1. The problem involves analyzing the commuter train distance over time given points on a graph. 2. The points provided are (10,15), (20,20), (30,15), (40,0), (50,15), (60,20), (70,15), and (80,0), where $x$ is time in minutes and $y$ is distance in miles. 3. Since the graph is a line plot connecting these points, the distance function is piecewise linear between these points. 4. To understand the train's movement, we calculate the slope between consecutive points using the formula for slope: $$m = \frac{y_2 - y_1}{x_2 - x_1}$$ 5. For example, between (10,15) and (20,20): $$m = \frac{20 - 15}{20 - 10} = \frac{5}{10} = 0.5$$ This means the train is moving away from the station at 0.5 miles per minute during this interval. 6. Between (20,20) and (30,15): $$m = \frac{15 - 20}{30 - 20} = \frac{-5}{10} = -0.5$$ The train is moving back toward the station at 0.5 miles per minute. 7. Similarly, calculate slopes for all intervals: - (30,15) to (40,0): $$m = \frac{0 - 15}{40 - 30} = -1.5$$ - (40,0) to (50,15): $$m = \frac{15 - 0}{50 - 40} = 1.5$$ - (50,15) to (60,20): $$m = \frac{20 - 15}{60 - 50} = 0.5$$ - (60,20) to (70,15): $$m = \frac{15 - 20}{70 - 60} = -0.5$$ - (70,15) to (80,0): $$m = \frac{0 - 15}{80 - 70} = -1.5$$ 8. These slopes indicate the train's speed and direction changes over time. 9. The distance function is not a single formula but a piecewise linear function connecting these points. 10. Understanding this helps interpret the train's movement pattern: moving away and toward the station at varying speeds. Final answer: The commuter train distance over time is represented by a piecewise linear function connecting the given points with slopes calculated above, showing intervals of moving away and toward the station.