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2. 2 4.4 Slope and Graphs of Linear Equations

3. 3 What You Will Learn  Determine the slope of a line through two points  Write linear equations in slope-intercept form and graph the equations  Use slopes to determine whether lines are parallel, perpendicular, or neither

4. 4 The Slope of a Line

5. 5 The slope of a nonvertical line is the number of units the line rises or falls vertically for each unit of horizontal change from left to right. For example, the line in Figure 4.21 rises two units for each unit of horizontal change from left to right, and so this line has a slope of m = 2.

6. 6 The slope m of a nonvertical line that passes through the points (x 1, y 1 ) and (x 2, y 2 ) is where x 1 ≠ x 2. The Slope of a Line

7. 7 When the formula for slope is used, the order of subtraction is important. Given two points on a line, you are free to label either of them (x 1, y 1 ) and the other (x 2, y 2 ). However, once this has been done, you must form the numerator and denominator using the same order of subtraction.

8. 8 Example 1 – Sketching the Graph of an Equation Find the slope of the line passing through each pair of points. a. (–2, 0) and (3, 1)b. (0, 0) and (1, –1) Solution: a. Let (x 1, y 1 ) = (–2, 0) and (x 2, y 2 ) = (3, 1). The slope of the line through these points is

9. 9 cont’d Difference in y-values Difference in x-values Simplify. Example 1 – Sketching the Graph of an Equation

10. 10 cont’d Example 1 – Sketching the Graph of an Equation b. The slope of the line through (0, 0) and (1, –1) is Here is the line of the graph. Difference in y-values Difference in x-values Simplify.

11. 11 The Slope of a Line

12. 12 The Slope of a Line From the slopes of the lines below, you can make several generalizations about the slope of a line.

13. 13 The Slope of a Line

14. 14 Example 2 – Finding the Slope of a Ladder Find the slope of the ladder leading up to the tree house.

15. 15 Example 2 – Finding the Slope of a Ladder Solution: Consider the tree trunk as the y-axis and the level ground as the x-axis. The endpoints of the ladder are (0, 12) and (5, 0). So, the slope of the ladder is

16. 16 Slope as a Graphing Aid

17. 17 Slopes as a Graphing Aid You have seen that before creating a table of values for an equation, it is helpful first to solve the equation for y. When you do this for a linear equation, you obtain some very useful information. Consider: 3x – 2y = 4 3x – 3x – 2y = –3x + 4 –2y = –3x + 4 Write original equation. Subtract 3x from each side. Simplify. Divide each side by –2. Simplify.

18. 18 Slopes as a Graphing Aid Observe that the coefficient of x is the slope of the graph of this equation. Moreover, the constant term, –2, gives the y-intercept of the graph. This form is called the slope-intercept form of the equation of the line.

19. 19 Slopes as a Graphing Aid

20. 20 So far, you have been plotting several points to sketch the equation of a line. However, now that you can recognize equations of lines, you don’t have to plot as many points—two points are enough. (You might remember from geometry that two points are all that are necessary to determine a line.) The next example shows how to use the slope to help sketch a line. Slopes as a Graphing Aid

21. 21 Example 3 – Using Slope-Intercept Form Use the slope and y-intercept to sketch the graph of x – 3y = –6. Solution: First, write the equation in slope-intercept form. x – 3y = –6 –3y = –x – 6 Write original equation. Subtract x from each side. Divide each side by –3. Simplify to slope-intercept form.

22. 22 cont’d So, the slope of the line is and the y-intercept is (0, b) = (0, 2). Now you can sketch the graph of the equation. First, plot the y-intercept. Example 3 – Using Slope-Intercept Form

23. 23 cont’d Example 3 – Using Slope-Intercept Form Then, using a slope of locate a second point on the line by moving three units to the right and one unit up (or one unit up and three units to the right).

24. 24 cont’d Example 3 – Using Slope-Intercept Form Finally, obtain the graph by drawing a line through the two points.

25. 25 Parallel and Perpendicular Lines

26. 26 You know from geometry that two lines in a plane are parallel if they do not intersect, and two lines in a plane are perpendicular if they intersect at right angles. Parallel and Perpendicular Lines

27. 27 Example 4 – Parallel and Perpendicular Lines (a) a. The lines y = 3x – 4 each have a slope of 3. So, the lines are parallel.

28. 28 Example 4 – Parallel and Perpendicular Lines (b) b. The lines y = 5x + 2 and y = – x – 4 have slopes that are negative reciprocals of each other. So, the lines are perpendicular.

29. 29 Example 5 – Parallel or Perpendicular? Determine whether the pairs of lines are parallel, perpendicular, or neither. a.y = –3x – 2, y = x + 1 b.y = x + 1, y = x – 1

30. 30 Example 5 – Parallel or Perpendicular? a.The first line has a slope of m 1 = –3 and the second line has a slope of m 2 =. Because these slopes are negative reciprocals of each other, the two lines must be perpendicular. cont’d

31. 31 Example 5 – Parallel or Perpendicular? b. Both lines have a slope of m =. So, the two lines must be parallel. cont’d