1. Pre-Algebra 11-2 Slope of a Line 11-2 Slope of a Line Pre-Algebra Homework & Learning Goal Homework & Learning Goal Lesson Presentation Lesson Presentation

2. Pre-Algebra 11-2 Slope of a Line 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 Pre-Algebra 11-2 Slope of a Line –8, –2, 4

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

4. Pre-Algebra 11-2 Slope of a Line HOMEWORK answers Page 543 #1-11

5. Pre-Algebra 11-2 Slope of a Line Pre-Algebra HOMEWORK Page 548 #1-5

6. Pre-Algebra 11-2 Slope of a Line Our Learning Goal Students will be able to graph lines using linear equations, understand the slope of a line and graph inequalities.

7. Pre-Algebra 11-2 Slope of a Line Our Learning Goal Assignments Learn to identify and graph linear equations. Learn to find the slope of a line and use slope to understand and draw graphs. Learn to use slopes and intercepts to graph linear equations. Learn to find the equation of a line given one point and the slope. Learn to recognize direct variation by graphing tables of data and checking for constant ratios. Learn to graph inequalities on the coordinate plane. Learn to recognize relationships in data and find the equation of a line of best fit.

8. Pre-Algebra 11-2 Slope of a Line Today’s Learning Goal Assignment Learn to find the slope of a line and use slope to understand and draw graphs.

9. Pre-Algebra 11-2 Slope of a Line You looked at slope on the coordinate plane in Lesson 5-5 (p. 244). Remember!

10. Pre-Algebra 11-2 Slope of a Line Linear equations have constant slope. For a line on the coordinate plane, slope is the following ratio: vertical change horizontal change change in y change 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. rise run

11. Pre-Algebra 11-2 Slope of a Line

12. Pre-Algebra 11-2 Slope of a Line

13. Pre-Algebra 11-2 Slope of a Line 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 (x 1, y 1 ) and (x 2, y 2 ) is as follows: y2 – y1y2 – y1x2 – x1x2 – x1y2 – y1y2 – y1x2 – x1x2 – x1

14. Pre-Algebra 11-2 Slope of a Line Find the slope of the line that passes through (–2, –3) and (4, 6). Additional Example 1: Finding Slope, Given Two Points Let (x 1, y 1 ) be (–2, –3) and (x 2, y 2 ) be (4, 6). 6 – (–3) 4 – (–2) Substitute 6 for y 2, –3 for y 1, 4 for x 2, and –2 for x 1. 9 6 = The slope of the line that passes through (–2, –3) and (4, 6) is. 3 2 = y 2 – y 1 x 2 – x 1 3 2 =

15. Pre-Algebra 11-2 Slope of a Line Find the slope of the line that passes through (–4, –6) and (2, 3). Try This: Example 1 Let (x 1, y 1 ) be (–4, –6) and (x 2, y 2 ) be (2, 3). 3 – (–6) 2 – (–4) Substitute 3 for y 2, –6 for y 1, 2 for x 2, and –4 for x 1. 9 6 = The slope of the line that passes through (–4, –6) and (2, 3) is. 3 2 = y 2 – y 1 x 2 – x 1 3 2 =

16. Pre-Algebra 11-2 Slope of a Line Use the graph of the line to determine its slope. Additional Example 2: Finding Slope from a Graph

17. Pre-Algebra 11-2 Slope of a Line Additional Example 2 Continued Choose two points on the line: (0, 1) and (3, –4). Guess by looking at the graph: rise run = –5 3 = – 5 3 Use the slope formula. Let (3, –4) be (x 1, y 1 ) and (0, 1) be (x 2, y 2 ). 1 – (–4) 0 – 3 = y 2 – y 1 x 2 – x 1 5 –3 = 5 3 = – –5 3

18. Pre-Algebra 11-2 Slope of a Line Notice that if you switch (x 1, y 1 ) and (x 2, y 2 ), you get the same slope: 5 3 The slope of the given line is –. Let (0, 1) be (x 1, y 1 ) and (3, –4) be (x 2, y 2 ). Additional Example 2 Continued –4 – 1 3 – 0 = y 2 – y 1 x 2 – x 1 –5 3 = 5 3 = –

19. Pre-Algebra 11-2 Slope of a Line Use the graph of the line to determine its slope. Try This: Example 2

20. Pre-Algebra 11-2 Slope of a Line Try This: Example 2 Continued Choose two points on the line: (1, 1) and (0, –1). Guess by looking at the graph: rise run = 2 1 = 2 Use the slope formula. Let (1, 1) be (x 1, y 1 ) and (0, –1) be (x 2, y 2 ). = y 2 – y 1 x 2 – x 1 –2 –1 = –1 – 1 0 – 1 = 2 1 2

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

22. Pre-Algebra 11-2 Slope of a Line 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: 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. 9 8 8 9 8 9 9 8 = y 2 – y 1 x 2 – x 1 –9 8 = –5 – 4 2 – (–6) 4 – (–4) 8 – (–1) = y 2 – y 1 x 2 – x 1 8 9 = 9 8 = –

23. Pre-Algebra 11-2 Slope of a Line 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) Both lines have a slope equal to –, so the lines are parallel. 7 6 slope of line 1: slope of line 2: = y 2 – y 1 x 2 – x 1 –7 6 = –2 – 5 6 – 0 = y 2 – y 1 x 2 – x 1 7 6 = – –7 6 = 7 6 = – –4 – 3 5 – (–1)

24. Pre-Algebra 11-2 Slope of a Line 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: 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. 9 8 8 9 8 9 9 8 = y 2 – y 1 x 2 – x 1 –9 8 = –7 – 2 0 – (–8) 2 – (–6) 6 – (–3) = y 2 – y 1 x 2 – x 1 8 9 = 9 8 = –

25. Pre-Algebra 11-2 Slope of a Line Try This: Example 3B B. line 1: (1, 1) and (2, 2); line 2: (1, –2) and (2, -1) Line 1 has a slope equal to 1 and line 2 has a slope equal to –1. 1 and –1 are negative reciprocals of each other, so the lines are perpendicular. slope of line 1: slope of line 2: = y 2 – y 1 x 2 – x 1 1 1 = 2 – 1 = y 2 – y 1 x 2 – x 1 –1 1 = –1 – (–2) 2 – (1) = 1 = –1

26. Pre-Algebra 11-2 Slope of a Line 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. 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

27. Pre-Algebra 11-2 Slope of a Line Additional Example 4 Continued 1 2 (3, 1)

28. Pre-Algebra 11-2 Slope of a Line 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. 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

29. Pre-Algebra 11-2 Slope of a Line Try This: Example 4 Continued 1 2 (1, 1)

30. Pre-Algebra 11-2 Slope of a Line 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. 2 55 3 – 3 4 –

31. Pre-Algebra 11-2 Slope of a Line 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) parallel perpendicular