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# 3 2 slopes of lines

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### 3 2 slopes of lines

1. 1. Slopes of Lines
2. 2. Slopes of Lines The slope of a line is a number. The slope of a line measures the amount of tilt, (inclination, steepness) of the line against the x-axis.
3. 3. Slopes of Lines The slope of a line is a number. The slope of a line measures the amount of tilt, (inclination, steepness) of the line against the x-axis. Steep lines have slopes with large absolute value. Gradual lines have slopes with small absolute value
4. 4. Slopes of Lines The slope of a line is a number. The slope of a line measures the amount of tilt, (inclination, steepness) of the line against the x-axis. Steep lines have slopes with large absolute value. Gradual lines have slopes with small absolute value Definition of Slope
5. 5. Slopes of Lines Definition of Slope Notation: The Greek capital letter Δ (delta) in general means “the difference” in mathematics. The slope of a line is a number. The slope of a line measures the amount of tilt, (inclination, steepness) of the line against the x-axis. Steep lines have slopes with large absolute value. Gradual lines have slopes with small absolute value
6. 6. Slopes of Lines Definition of Slope Notation: The Greek capital letter Δ (delta) in general means “the difference” in mathematics. Δy means the difference in the values of y’s, Δx means the difference the values of x’s. The slope of a line is a number. The slope of a line measures the amount of tilt, (inclination, steepness) of the line against the x-axis. Steep lines have slopes with large absolute value. Gradual lines have slopes with small absolute value
7. 7. Slopes of Lines Definition of Slope Notation: The Greek capital letter Δ (delta) in general means “the difference” in mathematics. Δy means the difference in the values of y’s, Δx means the difference the values of x’s. Example D. Let y1 = –2, y2 = 5, The slope of a line is a number. The slope of a line measures the amount of tilt, (inclination, steepness) of the line against the x-axis. Steep lines have slopes with large absolute value. Gradual lines have slopes with small absolute value
8. 8. Slopes of Lines Definition of Slope Notation: The Greek capital letter Δ (delta) in general means “the difference” in mathematics. Δy means the difference in the values of y’s, Δx means the difference the values of x’s. Example D. Let y1 = –2, y2 = 5, then Δy = y2 – y1 The slope of a line is a number. The slope of a line measures the amount of tilt, (inclination, steepness) of the line against the x-axis. Steep lines have slopes with large absolute value. Gradual lines have slopes with small absolute value
9. 9. Slopes of Lines Definition of Slope Notation: The Greek capital letter Δ (delta) in general means “the difference” in mathematics. Δy means the difference in the values of y’s, Δx means the difference the values of x’s. Example D. Let y1 = –2, y2 = 5, then Δy = y2 – y1 = 5 – (–2) = 7 The slope of a line is a number. The slope of a line measures the amount of tilt, (inclination, steepness) of the line against the x-axis. Steep lines have slopes with large absolute value. Gradual lines have slopes with small absolute value
10. 10. Slopes of Lines Definition of Slope Notation: The Greek capital letter Δ (delta) in general means “the difference” in mathematics. Δy means the difference in the values of y’s, Δx means the difference the values of x’s. Example D. Let y1 = –2, y2 = 5, then Δy = y2 – y1 = 5 – (–2) = 7 Let x1 = 7, x2 = –4, The slope of a line is a number. The slope of a line measures the amount of tilt, (inclination, steepness) of the line against the x-axis. Steep lines have slopes with large absolute value. Gradual lines have slopes with small absolute value
11. 11. Slopes of Lines Definition of Slope Notation: The Greek capital letter Δ (delta) in general means “the difference” in mathematics. Δy means the difference in the values of y’s, Δx means the difference the values of x’s. Example D. Let y1 = –2, y2 = 5, then Δy = y2 – y1 = 5 – (–2) = 7 Let x1 = 7, x2 = –4, then Δ x = x2 – x1 The slope of a line is a number. The slope of a line measures the amount of tilt, (inclination, steepness) of the line against the x-axis. Steep lines have slopes with large absolute value. Gradual lines have slopes with small absolute value
