Chapter 8 Section 2 and 3

1. Chapter 8Section 2 and 3 Click on the slide show tab Click from beginning (first option) Click the enter key for each movement If you go to a slide and need to go back, click the back button at the bottom of the screen PowerPoint Directions

2. When to write When you see this picture, you will write the problem, show your work, then give your final answer. You will create an assignment to turn in.

3. EQ: How are slope, y-intercept, x-intercept and equation used in graphing? • CC.8.F.4 • Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.

4. Just keep going on the same paper. Make sure you number each section/question Use graph paper for your graphing. There are cut pieces that you will tape into your work. How do I break up my questions

5. This will be a major grade Each question will be graded. There are 40 questions. Each question is worth about 2.5 points. How will I be graded

6. We will be in the computer lab Monday, Tuesday, and Wednesday and Thursday. This project is due on Thursday. If you do not finish in the 4 days we are in the computer lab, you will have to go to the library, work from home, or in Mrs. Acton’s room in the morning. When is it due

7. Turn in your work. You can play on coolmath.com You can play on other educational websites. What do I do when I am done

8. Starthere Warm Up Evaluate each equation for x = –1, 0, & 1. This means to make a function table, substitute the x and solve for y. 1.y = 3x 2.y = x – 7 3.y = 2x + 5 X equation y 1-3 -1 0 1

9. Learn to find the slope of a line and use slope to understand and draw graphs.

10. Vocabulary Define these words…definitions will be in the presentation 4. rise 5. run 6. slope 4-6

11. The constant rate of change of a line is called the slope of the line.

12. 2 4 1 2 1 2 slope = = The slope of the line is . Example 1: Finding the Slope of a Line Find the slope of the line. (5, 4) Begin at one point and count vertically to find the rise. (1, 2) Then count horizontally to the second point to find the run.

13. Work it out Find the slope of the line. (3, 2) 7 (–1, –2)

14. y2–y1 x2–x1 m = 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 m of a line through the points (x1, y1) and (x2, y2) is as follows: This is the “formal” formula. We use a T chart

15. y2 – y1 = x2 – x1 6 – (–3) 3 9 4 – (–2) 3 The slope of the line that passes through (–2, –3) and (4, 6) is . 6 2 2 = = Example 2: Finding Slope, Given Two Points Try their way Find the slope of the line that passes through (–2, –3) and (4, 6). Let (x1, y1) be (–2, –3) and (x2, y2) be (4, 6). Substitute 6 for y2, –3 for y1, 4 for x2, and –2 for x1.

16. Work it out 8. Find the slope of the line that passes through (–4, –6) and (2, 3). 8

17. Cost of Fruit Cost Pounds Example 3: Money Application The table shows the total cost of fruit per pound purchased at the grocery store. Use the data to make a graph. Find the slope of the line and explain what it shows. Graph the data.

18. Cost of Fruit y2 – y1 = x2 – x1 30 – 15 Cost 15 10 – 5 5 = = 3 Pounds Example 3 Continued Find the slope of the line: The slope of the line is 3. This means that for every pound of fruit, you will pay another $3.

19. Work it out 9. The table shows the total cost of gas per gallon. Use the data to make a graph. Find the slope of the line and explain what it shows. Graph the data. 9

20. 10 Draw the next 2 slides as #10.

21. 10

22. Find the slope of the line passing through each pair of points. 11.(4, 3) and (–1, 1) 12.(–1, 5) and (4, 2) 11 & 12

23. 13.The table shows how much money Susan earned as a house painter for one afternoon. Use the data to make a graph. Find the slope of the line and explain what it shows. 13

24. 14. Identify the slope of the line passing through the pair of points (5, 2) and (–2, 1). A. 7 B. –3 C. D. 14

25. 15 15. The table shows the number of hours a student works and her earnings. Identify the slope of the line and explain what it shows. A. The slope of the line is 15. This means that the student earns $15 for every hour that she works. B. The slope of the line is 30. This means that the student earns $30 for every hour that she works.

26. Find the slope of the line that passes through each pair of points. 16.(3, 6) and (–1, 4) 17. (1, 2) and (6, 1) 18. (4, 6) and (2, –1) 19. (–3, 0) and (–1, 1) 16- 19

27. Learn to use slopes and intercepts to graph linear equations.

28. Vocabulary 20. x-intercept 21. y-intercept 22. slope-intercept form 20- 22

29. You can graph a linear equation easily by finding the x-intercept and the y-intercept. The x-intercept of a line is the value of x where the line crosses the x-axis (where y = 0). The y-intercept of a line is the value of y where the line crosses the y-axis (where x = 0).

30. 4x = 12 4 4 Example: Finding x-intercepts and y-intercepts to Graph Linear Equations Find the x-intercept and y-intercept of the line 4x – 3y = 12. Use the intercepts to graph the equation. Find the x-intercept (y = 0). 4x – 3y = 12 4x – 3(0) = 12 4x = 12 x = 3 The x-intercept is 3.

