y-intercept4+ = − +b+ 5 5 5 to write the equation in slope- 26 intercept form. =b 5
- 131. You try...Write an equation in slope intercept • Find the slope.form of the line that passes through(-3, 7) and (2, 4). • Label points as 1’s and 2’s.( x2 , y2 ) ( x1, y1 ) Doesn’t matter which is which. y2 − y1 7−4 3 Will get the same slope.m= = = x2 − x1 −3 − 2 −5 • Substitute and simplify. y = mx + b • Use slope and one point to 3 ﬁnd y-intercept. Choose the 4 = − ⋅2 + b “easier” point to work with. 5 6 4 = − +b • Solve for b. 5 6 6 6 • Use the slope and y-intercept4+ = − +b+ 5 5 5 to write the equation in slope- 26 intercept form. =b 5
- 132. You try...Write an equation in slope intercept • Find the slope.form of the line that passes through(-3, 7) and (2, 4). • Label points as 1’s and 2’s.( x2 , y2 ) ( x1, y1 ) Doesn’t matter which is which. y2 − y1 7−4 3 Will get the same slope.m= = = x2 − x1 −3 − 2 −5 • Substitute and simplify. y = mx + b • Use slope and one point to 3 ﬁnd y-intercept. Choose the 4 = − ⋅2 + b “easier” point to work with. 5 6 4 = − +b • Solve for b. 5 6 6 6 • Use the slope and y-intercept4+ = − +b+ 5 5 5 to write the equation in slope- 3 26 intercept form. 26 y=− x+ =b 5 5 5
- 133. What does slope mean?
- 134. What does slope mean?• It measures the steepness of a line.
- 135. What does slope mean?• It measures the steepness of a line.• Also referred to as rate of change.
- 136. What does slope mean?• It measures the steepness of a line.• Also referred to as rate of change.• Slope is the ratio rise/run.
- 137. What does slope mean?• It measures the steepness of a line.• Also referred to as rate of change.• Slope is the ratio rise/run.• To ﬁnd the “meaning” of slope, identify the rise and run paying attention to the units. Image from http://regentsprep.org/REgents/math/ALGEBRA/AC1/Rate.htm
- 138. What does slope mean?• It measures the steepness of a line.• Also referred to as rate of change.• Slope is the ratio rise/run.• To ﬁnd the “meaning” of slope, identify the rise and run paying attention to the units.• Here the “rise” (red arrow) is -2 gallons because the line slopes downward and the y-axis is in gallons. Image from http://regentsprep.org/REgents/math/ALGEBRA/AC1/Rate.htm
- 139. What does slope mean?• It measures the steepness of a line.• Also referred to as rate of change.• Slope is the ratio rise/run.• To ﬁnd the “meaning” of slope, identify the rise and run paying attention to the units.• Here the “rise” (red arrow) is -2 gallons because the line slopes downward and the y-axis is in gallons.• The “run” (purple arrow) is 100 miles. Image from http://regentsprep.org/REgents/math/ALGEBRA/AC1/Rate.htm
- 140. What does slope mean?• It measures the steepness of a line.• Also referred to as rate of change.• Slope is the ratio rise/run.• To ﬁnd the “meaning” of slope, identify the rise and run paying attention to the units.• Here the “rise” (red arrow) is -2 gallons because the line slopes downward and the y-axis is in gallons.• The “run” (purple arrow) is 100 miles. Image from http://regentsprep.org/REgents/math/ALGEBRA/AC1/Rate.htm• So we have -2 gallons/100 miles but both numbers are even. Always simplify the slope before determining what it means.
- 141. What does slope mean?• It measures the steepness of a line.• Also referred to as rate of change.• Slope is the ratio rise/run.• To ﬁnd the “meaning” of slope, identify the rise and run paying attention to the units.• Here the “rise” (red arrow) is -2 gallons because the line slopes downward and the y-axis is in gallons.• The “run” (purple arrow) is 100 miles. Image from http://regentsprep.org/REgents/math/ALGEBRA/AC1/Rate.htm• So we have -2 gallons/100 miles but both numbers are even. Always simplify the slope before determining what it means.• Reduced the slope is -1 gallon/50 miles.
- 142. What does slope mean?• It measures the steepness of a line.• Also referred to as rate of change.• Slope is the ratio rise/run.• To ﬁnd the “meaning” of slope, identify the rise and run paying attention to the units.• Here the “rise” (red arrow) is -2 gallons because the line slopes downward and the y-axis is in gallons.• The “run” (purple arrow) is 100 miles. Image from http://regentsprep.org/REgents/math/ALGEBRA/AC1/Rate.htm• So we have -2 gallons/100 miles but both • When writing the meaning, numbers are even. Always simplify the use some common sense to slope before determining what it means. make a logical statement.• Reduced the slope is -1 gallon/50 miles.
- 143. What does slope mean?• It measures the steepness of a line.• Also referred to as rate of change.• Slope is the ratio rise/run.• To ﬁnd the “meaning” of slope, identify the rise and run paying attention to the units.• Here the “rise” (red arrow) is -2 gallons because the line slopes downward and the y-axis is in gallons.• The “run” (purple arrow) is 100 miles. Image from http://regentsprep.org/REgents/math/ALGEBRA/AC1/Rate.htm• So we have -2 gallons/100 miles but both • When writing the meaning, numbers are even. Always simplify the use some common sense to slope before determining what it means. make a logical statement.• Reduced the slope is -1 gallon/50 miles. • For every 50 miles traveled one gallon of gas is used.
- 144. You try... Image from http://www.algebra-class.com/rate-of-change.html
- 145. You try... What does the slope represent in the graph to the right? Image from http://www.algebra-class.com/rate-of-change.html
- 146. You try... What does the slope represent in the graph to the right?• John’s savings account balance increase $100 each month. OR Image from http://www.algebra-class.com/rate-of-change.html
- 147. You try... What does the slope represent in the graph to the right?• John’s savings account balance increase $100 each month. OR• Every one month, John’s savings account balance increases by $100. Image from http://www.algebra-class.com/rate-of-change.html

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