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# Unit 4 hw 8 - pointslope, parallel & perp

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• ### Unit 4 hw 8 - pointslope, parallel & perp

1. 1. FUNCTIONSUNIT 4 - HOMEWORK 8 Homework Help
2. 2. POINT-SLOPE FORMULA
3. 3. POINT-SLOPE FORMULA• Alternatemethod to ﬁnd the equation of a line with the slope and one point.
4. 4. POINT-SLOPE FORMULA• Alternatemethod to ﬁnd the equation of a line with the slope and one point. y − y1 = m ( x − x1 )
5. 5. POINT-SLOPE FORMULA• Alternatemethod to ﬁnd the equation of a line with the slope and one point. y − y1 = m ( x − x1 )• The x and y stay the same. Never substitute a value for these.
6. 6. POINT-SLOPE FORMULA• Alternatemethod to ﬁnd the equation of a line with the slope and one point. y − y1 = m ( x − x1 )• The x and y stay the same. Never substitute a value for these.• The x1 and y1 represent the given point. This is where you substitute the given point.
7. 7. USING POINT-SLOPE FORMULAWrite the point-slope form ofthe equation passing through(3, -2) with a slope of 5.
8. 8. USING POINT-SLOPE FORMULAWrite the point-slope form of • Write the general formula forthe equation passing through point-slope form.(3, -2) with a slope of 5.
9. 9. USING POINT-SLOPE FORMULAWrite the point-slope form of • Write the general formula forthe equation passing through point-slope form.(3, -2) with a slope of 5. y − y1 = m ( x − x1 )
10. 10. USING POINT-SLOPE FORMULAWrite the point-slope form of • Write the general formula forthe equation passing through point-slope form.(3, -2) with a slope of 5. • Label your given point. y − y1 = m ( x − x1 )
11. 11. USING POINT-SLOPE FORMULAWrite the point-slope form of • Write the general formula forthe equation passing through point-slope form.(3, -2) with a slope of 5. • Label your given point.( x1, y1 ) y − y1 = m ( x − x1 )
12. 12. USING POINT-SLOPE FORMULAWrite the point-slope form of • Write the general formula forthe equation passing through point-slope form.(3, -2) with a slope of 5. • Label your given point.( x1, y1 ) • Substitute the slope and point. y − y1 = m ( x − x1 )
13. 13. USING POINT-SLOPE FORMULAWrite the point-slope form of • Write the general formula forthe equation passing through point-slope form.(3, -2) with a slope of 5. • Label your given point.( x1, y1 ) • Substitute the slope and point. y − y1 = m ( x − x1 ) y − ( −2 ) = 5 ( x − 3)
14. 14. USING POINT-SLOPE FORMULAWrite the point-slope form of • Write the general formula forthe equation passing through point-slope form.(3, -2) with a slope of 5. • Label your given point.( x1, y1 ) • Substitute the slope and point. y − y1 = m ( x − x1 ) • When the question asks for the equation in point-slope form, the y − ( −2 ) = 5 ( x − 3) only simplifying done is to change any subtracting negatives to addition.
15. 15. USING POINT-SLOPE FORMULAWrite the point-slope form of • Write the general formula forthe equation passing through point-slope form.(3, -2) with a slope of 5. • Label your given point.( x1, y1 ) • Substitute the slope and point. y − y1 = m ( x − x1 ) • When the question asks for the equation in point-slope form, the y − ( −2 ) = 5 ( x − 3) only simplifying done is to change any subtracting negatives to addition. y + 2 = 5 ( x − 3)
16. 16. YOUR TURN...Write the point-slope form ofthe equation passing through(7, -3) with a slope of -2.
17. 17. YOUR TURN...Write the point-slope form of • Write the general formula forthe equation passing through point-slope form.(7, -3) with a slope of -2.
18. 18. YOUR TURN...Write the point-slope form of • Write the general formula forthe equation passing through point-slope form.(7, -3) with a slope of -2. y − y1 = m ( x − x1 )
19. 19. YOUR TURN...Write the point-slope form of • Write the general formula forthe equation passing through point-slope form.(7, -3) with a slope of -2. • Label your given point. y − y1 = m ( x − x1 )
20. 20. YOUR TURN...Write the point-slope form of • Write the general formula forthe equation passing through point-slope form.(7, -3) with a slope of -2. • Label your given point.( x1, y1 ) y − y1 = m ( x − x1 )
21. 21. YOUR TURN...Write the point-slope form of • Write the general formula forthe equation passing through point-slope form.(7, -3) with a slope of -2. • Label your given point.( x1, y1 ) • Substitute the slope and point. y − y1 = m ( x − x1 )
22. 22. YOUR TURN...Write the point-slope form of • Write the general formula forthe equation passing through point-slope form.(7, -3) with a slope of -2. • Label your given point.( x1, y1 ) • Substitute the slope and point. y − y1 = m ( x − x1 ) y − ( −3) = −2 ( x − 7 )
23. 23. YOUR TURN...Write the point-slope form of • Write the general formula forthe equation passing through point-slope form.(7, -3) with a slope of -2. • Label your given point.( x1, y1 ) • Substitute the slope and point. y − y1 = m ( x − x1 ) • When the question asks for the equation in point-slope form, the y − ( −3) = −2 ( x − 7 ) only simplifying is change to addition any subtraction negatives.
24. 24. YOUR TURN...Write the point-slope form of • Write the general formula forthe equation passing through point-slope form.(7, -3) with a slope of -2. • Label your given point.( x1, y1 ) • Substitute the slope and point. y − y1 = m ( x − x1 ) • When the question asks for the equation in point-slope form, the y − ( −3) = −2 ( x − 7 ) only simplifying is change to addition any subtraction negatives. y + 3 = −2 ( x − 7 )
25. 25. HORIZONTAL LINES• Look at the horizon at the right.
26. 26. HORIZONTAL LINES• Look at the horizon at the right.• The direction of the horizon is from left to right. This is how all horizontal lines appear on a graph.
27. 27. HORIZONTAL LINES• Look at the horizon at the right.• The direction of the horizon is from left to right. This is how all horizontal lines appear on a graph.• The x-coordinate changes but the y-coordinate remains constant.
28. 28. HORIZONTAL LINES• Look at the horizon at the right.• The direction of the horizon is from left to right. This is how all horizontal lines appear on a graph.• The x-coordinate changes but the y-coordinate remains constant.• Horizontal lines have the equation y = b.
29. 29. HORIZONTAL LINES• Look at the horizon at the right.• The direction of the horizon is from left to right. This is how all horizontal lines appear on a graph.• The x-coordinate changes but the y-coordinate remains constant.• Horizontal lines have the equation y = b.• Slope of all horizontal lines is zero. (Think, y = 0x + b.)
30. 30. HORIZONTAL LINES• Look at the horizon at the right.• The direction of the horizon is from left to right. This is how all horizontal lines appear on a graph.• The x-coordinate changes but the y-coordinate remains constant.• Horizontal lines have the equation y = b.• Slope of all horizontal lines is zero. (Think, y = 0x + b.)• Why is the slope zero?
31. 31. HORIZONTAL LINES• Look at the horizon at the right.• The direction of the horizon is from left to right. This is how all horizontal lines appear on a graph.• The x-coordinate changes but the y-coordinate remains constant.• Horizontal lines have the equation y = b.• Slope of all horizontal lines is zero. (Think, y = 0x + b.)• Why is the slope zero? • Suppose you have points (2, 5) and (3, 5). You know they are horizontal because the x-coordinate changes.
