Warm Up

1. Preview Warm Up California Standards Lesson Presentation

2. Warm Up Part I Add or subtract. 1. 4 + (–6) 2. –3 + 5 2 –2 3. –7 – 7 –14 4. 2 – (–1) 3 5. Find the x- and y-intercepts of 2x – 5y = 20. x-int.: 10; y-int.: –4

3. Warm Up Part II Describe the correlation shown by the scatter plot. 6. 7. • • • • • • • • • • • • • negative positive

4. California Standards Preparation for 8.0 Students understand the concepts of parallel lines and perpendicular lines and how their slopes are related. Students are able to find the equation of a line perpendicular to a given line that passes through a given point. Also covered:6.0

5. Vocabulary rate of change rise run slope

6. (y) (x) A rate of change is a ratio that compares the amount of change in a dependent variable to the amount of change in an independent variable. For any two points on a nonvertical line, this ratio is constant. The constant rate of change of a nonvertical line is call the slope of the line.

8. Rise = 3 Run = 9 Rise = –3 Run = –9 Additional Example 1: Finding Slope from a Graph Find the slope of the line. Begin at one point and count vertically to find the rise. Then count horizontally to the second point to find the run.

9. Run = –4 Rise = 2 Rise =–2 Run = 4 Check It Out! Example 1 Find the slope of the line. Begin at one point and count vertically to find the rise. Then count horizontally to the second point to find the run.

10. Additional Example 2: Finding Slopes of Horizontal and Vertical Lines Find the slope of each line. B. A. The slope is 0. The slope is undefined.

11. Check It Out! Example 2 Find the slope of each line. a. b. You cannot divide by 0. The slope is undefined. The slope is 0.

12. If you know 2 different points on a line, you can use the slope formula to find the slope of the line.

13. Additional Example 3: Finding Slope by Using the Slope Formula Find the slope of the line that contains (2, 5) and (8, 1). Use the slope formula. Substitute (2, 5) for (x1, y1) and (8, 1) for (x2,y2). Simplify.

14. Reading Math The small numbers to the bottom right of the variables are called subscripts. Read x1as “x sub one” and y2as “y sub two.”

15. Check It Out! Example 3a Find the slope of the line that contains (–2, –2) and (7, –2). Use the slope formula. Substitute (–2, –2) for (x1, y1) and (7, –2) for (x2,y2). Simplify.

16. Check It Out! Example 3b Find the slope of the line that contains (5, –7) and (6, –4). Use the slope formula. Substitute (5, –7) for (x1,y1) and (6, –4) for (x2,y2). Simplify.

17. As shown in the previous examples, slope can be positive, negative, zero, or undefined.

18. Additional Example 4: Describing Slope Tell whether the slope of each line is positive, negative, zero, or undefined. B. A. The line rises from left to right. The line falls from left to right. The slope is positive. The slope is negative.

19. Check It Out! Example 4 Tell whether the slope of each line is positive, negative, zero, or undefined. a. b. The line is vertical. The line rises from left to right. The slope is undefined. The slope is positive.

20. Remember that slope is a rate of change. In real-world problems, finding the slope can give you information about how a quantity is changing.

21. Additional Example 5: Application The graph shows the average electricity costs in dollars for operating a refrigerator for several months. Find the slope of the line. Then tell what the slope represents. Step 1 Use the slope formula.

22. In this situation, y represents the total cost of electricity and x represents time. So slope represents in units of . Additional Example 5 Continued Step 2 Tell what the slope represents. A slope of 6 means that the cost of electricity to run the refrigerator is increasing (positive change) at a rate of 6 dollars each month.

23. Check It Out! Example 5 The graph shows the height of a plant over a period of days. Find the slope of the line. Then tell what the slope represents. Step 1 Use the slope formula.

24. In this situation, y represents the height of the plant and x represents time. So slope represents in units of . A slope of means that the height of the plant is increasing (positive change) at a rate of 1 cm every two days. Check It Out! Example 5 Continued Step 2 Tell what the slope represents.

25. If you know the equation of a line, you can find its slope by using any two ordered-pair solutions. It is often easiest to use the ordered pairs that contain the intercepts.

26. Step 1 Step 2 Find the x-intercept. Find the y-intercept. 4x – 2y = 16 Additional Example 6: Finding Slope from an Equation Find the slope of the line given by 4x – 2y = 16. 4x – 2y = 16 4x – 2(0) = 16 Let y = 0. 4(0) – 2y = 16 Let x = 0. 4x = 16 –2y = 16 x = 4 y = –8

27. Additional Example 6 Continued Step 3 The line contains (4, 0) and (0, –8). Use the slope formula.

28. Step 1 Step 2 Find the x-intercept. Find the y-intercept. 2x + 3y = 12 Check It Out! Example 6 Find the slope of the line given by 2x + 3y = 12. 2x + 3y = 12 2x + 3(0) = 12 Let y = 0. 4(0) + 3y = 12 Let x = 0. 2x = 12 3y = 12 x = 6 y = 4

29. Check It Out! Example 6 Step 3 The line contains (6, 0) and (0, 4). Use the slope formula.

30. A line’s slope is a measure of its steepness. Some lines are steeper than others. As the absolute value of the slope increases, the line becomes steeper.

32. Lesson Quiz: Part I Find the slope of each line. 1. 2. undefined

33. Lesson Quiz: Part II 3. Find the slope of the line that contains (5, 3) and (–1, 4). 4. Find the slope. Then tell what the slope represents. 50 A slope of 50 means that the speed of the bus is 50 mi/h. 5. Find the slope of the line given by x – 2y = 8.