1. Evaluate and simplify the ratios when and. x2 = For use with pages xxx–xxx Daily Warm-Up Exercises For use with pages 281–289 y = 2 – 1. 5 – x 3 + y 2. 6 – x y – 1 A cross-country skier traveled 14 miles in 3.5 hours. Use the formula where d is distance, r is rate, and t is time, to find the average rate of speed. 3. drt =

2. Standard: Objective: Gr 7 AF 3.3 Graph linear functions, noting that the vertical change (change in y- value) per unit of horizontal change (change in x-value) is always the same and know that the ratio (rise over run) is called the slope of a graph Find a positive slope Find a negative slope Find the slope of a horizontal line Find the slope of a vertical line Find a rate of change Use a graph to find and compare rates of change Interpret a graph

5. Two points, (x 1, y 1 ) (x 2, y 2 ) X 1 : says “x sub 1”: means 1 st x value. y 1 : says “y sub 1”: means 1 st y value. X 2 : says “x sub 2”: means 2nd x value. y 2 : says “y sub 2”: means 2nd y value.

7. Example 1 Find the slope of the line shown. Find a positive slope x1x1 – x2x2 y1y1 – y2y2 m = Write formula for slope. = 2 – 6 – 2 () 4 – Substitute. = 6 4 = 3 2 Simplify. 4, 24, 2x1, y1x1, y1 Let = – = )( x2, y2x2, y2 2, 62, 6 and )()( ).). (

8. Guided Practice 1. for Example 1 Find the slope of the line that passes through (5, 2) and (4, 1). –

9. Example 2 Find the slope of the line shown. Find a negative slope x1x1 – x2x2 y1y1 – y2y2 m = Write formula for slope. = Substitute. – 63 5 – 1 – == Simplify. 3 6 – 2 – 3, 53, 5x1, y1x1, y1 Let == )( x2, y2x2, y2 6, 1 and )()( ).). ( –

10. Example 3 Find the slope of a horizontal line Find the slope of the line shown. x1x1 – x2x2 y1y1 – y2y2 m= Write formula for slope. = 4 – 4 – 4 () 2 – Substitute. = 6 0 = Simplify. 0 2, 42, 4x1, y1x1, y1 Let = – = )( x2, y2x2, y2 4, 44, 4 and )()( ).). (

11. Example 4 Find the slope of a vertical line Find the slope of the line shown. x1x1 – x2x2 y1y1 – y2y2 m = Write formula for slope. = 5 – 1 Substitute. – 33 = Division by zero is undefined. 0 4 – Because division by zero is undefined, the slope of a vertical line is undefined. ANSWER 3, 53, 5x1, y1x1, y1 Let == )( x2, y2x2, y2 3, 13, 1 and )()( ).). (

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

13. Example 5 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. INTERNET CAFE Find a rate of change 246 21147 Time (hours) Cost (dollars)

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

15. Guided Practice 5.EXERCISING The table shows the distance a person walks for exercise. Find the rate of change in distance with respect to time. for Example 5 306090 4.531.5 Time (minutes) Distance (miles)

16. Example 6 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. COMMUNITY THEATER Use a graph to find and compare rates of change SOLUTION Find the rates of change using the slope formula. Weeks 1–4 : 1 – 4 124 – 232 3 108 36 people per week ==

17. Use a graph to find and compare rates of change Example 6 Weeks 4–6 : 4 – 6 232 – 204 2 14 people per week == 28 – – Weeks 6–10 : 6 – 10 204 – 72 == 4 132 – 33 people per week – Attendance increased during the early weeks of performing the play. Then attendance decreased, slowly at first, then more rapidly. ANSWER

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

19. Interpret a graph Example 7 The third segment is steep, so the student is traveling far with respect to time. The student must be riding the bus.

20. Guided Practice 6.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. for Examples 6 and 7

21. Guided Practice for Examples 6 and 7 7.WHAT IF? Using the graph in Example 7, draw a graph that represents the student’s commute from school to home. ANSWER

22. Standard: Objective: Gr 7 AF 3.3 Graph linear functions, noting that the vertical change (change in y- value) per unit of horizontal change (change in x-value) is always the same and know that the ratio (rise over run) is called the slope of a graph Find a positive slope Find a negative slope Find the slope of a horizontal line Find the slope of a vertical line Find a rate of change Use a graph to find and compare rates of change Interpret a graph