32. 32. HORIZONTAL LINES• Look at the horizon at the right.• The direction of the horizon is from left to right. This is how all horizontal lines appear on a graph.• The x-coordinate changes but the y-coordinate remains constant.• Horizontal lines have the equation y = b.• Slope of all horizontal lines is zero. (Think, y = 0x + b.)• Why is the slope zero? • Suppose you have points (2, 5) and (3, 5). You know they are horizontal because the x-coordinate changes. • Substitute into the slope formula. y2 − y1 m= x2 − x1
33. 33. HORIZONTAL LINES• Look at the horizon at the right.• The direction of the horizon is from left to right. This is how all horizontal lines appear on a graph.• The x-coordinate changes but the y-coordinate remains constant.• Horizontal lines have the equation y = b.• Slope of all horizontal lines is zero. (Think, y = 0x + b.)• Why is the slope zero? • Suppose you have points (2, 5) and (3, 5). You know they are horizontal because the x-coordinate changes. • Substitute into the slope formula. y2 − y1 5 − 5 m= = x2 − x1 2 − 3
34. 34. HORIZONTAL LINES• Look at the horizon at the right.• The direction of the horizon is from left to right. This is how all horizontal lines appear on a graph.• The x-coordinate changes but the y-coordinate remains constant.• Horizontal lines have the equation y = b.• Slope of all horizontal lines is zero. (Think, y = 0x + b.)• Why is the slope zero? • Suppose you have points (2, 5) and (3, 5). You know they are horizontal because the x-coordinate changes. • Substitute into the slope formula. y2 − y1 5 − 5 0 m= = = x2 − x1 2 − 3 −1
35. 35. HORIZONTAL LINES• Look at the horizon at the right.• The direction of the horizon is from left to right. This is how all horizontal lines appear on a graph.• The x-coordinate changes but the y-coordinate remains constant.• Horizontal lines have the equation y = b.• Slope of all horizontal lines is zero. (Think, y = 0x + b.)• Why is the slope zero? • Suppose you have points (2, 5) and (3, 5). You know they are horizontal because the x-coordinate changes. • Substitute into the slope formula. y2 − y1 5 − 5 0 m= = = x2 − x1 2 − 3 −1 • Zero divided by anything is 0. Therefore, the slope is 0.
36. 36. VERTICAL LINES• Vertical lines go up and down.
37. 37. VERTICAL LINES• Vertical lines go up and down. • The x-value stays the same but the y-value changes.
38. 38. VERTICAL LINES• Vertical lines go up and down. • The x-value stays the same but the y-value changes. • Vertical lines have the equation x = point equation crosses x-axis
39. 39. VERTICAL LINES• Vertical lines go up and down. • The x-value stays the same but the y-value changes. • Vertical lines have the equation x = point equation crosses x-axis • The slope of all vertical lines is undeﬁned.
40. 40. VERTICAL LINES• Vertical lines go up and down. • The x-value stays the same but the y-value changes. • Vertical lines have the equation x = point equation crosses x-axis • The slope of all vertical lines is undeﬁned. • Why is the slope undeﬁned?
41. 41. VERTICAL LINES• Vertical lines go up and down. • The x-value stays the same but the y-value changes. • Vertical lines have the equation x = point equation crosses x-axis • The slope of all vertical lines is undeﬁned. • Why is the slope undeﬁned? • Suppose you have points (1, 5) and (1, 3). You know they are vertical because they have the same x-coordinate.
42. 42. VERTICAL LINES• Vertical lines go up and down. • The x-value stays the same but the y-value changes. • Vertical lines have the equation x = point equation crosses x-axis • The slope of all vertical lines is undeﬁned. • Why is the slope undeﬁned? • Suppose you have points (1, 5) and (1, 3). You know they are vertical because they have the same x-coordinate. • Substitute into the slope formula. y2 − y1 m= x2 − x1
43. 43. VERTICAL LINES• Vertical lines go up and down. • The x-value stays the same but the y-value changes. • Vertical lines have the equation x = point equation crosses x-axis • The slope of all vertical lines is undeﬁned. • Why is the slope undeﬁned? • Suppose you have points (1, 5) and (1, 3). You know they are vertical because they have the same x-coordinate. • Substitute into the slope formula. y2 − y1 5 − 3 m= = x2 − x1 1 − 1
44. 44. VERTICAL LINES• Vertical lines go up and down. • The x-value stays the same but the y-value changes. • Vertical lines have the equation x = point equation crosses x-axis • The slope of all vertical lines is undeﬁned. • Why is the slope undeﬁned? • Suppose you have points (1, 5) and (1, 3). You know they are vertical because they have the same x-coordinate. • Substitute into the slope formula. y2 − y1 5 − 3 2 m= = = x2 − x1 1 − 1 0
45. 45. VERTICAL LINES• Vertical lines go up and down. • The x-value stays the same but the y-value changes. • Vertical lines have the equation x = point equation crosses x-axis • The slope of all vertical lines is undeﬁned. • Why is the slope undeﬁned? • Suppose you have points (1, 5) and (1, 3). You know they are vertical because they have the same x-coordinate. • Substitute into the slope formula. y2 − y1 5 − 3 2 m= = = x2 − x1 1 − 1 0 • Can’t have division by 0. Therefore, the slope is undeﬁned.
46. 46. WRITE THE EQUATIONS ( 2, −5 )
47. 47. WRITE THE EQUATIONS ( 2, −5 )• Write the equation for the vertical line that goes through the above point.
48. 48. WRITE THE EQUATIONS ( 2, −5 )• Write the equation for the vertical line that goes through the above point. • Because the x-coordinate never changes, the equation is x = 2.
49. 49. WRITE THE EQUATIONS ( 2, −5 )• Write the equation for the vertical line that goes through the above point. • Because the x-coordinate never changes, the equation is x = 2.• Writethe equation for the horizontal line that goes through the above point.
50. 50. WRITE THE EQUATIONS ( 2, −5 )• Write the equation for the vertical line that goes through the above point. • Because the x-coordinate never changes, the equation is x = 2.• Writethe equation for the horizontal line that goes through the above point. • Because the y-coordinate never changes, the equation is y = -5.
51. 51. YOU TRY... ( −7, 9 )
52. 52. YOU TRY... ( −7, 9 )• Write the equation for the vertical line that goes through the above point.
53. 53. YOU TRY... ( −7, 9 )• Write the equation for the vertical line that goes through the above point. • Because the x-coordinate never changes, the equation is x = -7.
54. 54. YOU TRY... ( −7, 9 )• Write the equation for the vertical line that goes through the above point. • Because the x-coordinate never changes, the equation is x = -7.• Writethe equation for the horizontal line that goes through the above point.
55. 55. YOU TRY... ( −7, 9 )• Write the equation for the vertical line that goes through the above point. • Because the x-coordinate never changes, the equation is x = -7.• Writethe equation for the horizontal line that goes through the above point. • Because the y-coordinate never changes, the equation is y = 9.
56. 56. STANDARD FORM
57. 57. STANDARD FORM• An equation written in the form Ax + By = C
58. 58. STANDARD FORM• An equation written in the form Ax + By = C• Rules for standard form:
59. 59. STANDARD FORM• An equation written in the form Ax + By = C• Rules for standard form: • A, B, and C must be Integers.
60. 60. STANDARD FORM• An equation written in the form Ax + By = C• Rules for standard form: • A, B, and C must be Integers. •A must be positive.
61. 61. STANDARD FORM• An equation written in the form Ax + By = C• Rules for standard form: • A, B, and C must be Integers. •A must be positive. • Either A OR B can be 0. Both can NOT be 0.
62. 62. STANDARD FORM• An equation written in the form Ax + By = C• Rules for standard form: • A, B, and C must be Integers. •A must be positive. • Either A OR B can be 0. Both can NOT be 0. • As long as A ≠ 0, Ax must be the ﬁrst term.
63. 63. STANDARD FORM• An equation written in the form Ax + By = C• Rules for standard form: Examples • A, B, and C must be Integers. 2x − y = 5 •A must be positive. 5y = −3 • Either A OR B can be 0. Both can NOT be 0. x + 2y = 4 • As long as A ≠ 0, Ax must be the ﬁrst term.
64. 64. STANDARD FORM• An equation written in the form Ax + By = C• Rules for standard form: Examples • A, B, and C must be Integers. 2x − y = 5 •A must be positive. 5y = −3 • Either A OR B can be 0. Both can NOT be 0. x + 2y = 4 • As long as A ≠ 0, Ax must be the ﬁrst term. Non Examples −2x + y = 5 4 2x + 3y = 0.5y = −3.4 7
65. 65. WRITING AN EQUATION IN STANDARD FORMWrite the equation 2y − 1 = ( x + 3) 3in standard form.
66. 66. WRITING AN EQUATION IN STANDARD FORMWrite the equation • Distribute ﬁrst. 2y − 1 = ( x + 3) 3in standard form.
