Exploratory Activity Consider the story: Maya and Earl live at opposite ends of the hallway in their apartment building. Their doors are 50 ft. apart. Each starts at his or her own door and walks at a steady pace toward each other and stops when they meet. 1. A. What two aspects of their walk should we measure? Distance - y-axis Time - x-axis B. What would their graphing stories look like if we put them on the same graph? Two lines, one starting at 50ft with a negative slope and one starting at zero with a positive slope. 2. When the two people meet in the hallway, what would be happening on the graph? The two lines will intersect (meet). 3. Sketch a graph that shows their distance from Maya s door. S.31
Consider the story: Duke starts at the base of a ramp and walks up it at a constant rate. His elevation increases by 3 ft. every second. Just as Duke starts walking up the ramp, Shirley starts at the top of the same 25 ft. high ramp and begins walking down the ramp at a constant rate. Her elevation decreases 2 ft. every second. Understanding the Problem 4. Fill in the chart below to help you understand how quickly Duke and Shirley move and where they start. Duke s Path Time in Seconds Elevation in Feet Shirley s Path Time in Seconds 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 10 11 12 Elevation in Feet 5. Use the extra row in each table to write an expression that describes the elevation pattern. Elevation in Feet 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 The Story of Duke and Shirley Duke: Shirley: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Time in Seconds 6. Use the grid above to graph the data for Duke and Shirley. Be sure to finish the legend to help the reader understand your graph. S.32
Writing an Equation of a Line The equation of a line can be in the form y= mx+ b, where m represents the slope of the line and b represents the y-intercept. 7. Determine the slope of Duke s line and Shirley s line. Why is one of the slopes negative? Duke s slope: Shirley s slope: 8. What is the y-intercept for each line? Duke s y-intercept: Shirley s y-intercept: 9. Write the equation of Duke s line and Shirley s line. Duke s line: Shirley s line: 10. Where do the two lines intersect? This is the point of intersection. What time do Duke and Shirley pass each other? 11. How could you use the equations to find the point of intersection? The equation you wrote in Exercise 9 are in slope-intercept form or y =mx + b. Point-slope is another very useful form of a linear equation. For point-slope we need any point on the line, not just the y-intercept and the slope of the line. The general form looks like y y 1 = m(x x 1) where (x 1, y 1) is any point on the line and m is the slope of the line. 12. A. Use the points when t = 3 seconds and slope from Exercise 7 with the point-slope equation to get an equation for Duke and Shirley. B. How do these equations compare to the ones you wrote in Exercise 9? The equations are the same. S.33
Winter-Wonderland Party! Nancy and Jane want to throw a Winter-Wonderland party. They want to split the cost evenly between them. Nancy owes her parents $30 for the dress she wanted to have for the first day back to school. She must pay them back before she can contribute to the party expenses. Jane has been putting money away all summer and had $66 before she spent $6 on some new songs by her favorite band. Nancy earns $15 each week for keeping up on her chores. Jane earns $18 every week for allowance, but pays her brother half for doing her pooper-scooper duty and cleaning up after their two dogs. It is mid-august and the girls promise to not spend their earnings and plan to save for the party. They each try to figure out how long it will take to have the same amount of savings in order to maximize the party budget. 13. Create a table to help you organize the information you have so far. Then graph your data and include a legend. Does it make sense to connect the points? Finally, write an equation for each girl. Time in Weeks Since Mid-August 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Nancy s Savings Jane s Savings Money Saved in Dollars 230 220 210 200 190 180 170 160 150 140 130 120 110 100 90 80 70 60 50 40 30 20 10 0-10 -20-30 Winter-Wonderland Party 0 2 4 6 8 10 12 14 16 18 Time in Weeks n This is where you can write the expression for each girl s saving. S.34
