Big idea: A root means the output is zero.
In a word problem, ask: what does zero mean here?
| Model | Root means |
|---|---|
| Profit | Profit = 0 -> break-even |
| Height | Height = 0 -> object reaches the ground |
| Distance from home | Distance = 0 -> back home |
| Temperature change | Change = 0 -> no change |
Worked example
A profit graph has a root at x = 5.
Interpret this.
Step by step
- A root means the output is zero.
- For a profit graph, output = profit.
- So profit is 0 when x = 5.
Final answer
The business breaks even when x = 5.
Exam sentence: Do not just write x = 5.
Say what happens at x = 5 in the situation.
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Intersection = same value: If two graphs intersect, both models give the same output at the same input.
In context questions, read both coordinates.
The x-value tells when or where.
The y-value tells the shared amount.
| Situation | Intersection means |
|---|---|
| Two cost plans | Both plans cost the same |
| Cost and revenue | Break-even point |
| Two height graphs | Both objects have the same height |
| Two distance graphs | Both objects are the same distance from the start |
Worked example
Two phone plan graphs meet at (8, 56). Interpret this point.
In this context: x = number of months, y = total cost (in dollars).
Step by step
- Read the x-coordinate: x = 8. This means the time is 8 months.
- Read the y-coordinate: y = 56. This means the cost is $56.
- Interpret the intersection: because both graphs meet at this point, both phone plans have the same cost at that time.
Final answer
At 8 months, both phone plans cost $56, so neither plan is cheaper at that exact month.
Both coordinates matter: Do not just write one number.
Write a full sentence with both the time/location and the shared value.
Exam checklist
- State what x represents in context.
- State what y represents in context.
- Use both coordinates from the intersection point.
- Write a full interpretation sentence, not just numbers.
See how examiners mark answers
Answer exam-style questions with model answers. Learn exactly what earns marks and what doesn't.
Not every answer makes sense: A calculator may give several solutions, but the real situation may only allow some of them.
| Impossible value | Why reject it |
|---|---|
| Negative time | Time after an event cannot be negative |
| Negative population | Population cannot be negative |
| Negative length | Length cannot be negative |
| Decimal people | Number of people usually needs a whole number |
Worked example
A height graph has roots at t = −1 and t = 3.
Which root is meaningful if t is time after launch?
Step by step
- t = −1 means 1 second before launch, so it does not fit the context.
- t = 3 means 3 seconds after launch, so it makes sense.
Final answer
t = 3 is the relevant solution.
IB wording: If you reject a solution, say why in context.
Do not just cross it out.
Write a sentence, not just a number: IB often gives marks for interpretation.
That means your answer must use the context words from the question.
| Weak answer | Better answer |
|---|---|
| x = 5 | The company breaks even after 5 months. |
| (8, 56) | After 8 months, both plans cost 56. |
| t = 3 | The object hits the ground after 3 seconds. |
| y = 0 | The profit is zero, so the business breaks even. |
Simple interpretation formula
- Say what the x-value means.
- Say what the y-value means.
- Use context words like cost, profit, height, time, distance, or months.
Common mistake: A coordinate alone is usually not enough for an interpretation question.