67. 67. WRITING AN EQUATION IN STANDARD FORMWrite the equation • Distribute ﬁrst. 2y − 1 = ( x + 3) 3in standard form. 2 2 y −1= x + ⋅3 3 3
68. 68. WRITING AN EQUATION IN STANDARD FORMWrite the equation • Distribute ﬁrst. 2y − 1 = ( x + 3) • Simplify multiplication. 3in standard form. 2 2 y −1= x + ⋅3 3 3
69. 69. WRITING AN EQUATION IN STANDARD FORMWrite the equation • Distribute ﬁrst. 2y − 1 = ( x + 3) • Simplify multiplication. 3in standard form. 2 2 y −1= x + ⋅3 3 3 2 y −1= x + 2 3
70. 70. WRITING AN EQUATION IN STANDARD FORMWrite the equation • Distribute ﬁrst. 2y − 1 = ( x + 3) • Simplify multiplication. 3in standard form. • Use the properties of equality to get 2 2 the constants on the right and the y −1= x + ⋅3 variables on the left. 3 3 2 y −1= x + 2 3
71. 71. WRITING AN EQUATION IN STANDARD FORMWrite the equation • Distribute ﬁrst. 2y − 1 = ( x + 3) • Simplify multiplication. 3in standard form. • Use the properties of equality to get 2 2 the constants on the right and the y −1= x + ⋅3 variables on the left. 3 3 2 y −1= x + 2 +1 3 +1
72. 72. WRITING AN EQUATION IN STANDARD FORMWrite the equation • Distribute ﬁrst. 2y − 1 = ( x + 3) • Simplify multiplication. 3in standard form. • Use the properties of equality to get 2 2 the constants on the right and the y −1= x + ⋅3 variables on the left. 3 3 2 y −1= x + 2 +1 3 +1 2 2− x − x 3 3
73. 73. WRITING AN EQUATION IN STANDARD FORMWrite the equation • Distribute ﬁrst. 2y − 1 = ( x + 3) • Simplify multiplication. 3in standard form. • Use the properties of equality to get 2 2 the constants on the right and the y −1= x + ⋅3 variables on the left. 3 3 2 y −1= x + 2 +1 3 +1 2 2− x − x 3 3 2 − x+y=3 3
74. 74. WRITING AN EQUATION IN STANDARD FORMWrite the equation • Distribute ﬁrst. 2y − 1 = ( x + 3) • Simplify multiplication. 3in standard form. • Use the properties of equality to get 2 2 the constants on the right and the y −1= x + ⋅3 variables on the left. 3 3 2 • Multiplyby -3 so A is positive and y −1= x + 2 no fractions exist. +1 3 +1 2 2− x − x 3 3 2 − x+y=3 3
75. 75. WRITING AN EQUATION IN STANDARD FORMWrite the equation • Distribute ﬁrst. 2y − 1 = ( x + 3) • Simplify multiplication. 3in standard form. • Use the properties of equality to get 2 2 the constants on the right and the y −1= x + ⋅3 variables on the left. 3 3 2 • Multiplyby -3 so A is positive and y −1= x + 2 no fractions exist. +1 3 +1 2 2 ⎛ 2 ⎞− x − x −3 ⎜ − x ⎟ + −3 ⋅ y = −3 ⋅ 3 3 3 ⎝ 3 ⎠ 2 − x+y=3 3
76. 76. WRITING AN EQUATION IN STANDARD FORMWrite the equation • Distribute ﬁrst. 2y − 1 = ( x + 3) • Simplify multiplication. 3in standard form. • Use the properties of equality to get 2 2 the constants on the right and the y −1= x + ⋅3 variables on the left. 3 3 2 • Multiplyby -3 so A is positive and y −1= x + 2 no fractions exist. +1 3 +1 2 2 ⎛ 2 ⎞− x − x −3 ⎜ − x ⎟ + −3 ⋅ y = −3 ⋅ 3 3 3 ⎝ 3 ⎠ 2 − x+y=3 2x − 3y = −9 3
77. 77. PARALLEL• Parallel lines are 2 lines that never touch. The distance between them remains constant forever.
78. 78. PARALLEL• Parallel lines are 2 lines that never touch. The distance between them remains constant forever.• Symbol used to represent: ∕∕
79. 79. PARALLEL• Parallel lines are 2 lines that never touch. The distance between them remains constant forever.• Symbol used to represent: ∕∕• Real life examples...
80. 80. PARALLEL• Parallel lines are 2 lines that never touch. The distance between them remains constant forever.• Symbol used to represent: ∕∕• Real life examples... • The top of the Berlin Wall and ground in Berlin, Germany.
81. 81. PARALLEL• Parallel lines are 2 lines that never touch. The distance between them remains constant forever.• Symbol used to represent: ∕∕• Real life examples... • The top of the Berlin Wall and ground in Berlin, Germany.
82. 82. PARALLEL• Parallel lines are 2 lines that never touch. The distance between them remains constant forever.• Symbol used to represent: ∕∕• Real life examples... • The top of the Berlin Wall and ground in Berlin, Germany. • The towers of the Tower Bridge in London, England.
83. 83. PARALLEL• Parallel lines are 2 lines that never touch. The distance between them remains constant forever.• Symbol used to represent: ∕∕• Real life examples... • The top of the Berlin Wall and ground in Berlin, Germany. • The towers of the Tower Bridge in London, England.
84. 84. PERPENDICULAR• Perpendicular lines are 2 lines that intersect at a 90 degree angle.
85. 85. PERPENDICULAR• Perpendicular lines are 2 lines that intersect at a 90 degree angle.• Symbol used to represent: ⊥
86. 86. PERPENDICULAR• Perpendicular lines are 2 lines that intersect at a 90 degree angle.• Symbol used to represent: ⊥• Real life examples...
87. 87. PERPENDICULAR• Perpendicular lines are 2 lines that intersect at a 90 degree angle.• Symbol used to represent: ⊥• Real life examples... • The adjacent sides of a picture frame located in the Louvre Museum in Paris, France.
88. 88. PERPENDICULAR• Perpendicular lines are 2 lines that intersect at a 90 degree angle.• Symbol used to represent: ⊥• Real life examples... • The adjacent sides of a picture frame located in the Louvre Museum in Paris, France.
89. 89. PERPENDICULAR• Perpendicular lines are 2 lines that intersect at a 90 degree angle.• Symbol used to represent: ⊥• Real life examples... • The adjacent sides of a picture frame located in the Louvre Museum in Paris, France. • The frame of a doorway in the ruins of Pompeii, Italy.
90. 90. PERPENDICULAR• Perpendicular lines are 2 lines that intersect at a 90 degree angle.• Symbol used to represent: ⊥• Real life examples... • The adjacent sides of a picture frame located in the Louvre Museum in Paris, France. • The frame of a doorway in the ruins of Pompeii, Italy.
91. 91. EXPLORATION ON GRAPHING CALCULATOR
92. 92. EXPLORATION ON GRAPHING CALCULATOR• Parallel lines and Perpendicular lines can be VERY deceiving on a graphing calculator.
93. 93. EXPLORATION ON GRAPHING CALCULATOR• Parallel lines and Perpendicular lines can be VERY deceiving on a graphing calculator.• Enter these 2 equations in y= and graph on a standard window (Zoom - 6:ZStandard) • y1 = .1x - 3 • y2 = .11x + 3
94. 94. EXPLORATION ON GRAPHING CALCULATOR• Parallel lines and Perpendicular lines can be VERY deceiving on a graphing calculator.• Enter these 2 equations in y= and graph on a standard window (Zoom - 6:ZStandard) • y1 = .1x - 3 • y2 = .11x + 3• Are these parallel, perpendicular or neither based on the screen?
95. 95. EXPLORATION ON GRAPHING CALCULATOR• Parallel lines and Perpendicular lines can be VERY deceiving on a graphing calculator.• Enter these 2 equations in y= and graph on a standard window (Zoom - 6:ZStandard) • y1 = .1x - 3 • y2 = .11x + 3• Are these parallel, perpendicular or neither based on the screen?• Do the same with these equations. • y1 = 1.9x - 3 • y2 = -1.2x - 3
96. 96. EXPLORATION ON GRAPHING CALCULATOR• Parallel lines and Perpendicular lines can be VERY deceiving on a graphing calculator.• Enter these 2 equations in y= and graph on a standard window (Zoom - 6:ZStandard) • y1 = .1x - 3 • y2 = .11x + 3• Are these parallel, perpendicular or neither based on the screen?• Do the same with these equations. • y1 = 1.9x - 3 • y2 = -1.2x - 3• Keep your answers because you will need them shortly!
