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Warm Up

Find Slope and Rate of Change. Warm Up. Lesson Presentation. Lesson Quiz. 1. y + 3. 1. –. x 5. –. 3. 1 y. –. 2. x 6. –. 3. –. 4. Warm-Up. Evaluate and simplify the ratios when x = 2 and y = –2. ANSWER. ANSWER. Warm-Up.

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Warm Up

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  1. Find Slope and Rate of Change Warm Up Lesson Presentation Lesson Quiz

  2. 1. y+ 3 1 – x 5 – 3 1y – 2. x 6 – 3 – 4 Warm-Up Evaluate and simplify the ratios when x=2 and y= –2. ANSWER ANSWER

  3. Warm-Up 3.A cross-countryskier traveled 14 miles in 3.5 hours. Use the formula d=rt where d is distance, r is rate, and t is time, to find the average rate of speed. ANSWER 4 mi/h

  4. 6– 2 2– (–4) 4 2 3 6 y2 –y1 m = x2 – x1 = = = Example 1 Find the slope of the line shown. Let (x1, y1) = (–4, 2) = (x2, y2) = (2, 6). Write formula for slope. Substitute. Simplify.

  5. 3. ( , 5) and ( , –3) ANSWER ANSWER ANSWER 3 2 1 9 1 2 2 2 Guided Practice Find the slope of the line that passes through the points. 1. (5, 2) and (4, –1) 2. (–2, 3) and (4, 6)

  6. – 6 –1– 5 3 = 6– 3 –2 = = y2 –y1 m = x2 – x1 Example 2 Find the slope of the line shown. Let (x1,y1) = (3, 5) and (x2, y2) = (6, –1). Write formula for slope. Substitute. Simplify.

  7. 4– 4 = 4– (–2) 0 0 = = y2 –y1 6 m = x2 – x1 Example 3 Find the slope of the line shown. Let(x1, y1) = (–2, 4) and (x2,y2) = (4, 4). Write formula for slope. Substitute. Simplify.

  8. – 4 1– 5 0 = 3– 3 = y2 –y1 m = x2 – x1 ANSWER Because division by zero is undefined, the slope of a vertical line is undefined. Example 4 Find the slope of the line shown. Let(x1,y1) = (3, 5)and(x2, y2) = (3, 1). Write formula for slope. Substitute. Division by zero is undefined.

  9. ANSWER ANSWER ANSWER 0 –2 undefined Guided Practice Find the slope of the line that passes through the points. 4. (5, 2) and (5, –2) 5. (0, 4) and (–3, 4) 6. (0, 6) and (5, –4)

  10. Example 5 INTERNET CAFE The table shows the cost of using a computer at an Internet cafe for a given amount of time. Find the rate of change in cost with respect to time.

  11. change in cost = change in time 3.5 = 14– 7 = 4– 2 7 = 2 ANSWER The rate of change in cost is $3.50 per hour. Example 5 SOLUTION Rate of change

  12. 7. EXERCISE The table shows the distance a person walks for exercise. Find the rate of change in distance with respect to time. ANSWER 0.05 mi/min Guided Practice

  13. COMMUNITY THEATER A community theater performed a play each Saturday evening for 10 consecutive weeks. The graph shows the attendance for the performances in weeks 1, 4, 6, and 10. Describe the rates of change in attendance with respect to time. Example 6

  14. –28 108 232 – 124 = 1 4 – 3 2 204 – 232 = 4 6 – 72 – 204 –132 = 6 10 – 4 ANSWER Attendance increased during the early weeks of performing the play. Then attendance decreased, slowly at first, then more rapidly. Example 6 SOLUTION Find the rates of change using the slope formula. =36 people per week Weeks 1–4: Weeks4–6: = –14people per week =–33people per week Weeks 6–10:

  15. ANSWER Sample answer: The attendance did not decrease as rapidly between weeks 10 and 12. 8. WHAT IF?How would the answer to Example 6 change if you knew that attendance was 70 people in week 12? Guided Practice

  16. COMMUTING TO SCHOOL A student commutes from home to school by walking and by riding a bus. Describe the student’s commute in words. Example 7

  17. Example 7 SOLUTION The first segment of the graph is not very steep, so the student is not traveling very far with respect to time. The student must be walking. The second segment has a zero slope, so the student must not be moving. He or she is waiting for the bus. The last segment is steep, so the student is traveling far with respect to time. The student must be riding the bus.

  18. 9. WHAT IF?Using the graph in Example 7, draw a graph that represents the student’s commute from school to home. ANSWER Guided Practice

  19. ANSWER 0 3 – ANSWER 2 Lesson Quiz Find the slope of the line that passes through the points 1. (12, –1) and (–3, –1) 2.(–2, 6) and (4, –3)

  20. The graph shows the ticket sales for a school dance on day 1, day 3, day 6, and day 9 of ticket sales. Describe the rates of change in ticket sales with respect to time. 3. ANSWER Ticket sales grew moderately, declined slightly, and then had another moderate rate of increase. Lesson Quiz

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