1. 1 Warm UP 12-11-07 Graph each equation and tell whether it is linear. (create the table & graph) 1. y = 3x – 1 2. y = x 3. y = x 2 – 3 yes Insert Lesson Title Here no 1414 Course 3 11-1 Graphing Linear Equations

2. 2 Learn to find the slope of a line and use slope to understand and draw graphs. Course 3 11-2 Slope of a Line Essential Question Explain how to tell if two lines are parallel or perpendicular based off of their slope and not by graphing. Objective 5.01c, 5.02, 5.03 Identify and interpret slope; write an equation with a linear relationship, and solve problems using linear equations and inequalities.

3. 3 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 Course 3 11-2 Slope of a Line P u t I n N o t e s !

4. 4 3 More Voc Words – yes, write these down! From Chapter 5 (which we have not covered yet) Slope: a measure of a lines steepness or the “slant” of a line Rise: vertical change when slope is expressed as a ratio (y) Run: horizontal change when slope is expressed as a ratio (x)

5. 5 Course 3 11-2 Slope of a Line

6. 6 Course 3 11-2 Slope of a Line

7. 7 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 Course 3 11-2 Slope of a Line Yes! Put this in your notes!

8. 8 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 = Course 3 11-2 Slope of a Line

9. 9 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 = Course 3 11-2 Slope of a Line

10. 10 Use the graph of the line to determine its slope. Additional Example 2: Finding Slope from a Graph Course 3 11-2 Slope of a Line

11. 11 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 Course 3 11-2 Slope of a Line

12. 12 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 = – Course 3 11-2 Slope of a Line

13. 13 Use the graph of the line to determine its slope. Try This: Example 2 Course 3 11-2 Slope of a Line

14. 14 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 Course 3 11-2 Slope of a Line

15. 15 Recall that two parallel lines have the same slope. The slopes of two perpendicular lines are negative reciprocals of each other. Course 3 11-2 Slope of a Line Make sure you put this in your NOTES!

16. 16 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 = – Course 3 11-2 Slope of a Line

17. 17 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) Course 3 11-2 Slope of a Line

18. 18 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 = – Course 3 11-2 Slope of a Line

19. 19 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 Course 3 11-2 Slope of a Line

20. 20 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 Course 3 11-2 Slope of a Line

21. 21 Additional Example 4 Continued 1 2 (3, 1) Course 3 11-2 Slope of a Line

22. 22 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 Course 3 11-2 Slope of a Line

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

24. 24 Guided Practice p.548 TB In groups of 2, answer the following problems on page 548 of the text book: #’s 1, 2, 4, 5, 6, 7, 8, 9 1 Answer sheet per group will be turned in - a classroom participation grade will be given on this group work. You can use your book and notes You have 15-20 minutes

25. 25 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. Insert Lesson Title Here 2 55 3 – 3 4 – Course 3 11-2 Slope of a Line

26. 26 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 Insert Lesson Title Here Course 3 11-2 Slope of a Line

27. 27 Homework for 12-11-07 Answer the essential question Homework workbook section 11-2