97. 97. PARALLEL LINES• Becausethe distance never changes between parallel lines, their slopes must remain constant or eventually they will cross.
98. 98. PARALLEL LINES• Becausethe distance never changes between parallel lines, their slopes must remain constant or eventually they will cross.• To determine if 2 lines are parallel, check the slope.
99. 99. PARALLEL LINES• Becausethe distance never changes between parallel lines, their slopes must remain constant or eventually they will cross.• To determine if 2 lines are parallel, check the slope. • Same slope, different y-intercept = parallel lines
100. 100. PARALLEL LINES• Becausethe distance never changes between parallel lines, their slopes must remain constant or eventually they will cross.• To determine if 2 lines are parallel, check the slope. • Same slope, different y-intercept = parallel lines Parallel y = 2x − 5 y = 2x + 5
101. 101. PARALLEL LINES• Becausethe distance never changes between parallel lines, their slopes must remain constant or eventually they will cross.• To determine if 2 lines are parallel, check the slope. • Same slope, different y-intercept = parallel lines • Same slope, same y-intercept = SAME line Parallel y = 2x − 5 y = 2x + 5
102. 102. PARALLEL LINES• Becausethe distance never changes between parallel lines, their slopes must remain constant or eventually they will cross.• To determine if 2 lines are parallel, check the slope. • Same slope, different y-intercept = parallel lines • Same slope, same y-intercept = SAME line Same line Parallel (÷ second by 5) y = 2x − 5 y = −x + 3 y = 2x + 5 5y = −5x + 15
103. 103. PARALLEL LINES• Becausethe distance never changes between parallel lines, their slopes must remain constant or eventually they will cross.• To determine if 2 lines are parallel, check the slope. • Same slope, different y-intercept = parallel lines • Same slope, same y-intercept = SAME line • Different slope, same or different y-intercept = intersecting lines Same line Parallel (÷ second by 5) y = 2x − 5 y = −x + 3 y = 2x + 5 5y = −5x + 15
104. 104. PARALLEL LINES• Becausethe distance never changes between parallel lines, their slopes must remain constant or eventually they will cross.• To determine if 2 lines are parallel, check the slope. • Same slope, different y-intercept = parallel lines • Same slope, same y-intercept = SAME line • Different slope, same or different y-intercept = intersecting lines Same line Parallel (÷ second by 5) NOT parallel y = 2x − 5 y = −x + 3 y = 4x + 7 y = 2x + 5 5y = −5x + 15 y = 3x + 7
105. 105. BACK TO THE EXPLORATION• What did you say about the graph of these equations? Parallel, Perpendicular, or Neither? y1 = .1x −3 y2 = .11x + 3
106. 106. BACK TO THE EXPLORATION• What did you say about the graph of these equations? Parallel, Perpendicular, or Neither? y1 = .1x −3• Do you agree with your original answer? y2 = .11x + 3
107. 107. BACK TO THE EXPLORATION• What did you say about the graph of these equations? Parallel, Perpendicular, or Neither? y1 = .1x −3• Do you agree with your original answer? y2 = .11x + 3• Why do you still agree or why did you change you mind?
108. 108. BACK TO THE EXPLORATION• What did you say about the graph of these equations? Parallel, Perpendicular, or Neither? y1 = .1x −3• Do you agree with your original answer? y2 = .11x + 3• Why do you still agree or why did you change you mind?• These equations look parallel on the graphing calculator but what do you notice about the slopes?
109. 109. BACK TO THE EXPLORATION• What did you say about the graph of these equations? Parallel, Perpendicular, or Neither? y1 = .1x −3• Do you agree with your original answer? y2 = .11x + 3• Why do you still agree or why did you change you mind?• These equations look parallel on the graphing calculator but what do you notice about the slopes?• The slopes are different. They are very close, which is why they appear parallel on the graphing calculator.
110. 110. BACK TO THE EXPLORATION• What did you say about the graph of these equations? Parallel, Perpendicular, or Neither? y1 = .1x −3• Do you agree with your original answer? y2 = .11x + 3• Why do you still agree or why did you change you mind?• These equations look parallel on the graphing calculator but what do you notice about the slopes?• The slopes are different. They are very close, which is why they appear parallel on the graphing calculator.• Change your window settings to Xmin=-1100, Xmax=5, Ymin=-100, Ymax=5 and graph the 2 equations again.
111. 111. BACK TO THE EXPLORATION• What did you say about the graph of these equations? Parallel, Perpendicular, or Neither? y1 = .1x −3• Do you agree with your original answer? y2 = .11x + 3• Why do you still agree or why did you change you mind?• These equations look parallel on the graphing calculator but what do you notice about the slopes?• The slopes are different. They are very close, which is why they appear parallel on the graphing calculator.• Change your window settings to Xmin=-1100, Xmax=5, Ymin=-100, Ymax=5 and graph the 2 equations again.• Can you see the lines intersect now?
112. 112. BACK TO THE EXPLORATION• What did you say about the graph of these equations? Parallel, Perpendicular, or Neither? y1 = .1x −3• Do you agree with your original answer? y2 = .11x + 3• Why do you still agree or why did you change you mind?• These equations look parallel on the graphing calculator but what do you notice about the slopes?• The slopes are different. They are very close, which is why they appear parallel on the graphing calculator.• Change your window settings to Xmin=-1100, Xmax=5, Ymin=-100, Ymax=5 and graph the 2 equations again.• Can you see the lines intersect now?• The Window setting is crucial to “seeing” if the equations are parallel. It is easier to determine parallel lines by comparing the slopes.
113. 113. PERPENDICULAR LINES• Stand up nice and tall. What do you notice about your body in relation to the ﬂoor? Are you parallel or perpendicular with the ﬂoor?
114. 114. PERPENDICULAR LINES• Stand up nice and tall. What do you notice about your body in relation to the ﬂoor? Are you parallel or perpendicular with the ﬂoor?• Foryou to be parallel, you must lay your body on the ﬂoor. (This change is called a “rotation” in math.) Standing you meet the ﬂoor at a 90 degree angle so the “rotation” would be 90 degrees for you to be parallel to the ﬂoor.
115. 115. PERPENDICULAR LINES• Stand up nice and tall. What do you notice about your body in relation to the ﬂoor? Are you parallel or perpendicular with the ﬂoor?• Foryou to be parallel, you must lay your body on the ﬂoor. (This change is called a “rotation” in math.) Standing you meet the ﬂoor at a 90 degree angle so the “rotation” would be 90 degrees for you to be parallel to the ﬂoor.• Draw a coordinate plane on your paper. Place 2 pencils on the graph so they cross with a 90 degree angle. Don’t place them vertical and horizontal on the coordinate plane because these are a special case but do move them around maintaining the 90 degree angle.
116. 116. PERPENDICULAR LINES• Stand up nice and tall. What do you notice about your body in relation to the ﬂoor? Are you parallel or perpendicular with the ﬂoor?• Foryou to be parallel, you must lay your body on the ﬂoor. (This change is called a “rotation” in math.) Standing you meet the ﬂoor at a 90 degree angle so the “rotation” would be 90 degrees for you to be parallel to the ﬂoor.• Draw a coordinate plane on your paper. Place 2 pencils on the graph so they cross with a 90 degree angle. Don’t place them vertical and horizontal on the coordinate plane because these are a special case but do move them around maintaining the 90 degree angle.• What did you notice about the slopes of the pencils? Both positive? Both negative? One of each? ...
117. 117. PERPENDICULAR LINES (CONTINUED)• Noticeone is always positive and one is always negative.
118. 118. PERPENDICULAR LINES (CONTINUED)• Noticeone is always positive and one is always negative.• Perpendicular lines have slopes that are negative reciprocals.
119. 119. PERPENDICULAR LINES (CONTINUED)• Noticeone is always positive and one is always negative.• Perpendicular lines have slopes that are negative reciprocals.• Negative reciprocals mean the slopes have opposite signs and the number is ﬂipped.
120. 120. PERPENDICULAR LINES (CONTINUED)• Noticeone is always positive and one is always negative. Perpendicular y = 2x − 3• Perpendicular lines have slopes that are negative reciprocals. 1 y=− x+5• Negative reciprocals mean the slopes 2 have opposite signs and the number is ﬂipped.• Such as -1/2 and 2 are negative reciprocals.
