Represent the relationship among speed, time, and distance using the linear function y=ax+b, and solve problems by using graphs.
Example ① (Distance-Time Diagram)
A and B are 10 km apart. Taro leaves A and walks at 4 km per hour, while at the same time Jiro leaves B and walks at 6 km per hour. How many hours after Taro leaves will the two meet?
Answer: Taro: y=4x (distance from A), Jiro: y=10−6x (distance from A).
4x=10−6x → 10x=10 → x=1 hour later. 1 hour later
A and B are 10 km apart. Taro leaves A and walks at 4 km per hour, while at the same time Jiro leaves B and walks at 6 km per hour. How many hours after Taro leaves will the two meet?
Answer: Taro: y=4x (distance from A), Jiro: y=10−6x (distance from A).
4x=10−6x → 10x=10 → x=1 hour later. 1 hour later
Key Points
- Let y be the “distance from A” and x be the “time since departure”
- Meeting = the two people are at the same position = the intersection of the two equations
- The x-coordinate of the intersection on the graph = the time they meet
Practice Problems
- A person travels to a park 6 km from home. The person walks at 3 km per hour on the way there and runs at 6 km per hour on the way back. Express the relationship between x hours since departure and y km from home using separate equations for each interval.
Answers and Explanations
- Answer: Outbound (0≦x≦2): y=3x. Return (2≦x≦3): y=−6(x−2)+6=−6x+18.