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11-2

Slope of a Line. 11-2. Course 3. Warm Up. Problem of the Day. Lesson Presentation. Slope of a Line. 11-2. Course 3. Warm Up Evaluate each equation for x = –1, 0, and 1. 1. y = 3 x 2. y = x – 7 3. y = 2 x + 5 4. y = 6 x – 2. –3, 0, 3. –8, –7, –6. 3, 5, 7. –8, –2, 4.

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11-2

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  1. Slope of a Line 11-2 Course 3 Warm Up Problem of the Day Lesson Presentation

  2. Slope of a Line 11-2 Course 3 Warm Up Evaluate each equation for x = –1, 0, and 1. 1.y = 3x 2.y = x – 7 3.y = 2x + 5 4.y = 6x – 2 –3, 0, 3 –8, –7, –6 3, 5, 7 –8, –2, 4

  3. Slope of a Line 11-2 Course 3 Problem of the Day Write a linear equation that contains terms with x2. Possible answer: x2 + y = x2 + x + 4

  4. Slope of a Line 11-2 Course 3 Learn to find the slope of a line and use slope to understand and draw graphs.

  5. Slope of a Line 11-2 Remember! You looked at slope on the coordinate plane in Lesson 5-5 (p. 244). Course 3 Insert Lesson Title Here

  6. Slope of a Line 11-2 rise run vertical changehorizontal change change in ychange in x = This ratio is often referred to as , or “rise over run,” where rise indicates the number of units moved up or down and run indicates the number of units moved to the left or right. Slope can be positive, negative, zero, or undefined. A line with positive slope goes up from left to right. A line with negative slope goes down from left to right. Course 3 Linear equations have constant slope. For a line on the coordinate plane, slope is the following ratio:

  7. Slope of a Line 11-2 Course 3

  8. Slope of a Line 11-2 Course 3

  9. Slope of a Line 11-2 y2–y1 x2–x1 Course 3 If you know any two points on a line, or two solutions of a linear equation, you can find the slope of the line without graphing. The slope of a line through the points (x1, y1) and (x2, y2) is as follows:

  10. Slope of a Line 11-2 y2 – y1 = x2 – x1 6 – (–3) 3 9 4 – (–2) 3 The slope of the line that passes through (–2, –3) and (4, 6) is . 6 2 2 = = Course 3 Additional Example 1: Finding Slope, Given Two Points Find the slope of the line that passes through (–2, –3) and (4, 6). Let (x1, y1) be (–2, –3) and (x2, y2) be (4, 6). Substitute 6 for y2, –3 for y1, 4 for x2, and –2 for x1.

  11. Slope of a Line 11-2 y2 – y1 = x2 – x1 3 – (–6) 3 9 2 – (–4) 3 The slope of the line that passes through (–4, –6) and (2, 3) is . 6 2 2 = = Course 3 Try This: Example 1 Find the slope of the line that passes through (–4, –6) and (2, 3). Let (x1, y1) be (–4, –6) and (x2, y2) be (2, 3). Substitute 3 for y2, –6 for y1, 2 for x2, and –4 for x1.

  12. Slope of a Line 11-2 Course 3 Additional Example 2: Finding Slope from a Graph Use the graph of the line to determine its slope.

  13. Slope of a Line 11-2 rise = = – y2 – y1 run = x2 – x1 1 – (–4) 5 5 0 – 3 –5 3 –3 3 = 5 = – 3 Course 3 Additional Example 2 Continued Choose two points on the line: (0, 1) and (3, –4). Guess by looking at the graph: Use the slope formula. Let (3, –4) be (x1, y1) and (0, 1) be (x2, y2). –5 3

  14. Slope of a Line 11-2 y2 – y1 = x2 – x1 The slope of the given line is – . –4 – 1 5 –5 3 – 0 3 3 = 5 = – 3 Course 3 Additional Example 2 Continued Notice that if you switch (x1, y1) and (x2, y2), you get the same slope: Let (0, 1) be (x1, y1) and (3, –4) be (x2, y2).

  15. Slope of a Line 11-2 Course 3 Try This: Example 2 Use the graph of the line to determine its slope.

  16. Slope of a Line 11-2 rise = = 2 y2 – y1 run = x2 – x1 –1 – 1 –2 0 – 1 2 –1 1 = Course 3 Try This: Example 2 Continued Choose two points on the line: (1, 1) and (0, –1). Guess by looking at the graph: 1 Use the slope formula. 2 Let (1, 1) be (x1, y1) and (0, –1) be (x2, y2). = 2

  17. Slope of a Line 11-2 Course 3 Recall that two parallel lines have the same slope. The slopes of two perpendicular lines are negative reciprocals of each other.