121. 121. PERPENDICULAR LINES (CONTINUED)• Noticeone is always positive and one is always negative. Perpendicular y = 2x − 3• Perpendicular lines have slopes that are negative reciprocals. 1 y=− x+5• Negative reciprocals mean the slopes 2 have opposite signs and the number is ﬂipped.• Such as -1/2 and 2 are negative reciprocals.•3 and -3 are opposite but NOT reciprocals. NOT ⊥ y = 3x + 7 y = −3x + 7
122. 122. PERPENDICULAR LINES (CONTINUED)• Noticeone is always positive and one is always negative. Perpendicular y = 2x − 3• Perpendicular lines have slopes that are negative reciprocals. 1 y=− x+5• Negative reciprocals mean the slopes 2 have opposite signs and the number is ﬂipped.• Such as -1/2 and 2 are negative reciprocals. Perpendicular•3 and -3 are opposite 3 but NOT reciprocals. NOT ⊥ y = x −1 4• 3/4and -4/3 are negative y = 3x + 7 reciprocals. Can have 4 y = −3x + 7 y = − x −1 the same y-intercept. 3
123. 123. BACK TO THE EXPLORATION• What did you say about the graph of these equations? Parallel, Perpendicular, or Neither? y1 = 1.9x − 3 y2 = −1.2x − 3
124. 124. BACK TO THE EXPLORATION• What did you say about the graph of these equations? Parallel, Perpendicular, or Neither? y1 = 1.9x − 3• Do you agree with your answer? You may want to change these to fractions to make a better determination. y2 = −1.2x − 3
125. 125. BACK TO THE EXPLORATION• What did you say about the graph of these equations? Parallel, Perpendicular, or Neither? y1 = 1.9x − 3• Do you agree with your answer? You may want to change these to fractions to make a better determination. y2 = −1.2x − 3• Why do you still agree or why did you change you mind?
126. 126. BACK TO THE EXPLORATION• What did you say about the graph of these equations? Parallel, Perpendicular, or Neither? y1 = 1.9x − 3• Do you agree with your answer? You may want to change these to fractions to make a better determination. y2 = −1.2x − 3• Why do you still agree or why did you change you mind?• These equations look perpendicular on the graphing calculator but what do you notice about the slopes?
127. 127. BACK TO THE EXPLORATION• What did you say about the graph of these equations? Parallel, Perpendicular, or Neither? y1 = 1.9x − 3• Do you agree with your answer? You may want to change these to fractions to make a better determination. y2 = −1.2x − 3• Why do you still agree or why did you change you mind?• These equations look perpendicular on the graphing calculator but what do you notice about the slopes?• The slopes are NOT negative reciprocals! (1.9 = 19/10 and -1.2 = -6/5) They are very close, which is why they appear perpendicular on the graphing calculator.
128. 128. BACK TO THE EXPLORATION• What did you say about the graph of these equations? Parallel, Perpendicular, or Neither? y1 = 1.9x − 3• Do you agree with your answer? You may want to change these to fractions to make a better determination. y2 = −1.2x − 3• Why do you still agree or why did you change you mind?• These equations look perpendicular on the graphing calculator but what do you notice about the slopes?• The slopes are NOT negative reciprocals! (1.9 = 19/10 and -1.2 = -6/5) They are very close, which is why they appear perpendicular on the graphing calculator.• Change your window settings to Xmin=-5, Xmax=5 and graph the 2 equations again.
129. 129. BACK TO THE EXPLORATION• What did you say about the graph of these equations? Parallel, Perpendicular, or Neither? y1 = 1.9x − 3• Do you agree with your answer? You may want to change these to fractions to make a better determination. y2 = −1.2x − 3• Why do you still agree or why did you change you mind?• These equations look perpendicular on the graphing calculator but what do you notice about the slopes?• The slopes are NOT negative reciprocals! (1.9 = 19/10 and -1.2 = -6/5) They are very close, which is why they appear perpendicular on the graphing calculator.• Change your window settings to Xmin=-5, Xmax=5 and graph the 2 equations again.• Do they appear perpendicular?
130. 130. BACK TO THE EXPLORATION• What did you say about the graph of these equations? Parallel, Perpendicular, or Neither? y1 = 1.9x − 3• Do you agree with your answer? You may want to change these to fractions to make a better determination. y2 = −1.2x − 3• Why do you still agree or why did you change you mind?• These equations look perpendicular on the graphing calculator but what do you notice about the slopes?• The slopes are NOT negative reciprocals! (1.9 = 19/10 and -1.2 = -6/5) They are very close, which is why they appear perpendicular on the graphing calculator.• Change your window settings to Xmin=-5, Xmax=5 and graph the 2 equations again.• Do they appear perpendicular?• Like for parallel lines, the Window setting is crucial to “seeing” if the equations are perpendicular. It is easier to determine perpendicular lines by comparing the slopes.
131. 131. FINDING ∕∕ OR ⊥ SLOPE2x − y = 5
132. 132. FINDING ∕∕ OR ⊥ SLOPE • Always put equation in slope-2x − y = 5 intercept form (y = mx + b) before determining the slope of the given line.
133. 133. FINDING ∕∕ OR ⊥ SLOPE • Always put equation in slope- 2x − y = 5 intercept form (y = mx + b)−2x −2x before determining the slope of the given line.
134. 134. FINDING ∕∕ OR ⊥ SLOPE • Always put equation in slope- 2x − y = 5 intercept form (y = mx + b)−2x −2x before determining the slope of the given line. −y = −2x + 5
135. 135. FINDING ∕∕ OR ⊥ SLOPE • Always put equation in slope- 2x − y = 5 intercept form (y = mx + b)−2x −2x before determining the slope of the given line. −y = −2x + 5 −1 −1
136. 136. FINDING ∕∕ OR ⊥ SLOPE • Always put equation in slope- 2x − y = 5 intercept form (y = mx + b)−2x −2x before determining the slope of the given line. −y = −2x + 5 −1 −1 y = 2x − 5
137. 137. FINDING ∕∕ OR ⊥ SLOPE • Always put equation in slope- 2x − y = 5 intercept form (y = mx + b)−2x −2x before determining the slope of the given line. −y = −2x + 5 • Identify the slope. −1 −1 y = 2x − 5
138. 138. FINDING ∕∕ OR ⊥ SLOPE • Always put equation in slope- 2x − y = 5 intercept form (y = mx + b)−2x −2x before determining the slope of the given line. −y = −2x + 5 • Identify the slope. −1 −1 y = 2x − 5 m=2
139. 139. FINDING ∕∕ OR ⊥ SLOPE • Always put equation in slope- 2x − y = 5 intercept form (y = mx + b)−2x −2x before determining the slope of the given line. −y = −2x + 5 • Identify the slope. −1 −1 • Parallel slopes are the same. y = 2x − 5 m=2
140. 140. FINDING ∕∕ OR ⊥ SLOPE • Always put equation in slope- 2x − y = 5 intercept form (y = mx + b) −2x −2x before determining the slope of the given line. −y = −2x + 5 • Identify the slope. −1 −1 • Parallel slopes are the same. y = 2x − 5 m=2Parallel slope m=2
141. 141. FINDING ∕∕ OR ⊥ SLOPE • Always put equation in slope- 2x − y = 5 intercept form (y = mx + b) −2x −2x before determining the slope of the given line. −y = −2x + 5 • Identify the slope. −1 −1 • Parallel slopes are the same. y = 2x − 5 m=2 • Perpendicular slopes are negative reciprocals.Parallel slope m=2
142. 142. FINDING ∕∕ OR ⊥ SLOPE • Always put equation in slope- 2x − y = 5 intercept form (y = mx + b) −2x −2x before determining the slope of the given line. −y = −2x + 5 • Identify the slope. −1 −1 • Parallel slopes are the same. y = 2x − 5 m=2 • Perpendicular slopes are negative reciprocals. Perpendicular slopeParallel slope 1 m=2 m=− 2
143. 143. YOUR TURN TO FIND ∕∕ & ⊥ SLOPE...−3x + 2y = 7
144. 144. YOUR TURN TO FIND ∕∕ & ⊥ SLOPE...−3x + 2y = 7 • Always put equation in slope- intercept form (y = mx + b) before determining the slope of the given line.
145. 145. YOUR TURN TO FIND ∕∕ & ⊥ SLOPE...−3x + 2y = 7 • Always put equation in slope- intercept form (y = mx + b)+3x +3x before determining the slope of the given line.
