How to sketch graphs from word problems
A graph can tell a story without giving us an equation. The axes tell us what is changing, and the slope tells us how that change happens over time. Let’s use those ideas to read a graph and then sketch one from a description.
Read the axes before interpreting the slope
With time on the horizontal axis, a horizontal segment means the quantity on the vertical axis stays the same. A rising segment means the quantity increases, and a falling segment means it decreases. On graphs with the same scales, a steeper segment means a faster rate of change.
For example, if the vertical axis measures height, a rising line means the object is getting higher. If it measures the depth of a hole, a falling line means the hole is getting shallower. Each straight segment represents a constant rate during that part of the story.
Example: Write a story for a distance graph
Emily leaves home to drive to her grandmother’s house. This graph shows her distance from home over time. Write a possible story that follows its pieces.
The graph has five pieces. Emily first gets farther from home at a slower rate, then her distance stays constant. Next her distance increases more quickly, then more slowly, and finally stays constant again. For this trip, we can interpret the two horizontal parts as stops.
One possible story is that Emily drives along a local road, stops for gas, drives faster on a highway, slows down in a neighborhood, and stops when she arrives at her grandmother’s house. Any story that agrees with the order and relative rates shown by the graph can work.
Example: Sketch a graph for a bulldozer story
A bulldozer is filling a hole. It starts quickly, slows down, and then breaks. After repairs, it fills the hole quickly again. We want the depth of the hole on the vertical axis and time on the horizontal axis.
The hole gets shallower while the bulldozer works, so the graph falls. Start with a steep downward segment for the fast work, then a less steep downward segment as the machine slows. During repairs the depth stays constant, giving a horizontal segment. Finish with a steeper downward segment as the bulldozer works quickly again.
These are qualitative sketches, so their shapes describe the changes without assigning exact times or distances. If a problem supplies measurements, place those values on the axes and use them to set the segment endpoints.