  18. Slope of a Line 11-2 y2 – y1 y2 – y1 = = x2 – x1 x2 – x1 9 9 8 8 8 8 4 – (–4) –5 – 4 9 9 9 –9 8 2 – (–6) 8 – (–1) 8 9 8 Line 1 has a slope equal to – and line 2 has a slope equal to , – and are negative reciprocals of each other, so the lines are perpendicular. = = = – Course 3 Additional Example 3A: Identifying Parallel and Perpendicular Lines by Slope Tell whether the lines passing through the given points are parallel or perpendicular. A. line 1: (–6, 4) and (2, –5); line 2: (–1, –4) and (8, 4) slope of line 1: slope of line 2:

  19. Slope of a Line 11-2 y2 – y1 y2 – y1 = = x2 – x1 x2 – x1 7 –4 – 3 –2 – 5 6 7 –7 7 –7 5 – (–1) 6 – 0 6 6 6 6 Both lines have a slope equal to – , so the lines are parallel. = = = – = – Course 3 Additional Example 3B: Identifying Parallel and Perpendicular Lines by Slope B. line 1: (0, 5) and (6, –2); line 2: (–1, 3) and (5, –4) slope of line 1: slope of line 2:

  20. Slope of a Line 11-2 y2 – y1 y2 – y1 = = x2 – x1 x2 – x1 9 9 8 8 8 8 2 – (–6) –7 – 2 9 9 9 –9 8 0 – (–8) 6 – (–3) 8 9 8 Line 1 has a slope equal to – and line 2 has a slope equal to , – and are negative reciprocals of each other, so the lines are perpendicular. = = = – Course 3 Try This: Example 3A Tell whether the lines passing through the given points are parallel or perpendicular. A. line 1: (–8, 2) and (0, –7); line 2: (–3, –6) and (6, 2) slope of line 1: slope of line 2:

  21. Slope of a Line 11-2 y2 – y1 y2 – y1 = = x2 – x1 x2 – x1 –1 – (–2) 2 – (1) 2 – 1 1 1 2 – 1 1 1 = = Course 3 Try This: Example 3B B. line 1: (1, 1) and (2, 2); line 2: (1, –2) and (2, -1) = 1 slope of line 1: = 1 slope of line 2: Both lines have a slope equal to 1, so the lines are parallel.

  22. Slope of a Line 11-2 The slope is 2, or . So for every 2 units up, you will move right 1 unit, and for every 2 units down, you will move left 1 unit. 2 1 Course 3 Additional Example 4: Graphing a Line Using a Point and the Slope Graph the line passing through (3, 1) with slope 2. Plot the point (3, 1). Then move 2 units up and right 1 unit and plot the point (4, 3). Use a straightedge to connect the two points.

  23. Slope of a Line 11-2 Course 3 Additional Example 4 Continued 1 2 (3, 1)

  24. Slope of a Line 11-2 The slope is 2, or . So for every 2 units up, you will move right 1 unit, and for every 2 units down, you will move left 1 unit. 2 1 Course 3 Try This: Example 4 Graph the line passing through (1, 1) with slope 2. Plot the point (1, 1). Then move 2 units up and right 1 unit and plot the point (2, 3). Use a straightedge to connect the two points.

  25. Slope of a Line 11-2 Course 3 Try This: Example 4 Continued 1 2 (1, 1)

  26. Slope of a Line 11-2 2 5 3 4 – – 5 3 Course 3 Insert Lesson Title Here Lesson Quiz: Part 1 Find the slope of the line passing through each pair of points. 1. (4, 3) and (–1, 1) 2. (–1, 5) and (4, 2) 3. Use the graph of the line to determine its slope.

  27. Slope of a Line 11-2 Course 3 Insert Lesson Title Here Lesson Quiz: Part 2 Tell whether the lines passing through the given points are parallel or perpendicular. 4. line 1: (–2, 1), (2, –1); line 2: (0, 0), (–1, –2) 5. line 1: (–3, 1), (–2, 3); line 2: (2, 1), (0, –3) perpendicular parallel

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