146. 146. YOUR TURN TO FIND ∕∕ & ⊥ SLOPE...−3x + 2y = 7 • Always put equation in slope- intercept form (y = mx + b)+3x +3x before determining the slope of the given line. 2y = 3x + 7
147. 147. YOUR TURN TO FIND ∕∕ & ⊥ SLOPE...−3x + 2y = 7 • Always put equation in slope- intercept form (y = mx + b)+3x +3x before determining the slope of the given line. 2y = 3x + 7 2 2
148. 148. YOUR TURN TO FIND ∕∕ & ⊥ SLOPE...−3x + 2y = 7 • Always put equation in slope- intercept form (y = mx + b)+3x +3x before determining the slope of the given line. 2y = 3x + 7 2 2 3 7 y= x+ 2 2
149. 149. YOUR TURN TO FIND ∕∕ & ⊥ SLOPE...−3x + 2y = 7 • Always put equation in slope- intercept form (y = mx + b)+3x +3x before determining the slope of the given line. 2y = 3x + 7 2 2 • Identify the slope. 3 7 y= x+ 2 2
150. 150. YOUR TURN TO FIND ∕∕ & ⊥ SLOPE...−3x + 2y = 7 • Always put equation in slope- intercept form (y = mx + b)+3x +3x before determining the slope of the given line. 2y = 3x + 7 2 2 • Identify the slope. 3 7 y= x+ 2 2 3 m= 2
151. 151. YOUR TURN TO FIND ∕∕ & ⊥ SLOPE...−3x + 2y = 7 • Always put equation in slope- intercept form (y = mx + b)+3x +3x before determining the slope of the given line. 2y = 3x + 7 2 2 • Identify the slope. 3 7 • Parallel slopes are the same. y= x+ 2 2 3 m= 2 Parallel slope 3 m= 2
152. 152. YOUR TURN TO FIND ∕∕ & ⊥ SLOPE...−3x + 2y = 7 • Always put equation in slope- intercept form (y = mx + b)+3x +3x before determining the slope of the given line. 2y = 3x + 7 2 2 • Identify the slope. 3 7 • Parallel slopes are the same. y= x+ 2 2 • Perpendicular slopes are negative 3 m= reciprocals. 2 Perpendicular slope Parallel slope 3 2 m= m=− 2 3
153. 153. WRITING ∕∕ EQUATIONWrite the slope-intercept form of theequation parallel to y = -3x + 4,which passes through the point (2, -5).
154. 154. WRITING ∕∕ EQUATIONWrite the slope-intercept form of the • Always determine the slope of the givenequation parallel to y = -3x + 4, equation. (May need to put in slope-which passes through the point (2, -5). intercept form.)
155. 155. WRITING ∕∕ EQUATIONWrite the slope-intercept form of the • Always determine the slope of the givenequation parallel to y = -3x + 4, equation. (May need to put in slope-which passes through the point (2, -5). intercept form.) m = −3
156. 156. WRITING ∕∕ EQUATIONWrite the slope-intercept form of the • Always determine the slope of the givenequation parallel to y = -3x + 4, equation. (May need to put in slope-which passes through the point (2, -5). intercept form.) m = −3 • Parallel slopes are the same so use m = -3.
157. 157. WRITING ∕∕ EQUATIONWrite the slope-intercept form of the • Always determine the slope of the givenequation parallel to y = -3x + 4, equation. (May need to put in slope-which passes through the point (2, -5). intercept form.) m = −3 • Parallel slopes are the same so use m = -3. • Use point-slope formula to ﬁnd parallel equation.
158. 158. WRITING ∕∕ EQUATIONWrite the slope-intercept form of the • Always determine the slope of the givenequation parallel to y = -3x + 4, equation. (May need to put in slope-which passes through the point (2, -5). intercept form.) m = −3 • Parallel slopes are the same so use m = -3. y − y1 = m ( x − x1 ) • Use point-slope formula to ﬁnd parallel equation.
159. 159. WRITING ∕∕ EQUATIONWrite the slope-intercept form of the • Always determine the slope of the givenequation parallel to y = -3x + 4, equation. (May need to put in slope-which passes through the point (2, -5). intercept form.) m = −3 • Parallel slopes are the same so use m = -3. y − y1 = m ( x − x1 ) • Use point-slope formula to ﬁnd parallel equation. • Substitute given point and parallel slope.
160. 160. WRITING ∕∕ EQUATIONWrite the slope-intercept form of the • Always determine the slope of the givenequation parallel to y = -3x + 4, equation. (May need to put in slope-which passes through the point (2, -5). intercept form.) m = −3 • Parallel slopes are the same so use m = -3. y − y1 = m ( x − x1 ) • Use point-slope formula to ﬁnd parallel y − ( −5 ) = −3( x − 2 ) equation. • Substitute given point and parallel slope.
161. 161. WRITING ∕∕ EQUATIONWrite the slope-intercept form of the • Always determine the slope of the givenequation parallel to y = -3x + 4, equation. (May need to put in slope-which passes through the point (2, -5). intercept form.) m = −3 • Parallel slopes are the same so use m = -3. y − y1 = m ( x − x1 ) • Use point-slope formula to ﬁnd parallel y − ( −5 ) = −3( x − 2 ) equation. • Substitute given point and parallel slope. • Put equation in slope-intercept form.
162. 162. WRITING ∕∕ EQUATIONWrite the slope-intercept form of the • Always determine the slope of the givenequation parallel to y = -3x + 4, equation. (May need to put in slope-which passes through the point (2, -5). intercept form.) m = −3 • Parallel slopes are the same so use m = -3. y − y1 = m ( x − x1 ) • Use point-slope formula to ﬁnd parallel y − ( −5 ) = −3( x − 2 ) equation. y + 5 = −3x + 6 • Substitute given point and parallel slope. • Put equation in slope-intercept form.
163. 163. WRITING ∕∕ EQUATIONWrite the slope-intercept form of the • Always determine the slope of the givenequation parallel to y = -3x + 4, equation. (May need to put in slope-which passes through the point (2, -5). intercept form.) m = −3 • Parallel slopes are the same so use m = -3. y − y1 = m ( x − x1 ) • Use point-slope formula to ﬁnd parallel y − ( −5 ) = −3( x − 2 ) equation. y + 5 = −3x + 6 • Substitute given point and parallel slope. −5 −5 • Put equation in slope-intercept form.
164. 164. WRITING ∕∕ EQUATIONWrite the slope-intercept form of the • Always determine the slope of the givenequation parallel to y = -3x + 4, equation. (May need to put in slope-which passes through the point (2, -5). intercept form.) m = −3 • Parallel slopes are the same so use m = -3. y − y1 = m ( x − x1 ) • Use point-slope formula to ﬁnd parallel y − ( −5 ) = −3( x − 2 ) equation. y + 5 = −3x + 6 • Substitute given point and parallel slope. −5 −5 • Put equation in slope-intercept form. y = −3x + 1
165. 165. WRITING ∕∕ EQUATIONWrite the slope-intercept form of the • Always determine the slope of the givenequation parallel to y = -3x + 4, equation. (May need to put in slope-which passes through the point (2, -5). intercept form.) m = −3 • Parallel slopes are the same so use m = -3. y − y1 = m ( x − x1 ) • Use point-slope formula to ﬁnd parallel y − ( −5 ) = −3( x − 2 ) equation. y + 5 = −3x + 6 • Substitute given point and parallel slope. −5 −5 • Put equation in slope-intercept form. • Always check your equation to ensure it y = −3x + 1 makes sense. The lines are parallel so the slopes must be the same (they are) and y-intercepts different (they are).
166. 166. YOUR TURN...Write the slope-intercept form of theequation parallel to y = 4x + 7, whichpasses through the point (-3, 8).
167. 167. YOUR TURN...Write the slope-intercept form of the • Determine the slope of the givenequation parallel to y = 4x + 7, which equation. (No need to put in slope-passes through the point (-3, 8). intercept form here.)
168. 168. YOUR TURN...Write the slope-intercept form of the • Determine the slope of the givenequation parallel to y = 4x + 7, which equation. (No need to put in slope-passes through the point (-3, 8). intercept form here.) m=4
169. 169. YOUR TURN...Write the slope-intercept form of the • Determine the slope of the givenequation parallel to y = 4x + 7, which equation. (No need to put in slope-passes through the point (-3, 8). intercept form here.) m=4 • Parallel slopes are the same so use m = 4.
