Many graphs tell a story without any equation. You can read the story from the graph of a function, or sketch a graph from a story, by looking at its shape.
Why it works
A graph pairs each time with an amount. Its height is the amount, not a place on a map. So a rising part means the amount is growing, a flat part means it is not changing, and a falling part means it is shrinking. A steeper part means the change is happening faster.
Key idea
Read left to right, one piece at a time: rising (increasing), flat (constant) or falling (decreasing), and steep (fast) or gentle (slow).
Worked examples
Worked example: Tell the story
Describe Ana’s walk in the graph above.
- A: the distance rises gently, so she walks away from home slowly.
- B: flat, so she stops for 2 minutes.
- C: a steep rise, so she moves away fast (maybe she runs).
- D: the graph falls to 0, so she heads back and arrives home.
Answer: walk away, stop, run farther, return home
Worked example: Sketch from a story
A bathtub fills at a steady rate, stays full while someone bathes, then drains quickly. Describe the graph of water in the tub over time.
- Filling steadily: a straight part rising at a constant slant.
- Bathing: a flat part.
- Draining quickly: a steep part falling to 0.
Answer: rise, flat, steep fall
Worked example: Compare speeds
During which part of Ana’s walk was she moving fastest?
- Speed shows as steepness. Part C rises 6 blocks in 1 minute; part A rises 4 blocks in 4 minutes; part D falls 10 blocks in 4 minutes.
- C is the steepest.
Answer: part C
Common mistake
Reading a distance graph like a picture of the path, so that a rising part looks like walking uphill. The graph goes up because the distance from home grows; the ground could be flat. A part that falls to 0 means arriving home, not walking downhill.