14. How many weeks does it take for the girls to have the same amount of money? Check that this answer makes sense on the graph and in each equation. 15 weeks 15. If the girls determine that they need $150 each for the party, which week would they have enough money? Remember that they each have to put in the same amount. Justify your answer using the tables, graph or equations. Week 12 S.35
Lesson Summary Linear Equations: If you know the slope and y-intercept you can use the equation y= mx+ b to write an equation for the line. If you have the slope and any point on line you can use the pointslope equation, y y 1 = m(x x 1) where (x 1, y 1) is any point on the line and m is the slope of the line. Example 1: The slope of a line is -3 and the y-intercept is 5. An equation of this line is. Example 2: The slope of a line is ½ and the point (2, 4) is on the line. The equation of this line is. Point of Intersection: The point of intersection is an ordered pair that is a solution to both equations. On a graph this is where the two graphs cross each other. Example: Determine the point of intersection for the graph below. Point of intersection: S.36
Homework Problem Set 1. Maya and Earl live at opposite ends of the hallway in their apartment building. Their doors are 50 ft. apart. Each starts at his or her own door and walks at a steady pace toward each other and stops when they meet. Suppose that Maya walks at a constant rate of 2 ft. every second and Earl walks at a constant rate of 4 ft. every second starting from 50 ft. away. Create equations for each person s distance from Maya s door and determine exactly when they meet in the hallway. How far are they from Maya s door at this time? Let y = distance from Maya s door in feet Let x = time in seconds 2. Suppose two cars are travelling north along a road. Car 1 travels at a constant speed of 50 mph for two hours, then speeds up and drives at a constant speed of 100 mph for the next hour. The car breaks down and the driver has to stop and work on it for two hours. When he gets it running again, he continues driving recklessly at a constant speed of 100 mph. Car 2 starts at the same time that Car 1 starts, but Car 2 starts 100 mi. farther north than Car 1 and travels at a constant speed of 25 mph throughout the trip. A. Sketch the distance-versus-time graphs for Car 1 and Car 2 on a coordinate plane at the right. Be sure to include a legend. B. Approximately when do the cars pass each other? C. Tell the entire story of the graph from the point of view of Car 2. (What does the driver of Car 2 see along the way and when?) S.37
3. Challenge Problem A. Write linear equations representing each car s distance in terms of time (in hours). Note that you will need four equations for Car 1 and only one for Car 2. B. Use these equations to find the exact coordinates of when the cars meet. 4. Suppose that in Problem 2 above, Car 1 travels at the constant speed of 25 mph the entire time. A. Sketch the distance-versus-time graphs for the two cars on a graph below. B. Do the cars ever pass each other? Explain. C. What is the linear equation for Car 1 in this case? S.38
5. The graph at the right shows the revenue (or income) a company makes from designer coffee mugs and the total cost (including overhead, maintenance of machines, etc.) that the company spends to make the coffee mugs. A. How are revenue and total cost related to the number of units of coffee mugs produced? Dollars 14000 12000 10000 8000 6000 4000 2000 0 0 100 200 300 400 500 600 700 800 900 1000 Units Produced and Sold Total Cost Revenue B. What is the meaning of the point (0, 4000) on the total cost line? C. What are the coordinates of the intersection point? What is the meaning of this point in this situation? S.39
6. Challenge Problem Consider the story: May, June, and July were running at the track. May started first and ran at a steady pace of 1 mi. every 11 min. June started 5 min. later than May and ran at a steady pace of 1 mi. every 9 min. July started 2 min. after June and ran at a steady pace, running the first lap 1 mi. in 1.5 min. She maintained this 4 steady pace for 3 more laps and then slowed down to 1 lap every 3 min. A. Sketch May, June, and July s distance-versus-time graphs on a coordinate plane below. Be sure to label your axes, put a title on your graph, and include a legend. B. Write linear equations that represent each girl s mileage in terms of time in minutes. You will need two equations for July since her pace changes after 4 laps (1 mi.). C. Who was the first person to run 3 mi.? D. Did June and July pass May on the track? If they did, when and at what mileage? E. Did July pass June on the track? If she did, when and at what mileage? S.40