170. 170. YOUR TURN...Write the slope-intercept form of the • Determine the slope of the givenequation parallel to y = 4x + 7, which equation. (No need to put in slope-passes through the point (-3, 8). intercept form here.) m=4 • Parallel slopes are the same so use m = 4. • Use point-slope formula to ﬁnd parallel equation.
171. 171. YOUR TURN...Write the slope-intercept form of the • Determine the slope of the givenequation parallel to y = 4x + 7, which equation. (No need to put in slope-passes through the point (-3, 8). intercept form here.) m=4 • Parallel slopes are the same so use m = 4. y − y1 = m ( x − x1 ) • Use point-slope formula to ﬁnd parallel equation.
172. 172. YOUR TURN...Write the slope-intercept form of the • Determine the slope of the givenequation parallel to y = 4x + 7, which equation. (No need to put in slope-passes through the point (-3, 8). intercept form here.) m=4 • Parallel slopes are the same so use m = 4. y − y1 = m ( x − x1 ) • Use point-slope formula to ﬁnd parallel equation. • Substitute given point and parallel slope.
173. 173. YOUR TURN...Write the slope-intercept form of the • Determine the slope of the givenequation parallel to y = 4x + 7, which equation. (No need to put in slope-passes through the point (-3, 8). intercept form here.) m=4 • Parallel slopes are the same so use m = 4. y − y1 = m ( x − x1 ) • Use point-slope formula to ﬁnd parallel y − 8 = 4 ( x − ( −3)) equation. • Substitute given point and parallel slope.
174. 174. YOUR TURN...Write the slope-intercept form of the • Determine the slope of the givenequation parallel to y = 4x + 7, which equation. (No need to put in slope-passes through the point (-3, 8). intercept form here.) m=4 • Parallel slopes are the same so use m = 4. y − y1 = m ( x − x1 ) • Use point-slope formula to ﬁnd parallel y − 8 = 4 ( x − ( −3)) equation. • Substitute given point and parallel slope. • Put equation in slope-intercept form.
175. 175. YOUR TURN...Write the slope-intercept form of the • Determine the slope of the givenequation parallel to y = 4x + 7, which equation. (No need to put in slope-passes through the point (-3, 8). intercept form here.) m=4 • Parallel slopes are the same so use m = 4. y − y1 = m ( x − x1 ) • Use point-slope formula to ﬁnd parallel y − 8 = 4 ( x − ( −3)) equation. y − 8 = 4 ( x + 3) • Substitute given point and parallel slope. • Put equation in slope-intercept form.
176. 176. YOUR TURN...Write the slope-intercept form of the • Determine the slope of the givenequation parallel to y = 4x + 7, which equation. (No need to put in slope-passes through the point (-3, 8). intercept form here.) m=4 • Parallel slopes are the same so use m = 4. y − y1 = m ( x − x1 ) • Use point-slope formula to ﬁnd parallel y − 8 = 4 ( x − ( −3)) equation. y − 8 = 4 ( x + 3) • Substitute given point and parallel slope. y − 8 = 4x + 12 • Put equation in slope-intercept form.
177. 177. YOUR TURN...Write the slope-intercept form of the • Determine the slope of the givenequation parallel to y = 4x + 7, which equation. (No need to put in slope-passes through the point (-3, 8). intercept form here.) m=4 • Parallel slopes are the same so use m = 4. y − y1 = m ( x − x1 ) • Use point-slope formula to ﬁnd parallel y − 8 = 4 ( x − ( −3)) equation. y − 8 = 4 ( x + 3) • Substitute given point and parallel slope. y − 8 = 4x + 12 • Put equation in slope-intercept form. +8 +8
178. 178. YOUR TURN...Write the slope-intercept form of the • Determine the slope of the givenequation parallel to y = 4x + 7, which equation. (No need to put in slope-passes through the point (-3, 8). intercept form here.) m=4 • Parallel slopes are the same so use m = 4. y − y1 = m ( x − x1 ) • Use point-slope formula to ﬁnd parallel y − 8 = 4 ( x − ( −3)) equation. y − 8 = 4 ( x + 3) • Substitute given point and parallel slope. y − 8 = 4x + 12 • Put equation in slope-intercept form. +8 +8 y = 4x + 20
179. 179. YOUR TURN...Write the slope-intercept form of the • Determine the slope of the givenequation parallel to y = 4x + 7, which equation. (No need to put in slope-passes through the point (-3, 8). intercept form here.) m=4 • Parallel slopes are the same so use m = 4. y − y1 = m ( x − x1 ) • Use point-slope formula to ﬁnd parallel y − 8 = 4 ( x − ( −3)) equation. y − 8 = 4 ( x + 3) • Substitute given point and parallel slope. y − 8 = 4x + 12 • Put equation in slope-intercept form. +8 +8 • Check that equation makes sense. The lines are parallel so the slopes must be the same (they are) and y-intercepts y = 4x + 20 different (they are).
180. 180. WRITING ⊥ EQUATIONWrite the slope-intercept form of theequation perpendicular to y = -5x + 2,which passes through the point (10, 3).
181. 181. WRITING ⊥ EQUATIONWrite the slope-intercept form of the • Always determine the slope of the givenequation perpendicular to y = -5x + 2, equation. (May need to put in slope-which passes through the point (10, 3). intercept form.)
182. 182. WRITING ⊥ EQUATIONWrite the slope-intercept form of the • Always determine the slope of the givenequation perpendicular to y = -5x + 2, equation. (May need to put in slope-which passes through the point (10, 3). intercept form.) m = −5
183. 183. WRITING ⊥ EQUATIONWrite the slope-intercept form of the • Always determine the slope of the givenequation perpendicular to y = -5x + 2, equation. (May need to put in slope-which passes through the point (10, 3). intercept form.) m = −5 • Perpendicular slopes are negative reciprocals.
184. 184. WRITING ⊥ EQUATIONWrite the slope-intercept form of the • Always determine the slope of the givenequation perpendicular to y = -5x + 2, equation. (May need to put in slope-which passes through the point (10, 3). intercept form.) m = −5 1 • Perpendicular slopes are negative m⊥ = 5 reciprocals.
185. 185. WRITING ⊥ EQUATIONWrite the slope-intercept form of the • Always determine the slope of the givenequation perpendicular to y = -5x + 2, equation. (May need to put in slope-which passes through the point (10, 3). intercept form.) m = −5 1 • Perpendicular slopes are negative m⊥ = 5 reciprocals. • Use point-slope formula to ﬁnd perpendicular equation.
186. 186. WRITING ⊥ EQUATIONWrite the slope-intercept form of the • Always determine the slope of the givenequation perpendicular to y = -5x + 2, equation. (May need to put in slope-which passes through the point (10, 3). intercept form.) m = −5 1 • Perpendicular slopes are negative m⊥ = 5 reciprocals. y − y1 = m ( x − x1 ) • Use point-slope formula to ﬁnd perpendicular equation.
187. 187. WRITING ⊥ EQUATIONWrite the slope-intercept form of the • Always determine the slope of the givenequation perpendicular to y = -5x + 2, equation. (May need to put in slope-which passes through the point (10, 3). intercept form.) m = −5 1 • Perpendicular slopes are negative m⊥ = 5 reciprocals. y − y1 = m ( x − x1 ) • Use point-slope formula to ﬁnd perpendicular equation. • Substitute given point and perpendicular slope.
188. 188. WRITING ⊥ EQUATIONWrite the slope-intercept form of the • Always determine the slope of the givenequation perpendicular to y = -5x + 2, equation. (May need to put in slope-which passes through the point (10, 3). intercept form.) m = −5 1 • Perpendicular slopes are negative m⊥ = 5 reciprocals. y − y1 = m ( x − x1 ) • Use point-slope formula to ﬁnd 1 perpendicular equation. y − 3 = ( x − 10 ) 5 • Substitute given point and perpendicular slope.
189. 189. WRITING ⊥ EQUATIONWrite the slope-intercept form of the • Always determine the slope of the givenequation perpendicular to y = -5x + 2, equation. (May need to put in slope-which passes through the point (10, 3). intercept form.) m = −5 1 • Perpendicular slopes are negative m⊥ = 5 reciprocals. y − y1 = m ( x − x1 ) • Use point-slope formula to ﬁnd 1 perpendicular equation. y − 3 = ( x − 10 ) 5 • Substitute given point and perpendicular slope. • Put equation in slope-intercept form.