31. = –3y 12 –3 –3 Example Continued Find the y-intercept (x = 0). 4x – 3y = 12 4(0) – 3y = 12 –3y = 12 y = –4 The y-intercept is –4.

32. Example Continued The graph of 4x – 3y = 12 is the line that crosses the x-axis at the point (3, 0) and the y-axis at the point (0, –4).

33. Helpful Hint The form Ax + By = C, where A, B, C are real numbers, is called the Standard Form of a Linear Equation.

34. Work it out 23. Find the x-intercept and y-intercept of the line 8x – 6y = 24. Use the intercepts to graph the equation. 23

35. y = mx + b Slope y-intercept In an equation written in slope-intercept form, y = mx + b, m is the slope and b is the y-intercept.

36. Using Slope-Intercept Form to Find Slopes and y-intercepts Write each equation in slope-intercept form, and then find the slope and y-intercept. 2x + y = 3 2x + y = 3 –2x–2x Subtract 2x from both sides. y = 3 – 2x Rewrite to match slope-intercept form. y = –2x + 3 The equation is in slope-intercept form. m = –2 b= 3 The slope of the line 2x + y = 3 is –2, and the y-intercept is 3.

37. 5y = x 5 3 3 5 5 y = x + 0 3 m = 5 3 The slope of the line 5y = 3x is , and the y-intercept is 0. 5 Example: Using Slope-Intercept Form to Find Slopes and y-intercepts 5y = 3x 5y = 3x Divide both sides by 5 to solve for y. The equation is in slope-intercept form. b = 0

38. 24: Write the equation in slope-intercept form. Name the slope and the y-intercept. 4x + 3y = 9 24 Subtract 4x from both sides. Divide both sides by 3. The equation is in slope-intercept form.

39. 25. Write each equation in slope-intercept form, and then find the slope and y-intercept. 4x + y = 4 25 Subtract 4x from both sides. The equation is in slope-intercept form.

40. 26. Write each equation in slope-intercept form, and then find the slope and y-intercept. 7y = 2x Divide both sides by 7 to solve for y. The equation is in slope-intercept form. 26

41. Example: Entertainment Application A video club charges $8 to join, and $1.25 for each DVD that is rented. The linear equation y = 1.25x + 8 represents the amount of money y spent after renting x DVDs. Graph the equation by first identifying the slope and y-intercept. The equation is in slope-intercept form. y = 1.25x + 8 m =1.25 b = 8

42. Example Continued Cost of DVDs Cost The slope of the line is 1.25, and the y-intercept is 8. The line crosses the y-axis at the point (0, 8) and moves up 1.25 units for every 1 unit it moves to the right. Number of DVDs

43. y2 – y1 = x2 – x1 4 – (–4) 8 –1 – 3 –4 = Example: Writing Slope-Intercept Form Write the equation of the line that passes through (3, –4) and (–1, 4) in slope-intercept form. Find the slope. This is their way, we can do a T chart = –2 The slope is –2. Substitute either point and the slope into the slope-intercept form. y = mx + b Substitute –1 for x, 4 for y, and –2 for m. 4 = –2(–1) + b 4 = 2 + b Simplify.

44. Example Continued Solve for b. 4 = 2 + b –2–2 Subtract 2 from both sides. 2 = b Write the equation of the line, using –2 for m and 2 for b. y = –2x + 2

45. Work it out 27. Write the equation of the line that passes through (1, 2) and (2, 6) in slope-intercept form. Find the slope. Substitute either point and the slope into the slope-intercept form. y = mx + b 27 Substitute x, y, and m. Simplify.

46. Write each equation in slope-intercept form, and then find the slope and y-intercept. 28. 2y – 6x = –10 29. –5y – 15x = 30 Write the equation of the line that passes through each pair of points in slope-intercept form. 30. (0, 2) and (4, –1) 31. (–2, 2) and (4, –4) 28-31

47. 32. Identify the slope-intercept form of the equation 3y – 9x = –12, and then find the slope and y-intercept. A. y = 3x + 4; m = 3, b = –4 B. y = 3x + 4; m = 3, b = 4 C. y = 3x – 4; m = 3, b = –4 D. y = 3x – 4; m = 3, b = 4 32

48. 33. Identify the slope-intercept form of the equation –3y – 15x = 45, and then find the slope and y-intercept. A. y = –5x – 15; m = –5, b = –15 B. y = 5x – 15; m = 5, b = –15 C. y = –5x – 15; m = –5, b = 15 D. y = 5x – 15; m = –5, b = 15 33

49. 34. Identify the equation of the line that passes through the pair of points (–1, 4) and (2, –8) in slope-intercept form. A. y = 4x B. y = –4x C. y = 4x + 2 D. y = –4x + 2 34

50. 35. Find the slope of the line that passes through the points (9,7) and (2,9) 35