1. 3.3 Slopes of Lines

2. Objectives Find slopes of lines Use slope to identify parallel and perpendicular lines

3. What is Slope? What is Slope? The slope (m) of a line is the number of units the line rises or falls for each unit of horizontal change from left to right. In other words, slope is the ratio of vertical rise or fall to its horizontal run. m = rise run

4. y2 -y2 -y2 -y2 - m = x 2 - x 1 y2 - y1y2 - y1y2 - y1y2 - y1 x2 - x1x2 - x1x2 - x1x2 - x1 In algebra, we also learned a formula for slope called the In algebra, we also learned a formula for slope called the slope formula. Subtraction order is the same CORRECT x1 - x2x1 - x2x1 - x2x1 - x2 y2 - y1y2 - y1y2 - y1y2 - y1 Subtraction order is different INCORRECT The Slope Formula Remember to keep your subtraction of the coordinates in the proper order. y1y1y1y1

5. From (–3, 7) to (–1, –1), go down 8 units and right 2 units. Find the slope of the line. Answer: – 4 Example 1a:

6. Use the slope formula. Answer: undefined Find the slope of the line. Let be and be. Example 1b:

7. Find the slope of the line. Answer: Example 1c:

8. Find the slope of the line. Answer: 0 Example 1d:

9. Rate of Change In Cartesian math, we think of the slope of a line as a method to identify coordinates on a line; but in the physical world, slope is often thought of as a way to describe the. In Cartesian math, we think of the slope of a line as a method to identify coordinates on a line; but in the physical world, slope is often thought of as a way to describe the rate of change.

10. RECREATION For one manufacturer of camping equipment, between 1990 and 2000, annual sales increased by $7.4 million per year. In 2000, the total sales were $85.9 million. If sales increase at the same rate, what will be the total sales in 2010? Slope formula Example 2:

11. Multiply each side by 10. Add 85.9 to each side. Simplify. The coordinates of the point representing the sales for 2010 are (2010, 159.9). Answer: The total sales in 2010 will be about $159.9 million. Example 2:

12. Slopes of ║ and  Lines Finally, recall from algebra that lines which are║ or  have mathematical relationships. ║ lines have the same slope. i.e. If line l has a slope of ¾ and line m is ║to line l then it also has a slope of ¾.  lines have opposite reciprocal slopes. i.e. If line a has a slope of 2 and line b is  to line a then it has a slope of – ½.

13. Slope Postulates Postulate 3.2 Two nonvertical lines have the same slope iff they are ║. Postulate 3.3 Two nonvertical lines are  iff the product of their slopes is -1.

14. Determine whether and are parallel, perpendicular, or neither. Example 3a:

15. The slopes are not the same, The product of the slopes is are neither parallel nor perpendicular. Answer: Example 3a:

16. Answer: The slopes are the same, so Determine whether and are parallel, perpendicular, or neither. Example 3b: are | |.

17. Answer: perpendicular Answer: neither a. b. Determine whether and are parallel, perpendicular, or neither. Your Turn:

18. Graph the line that contains Q(5, 1) and is parallel to with M(–2, 4) and N(2, 1). Substitution Simplify. Slope formula Example 4:

19. The slopes of two parallel lines are the same. Graph the line. Answer: The slope of the line parallel to Start at (5, 1). Move up 3 units and then move left 4 units. Label the point R. Example 4:

20. Graph the line that contains R(2, –1) and is parallel to with O(1, 6) and P(–3, 1). Answer: Your Turn:

21. Assignment Geometry: Pg.142 – 144 #6 – 36 evens Pre – AP Geometry: Pg. 142 – 144 #6 – 36 evens Would you do it?it