Context changes the wording: A root is an x-value where the model output is zero. In context, that can mean profit is zero, height is zero, velocity is zero, and so on.
| Graph/model | Possible meaning of root |
|---|---|
| Profit graph | break-even point |
| Height graph | object reaches the ground |
| Population-change model | no change if output is zero |
The graph value zero matters: Roots tell you where the output becomes zero.
Intersection = equality of models: If two graphs intersect, it means both models give the same output at that input value.
Worked example
Two cost graphs meet at (8, 56). Interpret this.
Step by step
- At x = 8, both cost models give y = 56.
Final answer
At 8 units, both options cost 56.
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Not every algebraic/graphical solution makes sense: A negative time, a negative population, or an impossible context value may need to be rejected.
Worked example
A height graph has roots at t = -1 and t = 3. Which is physically meaningful if t is time after launch?
Step by step
- Negative time before launch is not relevant.
- The meaningful root is t = 3.
Final answer
t = 3 is the relevant solution.
Write a sentence, not just a coordinate: For context questions, explain what the number means in the situation.
| Weak answer | Better answer |
|---|---|
| x = 5 | The company breaks even after 5 months. |
| (8, 56) | At 8 units, both plans cost 56. |
Use the context words: Say profit, cost, distance, time, or whichever variable the graph is representing.