12. 12. Slopes of Lines Definition of Slope Notation: The Greek capital letter Δ (delta) in general means “the difference” in mathematics. Δy means the difference in the values of y’s, Δx means the difference the values of x’s. Example D. Let y1 = –2, y2 = 5, then Δy = y2 – y1 = 5 – (–2) = 7 Let x1 = 7, x2 = –4, then Δ x = x2 – x1 = –4 – 7 = –11 The slope of a line is a number. The slope of a line measures the amount of tilt, (inclination, steepness) of the line against the x-axis. Steep lines have slopes with large absolute value. Gradual lines have slopes with small absolute value
13. 13. Definition of Slope Slopes of Lines
14. 14. Definition of Slope Let (x1, y1) and (x2, y2) be two points on a line, Slopes of Lines (x1, y1) (x2, y2)
15. 15. Definition of Slope Let (x1, y1) and (x2, y2) be two points on a line, then the slope m of the line is Δy Δx m = Slopes of Lines (x1, y1) (x2, y2)
16. 16. Definition of Slope Let (x1, y1) and (x2, y2) be two points on a line, then the slope m of the line is Δy Δx y2 – y1 x2 – x1 m = = Slopes of Lines (x1, y1) (x2, y2)
17. 17. Definition of Slope Let (x1, y1) and (x2, y2) be two points on a line, then the slope m of the line is Δy Δx y2 – y1 x2 – x1 m = = Slopes of Lines Geometry of Slope (x1, y1) (x2, y2)
18. 18. Definition of Slope Let (x1, y1) and (x2, y2) be two points on a line, then the slope m of the line is Δy Δx y2 – y1 x2 – x1 m = = Slopes of Lines (x1, y1) (x2, y2) Δy=y2–y1=rise Geometry of Slope Δy = y2 – y1 = the difference in the heights of the points.
19. 19. Definition of Slope Let (x1, y1) and (x2, y2) be two points on a line, then the slope m of the line is Δy Δx y2 – y1 x2 – x1 m = = Slopes of Lines (x1, y1) (x2, y2) Δy=y2–y1=rise Δx=x2–x1=run Geometry of Slope Δy = y2 – y1 = the difference in the heights of the points. Δx = x2 – x1 = the difference in the runs of the points.
20. 20. Definition of Slope Let (x1, y1) and (x2, y2) be two points on a line, then the slope m of the line is Δy Δx y2 – y1 x2 – x1 m = = Slopes of Lines (x1, y1) (x2, y2) Δy=y2–y1=rise Δx=x2–x1=run Geometry of Slope Δy = y2 – y1 = the difference in the heights of the points. Δx = x2 – x1 = the difference in the runs of the points. Δy Δx =Therefore m is the ratio of the “rise” to the “run”.
21. 21. Definition of Slope Let (x1, y1) and (x2, y2) be two points on a line, then the slope m of the line is Δy Δx y2 – y1 x2 – x1 m = = rise run= Slopes of Lines (x1, y1) (x2, y2) Δy=y2–y1=rise Δx=x2–x1=run Geometry of Slope Δy = y2 – y1 = the difference in the heights of the points. Δx = x2 – x1 = the difference in the runs of the points. Δy Δx =Therefore m is the ratio of the “rise” to the “run”. m = Δy Δx y2 – y1 x2 – x1 =
22. 22. Definition of Slope Let (x1, y1) and (x2, y2) be two points on a line, then the slope m of the line is Δy Δx y2 – y1 x2 – x1 m = = rise run= Slopes of Lines (x1, y1) (x2, y2) Δy=y2–y1=rise Δx=x2–x1=run Geometry of Slope Δy = y2 – y1 = the difference in the heights of the points. Δx = x2 – x1 = the difference in the runs of the points. Δy Δx =Therefore m is the ratio of the “rise” to the “run”. m = Δy Δx y2 – y1 x2 – x1 = easy to memorize
23. 23. Definition of Slope Let (x1, y1) and (x2, y2) be two points on a line, then the slope m of the line is Δy Δx y2 – y1 x2 – x1 m = = rise run= Slopes of Lines (x1, y1) (x2, y2) Δy=y2–y1=rise Δx=x2–x1=run Geometry of Slope Δy = y2 – y1 = the difference in the heights of the points. Δx = x2 – x1 = the difference in the runs of the points. Δy Δx =Therefore m is the ratio of the “rise” to the “run”. m = Δy Δx y2 – y1 x2 – x1 = easy to memorize the exact formula
24. 24. Definition of Slope Let (x1, y1) and (x2, y2) be two points on a line, then the slope m of the line is Δy Δx y2 – y1 x2 – x1 m = = rise run= Slopes of Lines (x1, y1) (x2, y2) Δy=y2–y1=rise Δx=x2–x1=run Geometry of Slope Δy = y2 – y1 = the difference in the heights of the points. Δx = x2 – x1 = the difference in the runs of the points. Δy Δx =Therefore m is the ratio of the “rise” to the “run”. m = Δy Δx y2 – y1 x2 – x1 = easy to memorize the exact formula geometric meaning
25. 25. Example E. Find the slope of the line that passes through (3, –2) and (–2, 8). Draw the line. Slopes of Lines
26. 26. Example E. Find the slope of the line that passes through (3, –2) and (–2, 8). Draw the line. Slopes of Lines
27. 27. Example E. Find the slope of the line that passes through (3, –2) and (–2, 8). Draw the line. Slopes of Lines It’s easier to find Δx and Δy vertically.
