The big idea: A linear model uses a straight line for a real situation.
We write it as y = mx + c.
Here, x and y have real units (time, money, distance, temperature).
A situation is linear when the quantity changes by a fixed amount per unit.
If you earn $15 per hour, your earnings increase by exactly that amount for every additional hour — that is a linear relationship.
| Situation | Input x | Output y | Linear? |
|---|---|---|---|
| Phone plan: 0.10/min + 10 base fee | Minutes used | Cost ($) | Yes |
| Population doubling each year | Years | Population | No — exponential |
| Taxi: 3 flag-fall + 2.50/km | Distance (km) | Fare ($) | Yes |
| Compound interest bank account | Years | Balance ($) | No |
IB exam pattern: IB often gives a table of values or a context description and asks you to identify or build a linear model.
Check whether the rate of change is constant — if yes, use y = mx + c.
Interactive graph: linear model and parameters
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The big idea: In any real-world linear model y = mx + c: the gradient m is the rate of change (how much y changes per 1-unit increase in x), and the y-intercept c is the starting value (what y equals when x = 0).
Always include units in your interpretation.
Interpreting a model
A car rental model is C = 0.20d + 30, where C is cost ($) and d is distance (km).
Interpret m and c.
Step by step
- Identify m and c.
- Interpret m: rate of change.
- Interpret c: starting value.
Final answer
Gradient: 0.20 per km. y-intercept: 30 fixed fee.
| Symbol | Role in context | Example interpretation |
|---|---|---|
| m (positive) | Constant rate of increase | Temperature rises by 2°C per hour |
| m (negative) | Constant rate of decrease | Water drains at 50 L/hour |
| c | Starting/initial value | Tank starts with 800 L at t = 0 |
Always include units: Saying 'the gradient is 0.20' is incomplete.
You must say 'the gradient is $0.20 per km' — IB expects units for full marks on an interpretation question.
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The big idea: To write a linear model: identify the fixed starting value (c = y-intercept) and the constant rate of change (m = gradient).
Then write y = mx + c.
If two data points are given instead, find m first, then solve for c.
Model from context description
A factory has 200 widgets in stock.
It produces 50 more per hour.
Write a model for the total W after t hours.
Step by step
- Fixed start: 200 widgets → c = 200.
- Rate of change: 50 widgets/hour → m = 50.
- Model.
Final answer
W = 50t + 200
Model from two data points
A taxi costs $5.50 at 1 km and $9.00 at 3 km.
Find the linear model for fare F in terms of distance d.
Step by step
- Find the gradient.
- Substitute (1, 5.50) to find c.
- Model.
Final answer
F = 1.75d + 3.75. Interpretation: 1.75 per km, 3.75 flag fall.
Context clues for m and c: 'Per hour', 'per km', 'per unit' → these signal the gradient. 'Fixed fee', 'initial value', 'starting amount', 'at time zero' → these signal the y-intercept.
Step through the taxi fare model C = 15d + 20. See how gradient and y-intercept connect to the real-world context.
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The big idea: Once you have a linear model, you can predict values by substituting.
You can also evaluate how realistic the model is by considering domain limits and whether a constant rate of change is reasonable.
Prediction from a model
A candle of initial length 24 cm burns at 3 cm per hour.
The model is L = −3t + 24.
Find:
(a) length after 5 hours;
(b) when the candle burns out.
Step by step
- (a) Substitute t = 5.
- (b) Set L = 0 and solve for t.
Final answer
(a) 9 cm. (b) The candle burns out after 8 hours.
Evaluating a linear model — what to consider
- Domain: the range of x-values for which the model makes sense (e.g., t ≥ 0)
- Range: is the predicted y-value realistic? (e.g., volume cannot be negative)
- Constant rate assumption: does the real situation actually change at a constant rate?
- Extrapolation risk: predicting far beyond the data range may be unreliable
Commenting on validity: IB often asks you to comment on the validity of a linear model. Two points are expected:
1. Constant rate assumption — in reality, the rate of change is unlikely to stay exactly constant throughout.
2. Domain limitation — the model only makes sense within a realistic range of values (e.g., time cannot be negative, length cannot go below zero).