4 4 Find Slope and Rate of Change

4. 4 Find Slope and Rate of Change Warm Up Lesson Presentation Lesson Quiz

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

4. 4 Warm-Up 3. A cross-country skier 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. 4 Example 1 Find the slope of the line shown. Let (x 1, y 1) = (– 4, 2) = (x 2, y 2) = (2, 6). y 2 – y 1 m= x 2 – x 1 Write formula for slope. 6– 2 = 2 – (– 4) Substitute. 4 2 = 6= 3 Simplify.

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

4. 4 Example 2 Find the slope of the line shown. Let (x 1, y 1) = (3, 5) and (x 2, y 2) = (6, – 1). y 2 – y 1 Write formula for slope. m= x 2 – x 1 – 5 Substitute. = 6– 3 – 6 – 2 Simplify. = 3 =

4. 4 Example 3 Find the slope of the line shown. Let (x 1, y 1) = (– 2, 4) and (x 2, y 2) = (4, 4). y 2 – y 1 m= x 2 – x 1 4– 4 = 4 – (– 2) 0 = = 0 6 Write formula for slope. Substitute. Simplify.

4. 4 Example 4 Find the slope of the line shown. Let (x 1, y 1) = (3, 5) and (x 2, y 2) = (3, 1). y 2 – y 1 m= x 2 – x 1 1– 5 = 3– 3 – 4 = 0 Write formula for slope. Substitute. Division by zero is undefined. ANSWER Because division by zero is undefined, the slope of a vertical line is undefined.

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

4. 4 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.

4. 4 Example 5 SOLUTION Rate of change = change in cost change in time 14 – 7 7 = = = 3. 5 2 4– 2 ANSWER The rate of change in cost is $3. 50 per hour.

4. 4 7. Guided Practice 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

4. 4 Example 6 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.

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

4. 4 8. Guided Practice WHAT IF? How would the answer to Example 6 change if you knew that attendance was 70 people in week 12? ANSWER Sample answer: The attendance did not decrease as rapidly between weeks 10 and 12.

4. 4 Example 7 COMMUTING TO SCHOOL A student commutes from home to school by walking and by riding a bus. Describe the student’s commute in words.

4. 4 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.

4. 4 9. Guided Practice WHAT IF? Using the graph in Example 7, draw a graph that represents the student’s commute from school to home. ANSWER

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

4. 4 3. Lesson Quiz 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. ANSWER Ticket sales grew moderately, declined slightly, and then had another moderate of increase.