190. 190. WRITING ⊥ EQUATIONWrite the slope-intercept form of the • Always determine the slope of the givenequation perpendicular to y = -5x + 2, equation. (May need to put in slope-which passes through the point (10, 3). intercept form.) m = −5 1 • Perpendicular slopes are negative m⊥ = 5 reciprocals. y − y1 = m ( x − x1 ) • Use point-slope formula to ﬁnd 1 perpendicular equation. y − 3 = ( x − 10 ) 5 • Substitute given point and perpendicular 1 slope. y−3= x−2 5 • Put equation in slope-intercept form.
191. 191. WRITING ⊥ EQUATIONWrite the slope-intercept form of the • Always determine the slope of the givenequation perpendicular to y = -5x + 2, equation. (May need to put in slope-which passes through the point (10, 3). intercept form.) m = −5 1 • Perpendicular slopes are negative m⊥ = 5 reciprocals. y − y1 = m ( x − x1 ) • Use point-slope formula to ﬁnd 1 perpendicular equation. y − 3 = ( x − 10 ) 5 • Substitute given point and perpendicular 1 slope. y−3= x−2 +3 5 +3 • Put equation in slope-intercept form.
192. 192. WRITING ⊥ EQUATIONWrite the slope-intercept form of the • Always determine the slope of the givenequation perpendicular to y = -5x + 2, equation. (May need to put in slope-which passes through the point (10, 3). intercept form.) m = −5 1 • Perpendicular slopes are negative m⊥ = 5 reciprocals. y − y1 = m ( x − x1 ) • Use point-slope formula to ﬁnd 1 perpendicular equation. y − 3 = ( x − 10 ) 5 • Substitute given point and perpendicular 1 slope. y−3= x−2 +3 5 +3 • Put equation in slope-intercept form. 1 y = x +1 5
193. 193. WRITING ⊥ EQUATIONWrite the slope-intercept form of the • Always determine the slope of the givenequation perpendicular to y = -5x + 2, equation. (May need to put in slope-which passes through the point (10, 3). intercept form.) m = −5 1 • Perpendicular slopes are negative m⊥ = 5 reciprocals. y − y1 = m ( x − x1 ) • Use point-slope formula to ﬁnd 1 perpendicular equation. y − 3 = ( x − 10 ) 5 • Substitute given point and perpendicular 1 slope. y−3= x−2 +3 5 +3 • Put equation in slope-intercept form. • Always check your equation to ensure it 1 makes sense. The lines are perpendicular y = x +1 so the slopes must be negative 5 reciprocals (they are).
194. 194. YOUR TURN...Write the slope-intercept form of theequation perpendicular to y = 3x - 1,which passes through the point (6, 9).
195. 195. YOUR TURN...Write the slope-intercept form of the • Determine the slope of the givenequation perpendicular to y = 3x - 1, equation. (No need to put in slope-which passes through the point (6, 9). intercept form here.)
196. 196. YOUR TURN...Write the slope-intercept form of the • Determine the slope of the givenequation perpendicular to y = 3x - 1, equation. (No need to put in slope-which passes through the point (6, 9). intercept form here.) m=3
197. 197. YOUR TURN...Write the slope-intercept form of the • Determine the slope of the givenequation perpendicular to y = 3x - 1, equation. (No need to put in slope-which passes through the point (6, 9). intercept form here.) m=3 • Perpendicular slopes are negative reciprocals.
198. 198. YOUR TURN...Write the slope-intercept form of the • Determine the slope of the givenequation perpendicular to y = 3x - 1, equation. (No need to put in slope-which passes through the point (6, 9). intercept form here.) m=3 1 m⊥ = − • Perpendicular slopes are negative 3 reciprocals.
199. 199. YOUR TURN...Write the slope-intercept form of the • Determine the slope of the givenequation perpendicular to y = 3x - 1, equation. (No need to put in slope-which passes through the point (6, 9). intercept form here.) m=3 1 m⊥ = − • Perpendicular slopes are negative 3 reciprocals. • Use point-slope formula to ﬁnd perpendicular equation.
200. 200. YOUR TURN...Write the slope-intercept form of the • Determine the slope of the givenequation perpendicular to y = 3x - 1, equation. (No need to put in slope-which passes through the point (6, 9). intercept form here.) m=3 1 m⊥ = − • Perpendicular slopes are negative 3 reciprocals. y − y1 = m ( x − x1 ) • Use point-slope formula to ﬁnd perpendicular equation.
201. 201. YOUR TURN...Write the slope-intercept form of the • Determine the slope of the givenequation perpendicular to y = 3x - 1, equation. (No need to put in slope-which passes through the point (6, 9). intercept form here.) m=3 1 m⊥ = − • Perpendicular slopes are negative 3 reciprocals. y − y1 = m ( x − x1 ) • Use point-slope formula to ﬁnd perpendicular equation. • Substitute given point and perpendicular slope.
202. 202. YOUR TURN...Write the slope-intercept form of the • Determine the slope of the givenequation perpendicular to y = 3x - 1, equation. (No need to put in slope-which passes through the point (6, 9). intercept form here.) m=3 1 m⊥ = − • Perpendicular slopes are negative 3 reciprocals. y − y1 = m ( x − x1 ) • Use point-slope formula to ﬁnd 1 perpendicular equation. y − 9 = − ( x − 6) 3 • Substitute given point and perpendicular slope.
203. 203. YOUR TURN...Write the slope-intercept form of the • Determine the slope of the givenequation perpendicular to y = 3x - 1, equation. (No need to put in slope-which passes through the point (6, 9). intercept form here.) m=3 1 m⊥ = − • Perpendicular slopes are negative 3 reciprocals. y − y1 = m ( x − x1 ) • Use point-slope formula to ﬁnd 1 perpendicular equation. y − 9 = − ( x − 6) 3 • Substitute given point and perpendicular slope. • Put equation in slope-intercept form.
204. 204. YOUR TURN...Write the slope-intercept form of the • Determine the slope of the givenequation perpendicular to y = 3x - 1, equation. (No need to put in slope-which passes through the point (6, 9). intercept form here.) m=3 1 m⊥ = − • Perpendicular slopes are negative 3 reciprocals. y − y1 = m ( x − x1 ) • Use point-slope formula to ﬁnd 1 perpendicular equation. y − 9 = − ( x − 6) 3 • Substitute given point and 1 perpendicular slope. y−3= − x+2 3 • Put equation in slope-intercept form.
205. 205. YOUR TURN...Write the slope-intercept form of the • Determine the slope of the givenequation perpendicular to y = 3x - 1, equation. (No need to put in slope-which passes through the point (6, 9). intercept form here.) m=3 1 m⊥ = − • Perpendicular slopes are negative 3 reciprocals. y − y1 = m ( x − x1 ) • Use point-slope formula to ﬁnd 1 perpendicular equation. y − 9 = − ( x − 6) 3 • Substitute given point and 1 perpendicular slope. y−3= − x+2 +3 3 +3 • Put equation in slope-intercept form.
206. 206. YOUR TURN...Write the slope-intercept form of the • Determine the slope of the givenequation perpendicular to y = 3x - 1, equation. (No need to put in slope-which passes through the point (6, 9). intercept form here.) m=3 1 m⊥ = − • Perpendicular slopes are negative 3 reciprocals. y − y1 = m ( x − x1 ) • Use point-slope formula to ﬁnd 1 perpendicular equation. y − 9 = − ( x − 6) 3 • Substitute given point and 1 perpendicular slope. y−3= − x+2 +3 3 +3 • Put equation in slope-intercept form. 1 y=− x+5 3
207. 207. YOUR TURN...Write the slope-intercept form of the • Determine the slope of the givenequation perpendicular to y = 3x - 1, equation. (No need to put in slope-which passes through the point (6, 9). intercept form here.) m=3 1 m⊥ = − • Perpendicular slopes are negative 3 reciprocals. y − y1 = m ( x − x1 ) • Use point-slope formula to ﬁnd 1 perpendicular equation. y − 9 = − ( x − 6) 3 • Substitute given point and 1 perpendicular slope. y−3= − x+2 +3 3 +3 • Put equation in slope-intercept form. 1 • Check that equation makes sense. The y=− x+5 lines are perpendicular so the slopes must be negative reciprocals (they are). 3