28. 28. (–2 , 8) ( 3 , –2) Example E. Find the slope of the line that passes through (3, –2) and (–2, 8). Draw the line. Slopes of Lines It’s easier to find Δx and Δy vertically.
29. 29. (–2 , 8) ( 3 , –2) –5 , 10 Example E. Find the slope of the line that passes through (3, –2) and (–2, 8). Draw the line. Slopes of Lines It’s easier to find Δx and Δy vertically.
30. 30. Δy (–2 , 8) ( 3 , –2) –5 , 10 Δx Example E. Find the slope of the line that passes through (3, –2) and (–2, 8). Draw the line. Slopes of Lines It’s easier to find Δx and Δy vertically.
31. 31. Δy Δx (–2 , 8) ( 3 , –2) –5 , 10 Δy Δx Hence the slope is Example E. Find the slope of the line that passes through (3, –2) and (–2, 8). Draw the line. Slopes of Lines It’s easier to find Δx and Δy vertically. m =
32. 32. Δy Δx (–2 , 8) ( 3 , –2) –5 , 10 Δy Δx Hence the slope is 10 –5 Example E. Find the slope of the line that passes through (3, –2) and (–2, 8). Draw the line. Slopes of Lines It’s easier to find Δx and Δy vertically. m = = = –2
33. 33. Δy Δx (–2 , 8) ( 3 , –2) –5 , 10 Δy Δx Hence the slope is 10 –5 Example E. Find the slope of the line that passes through (3, –2) and (–2, 8). Draw the line. Slopes of Lines It’s easier to find Δx and Δy vertically. m = = = –2 Example F. Find the slope of the line that passes through (3, 5) and (-2, 5). Draw the line.
34. 34. Δy Δx (–2 , 8) ( 3 , –2) –5 , 10 Δy Δx Hence the slope is 10 –5 Example E. Find the slope of the line that passes through (3, –2) and (–2, 8). Draw the line. Slopes of Lines It’s easier to find Δx and Δy vertically. m = = = –2 Example F. Find the slope of the line that passes through (3, 5) and (-2, 5). Draw the line.
