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
Access past paper 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.