35. 35. Δy Δx (–2 , 8) ( 3 , –2) –5 , 10 Δy Δx Hence the slope is 10 –5 Example E. Find the slope of the line that passes through (3, –2) and (–2, 8). Draw the line. Slopes of Lines It’s easier to find Δx and Δy vertically. m = = = –2 Example F. Find the slope of the line that passes through (3, 5) and (-2, 5). Draw the line. (–2, 5) ( 3, 5)
36. 36. Δy Δx (–2 , 8) ( 3 , –2) –5 , 10 Δy Δx Hence the slope is 10 –5 Example E. Find the slope of the line that passes through (3, –2) and (–2, 8). Draw the line. Slopes of Lines It’s easier to find Δx and Δy vertically. m = = = –2 Example F. Find the slope of the line that passes through (3, 5) and (-2, 5). Draw the line. Δy (–2, 5) ( 3, 5) –5, 0 Δx
37. 37. Δy Δx (–2 , 8) ( 3 , –2) –5 , 10 Δy Δx Hence the slope is 10 –5 Example E. Find the slope of the line that passes through (3, –2) and (–2, 8). Draw the line. Slopes of Lines It’s easier to find Δx and Δy vertically. m = = = –2 Example F. Find the slope of the line that passes through (3, 5) and (-2, 5). Draw the line. Δy (–2, 5) ( 3, 5) –5, 0 Δx So the slope is Δx Δy m =
38. 38. Δy Δx (–2 , 8) ( 3 , –2) –5 , 10 Δy Δx Hence the slope is 10 –5 Example E. Find the slope of the line that passes through (3, –2) and (–2, 8). Draw the line. Slopes of Lines It’s easier to find Δx and Δy vertically. m = = = –2 Example F. Find the slope of the line that passes through (3, 5) and (-2, 5). Draw the line. Δy (–2, 5) ( 3, 5) –5, 0 Δx So the slope is Δx Δy 0 –5 m = = = 0
39. 39. As shown in example F, the slope of a horizontal line is 0, i.e. it’s “tilt” is 0. Slopes of Lines
40. 40. As shown in example F, the slope of a horizontal line is 0, i.e. it’s “tilt” is 0. Slopes of Lines Example G. Find the slope of the line that passes through (3, 2) and (3, 5). Draw the line.
41. 41. As shown in example F, the slope of a horizontal line is 0, i.e. it’s “tilt” is 0. Slopes of Lines Example G. Find the slope of the line that passes through (3, 2) and (3, 5). Draw the line.
42. 42. As shown in example F, the slope of a horizontal line is 0, i.e. it’s “tilt” is 0. Slopes of Lines Example G. Find the slope of the line that passes through (3, 2) and (3, 5). Draw the line. Δy (3, 5) (3, 2) 0, 3 Δx
43. 43. As shown in example F, the slope of a horizontal line is 0, i.e. it’s “tilt” is 0. Slopes of Lines Example G. Find the slope of the line that passes through (3, 2) and (3, 5). Draw the line. Δy (3, 5) (3, 2) 0, 3 Δx So the slope Δx Δy 3 0 m = =
44. 44. As shown in example F, the slope of a horizontal line is 0, i.e. it’s “tilt” is 0. Slopes of Lines Example G. Find the slope of the line that passes through (3, 2) and (3, 5). Draw the line. Δy (3, 5) (3, 2) 0, 3 Δx So the slope Δx Δy 3 0 m = = is undefined!
45. 45. As shown in example F, the slope of a horizontal line is 0, i.e. it’s “tilt” is 0. Slopes of Lines Example G. Find the slope of the line that passes through (3, 2) and (3, 5). Draw the line. Δy (3, 5) (3, 2) 0, 3 Δx So the slope Δx Δy 3 0 m = = is undefined! As shown in example G, the slope of a vertical line is undefined.
46. 46. Summary of Slope The slope of the line that passes through (x1, y1) and (x2, y2) is Horizontal line Slope = 0 Vertical line Slope is UDF. Tilted line Slope = –2 0 rise run =m = Δy Δx y2 – y1 x2 – x1 =
47. 47. Exercise A. Select two points and estimate the slope of each line. 1. 2. 3. 4. Slopes of Lines 5. 6. 7. 8.
48. 48. Exercise B. Draw and find the slope of the line that passes through the given two points. Identify the vertical line and the horizontal lines by inspection first. 9. (0, –1), (–2, 1) 10. (1, –2), (–2, 0) 11. (1, –2), (–2, –1) 12. (3, –1), (3, 1) 13. (1, –2), (–2, 3) 14. (2, –1), (3, –1) 15. (4, –2), (–3, 1) 16. (4, –2), (4, 0) 17. (7, –2), (–2, –6) 18. (3/2, –1), (3/2, 1) 19. (3/2, –1), (1, –3/2) 20. (–5/2, –1/2), (1/2, 1) 21. (3/2, 1/3), (1/3, 1/3) 22. (–2/3, –1/4), (1/2, 2/3) 23. (3/4, –1/3), (1/3, 3/2) Slopes of Lines