Constant rate of change → linear model: A linear model fits when the dependent variable increases (or decreases) by the same amount for every one-unit increase in the independent variable.
This constant rate of change is the gradient.
Three signals in a word problem that suggest a linear model:
- Language: "increases by … per …", "costs … for each …", "charges a fixed … plus … per …"
- Data: the differences between consecutive y-values are (roughly) equal.
- Scatter plot: the points lie close to a straight line.
| Scenario | Linear? | Why |
|---|---|---|
| Bicycle hire: $60 per day + $10 fixed fee | Yes | Cost increases by exactly $60 per day — constant rate. |
| Ant colony growing 15% each week | No | Growth by percentage → exponential, not linear. |
| Car travelling at constant speed: d = 80t | Yes | Distance increases by 80 km for each extra hour. |
| Ball thrown upward: h = −5t² + 20t | No | The t² term makes it quadratic. |
| Balloon volume vs. passengers (r = 0.998) | Yes (strong linear) | Pearson r close to ±1 → linear regression appropriate. |
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- Dependent variable (output)
- Independent variable (input)
- Gradient — rate of change of y per unit of x
- y-intercept — value of y when x = 0
A hot-air balloon company finds that the recommended minimum volume V (m³) depends on the number of passengers p.
Data: p = 1 → V = 1000; p = 15 → V = 5800.
The data lies very close to a straight line.
Write a linear model and use it to predict V when p = 10.
Step by step
- Calculate the gradient using two data points.
- Find c using y = mx + c with point (1, 1000).
- Write the model.
- Substitute p = 10.
Final answer
Predicted minimum volume for 10 passengers ≈ 4086 m³. (In practice, use GDC linear regression for the exact line.)
GDC LinReg — faster and more accurate: For more than two data points, do LinReg on the GDC instead of computing the gradient by hand.
TI-84: 1. STAT → 1:Edit — type x-values into L1 and y-values into L2. 2. STAT → CALC → 4:LinReg(ax+b) — press ENTER. 3. Read off a (gradient) and b (y-intercept). The screen also shows r (correlation) when DiagnosticOn is set.
Casio fx-CG50: 1. MENU → Statistics — type the x-values into List 1 and y-values into List 2. 2. CALC → REG → X → aX+b — read a and b.
LinReg gives the BEST-fit line through ALL data points (least-squares); the two-point method only fits a line through two of them and ignores the rest. Use LinReg whenever the question gives you a full table of data.
Build a linear model from data — tutor fee f(h) = 25h + 50.
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Every coefficient has a real-world meaning: IB questions frequently ask: "State what the value of m represents in this context." Never write "m is the gradient" — always say what the gradient means in terms of the variables.
Bicycle hire context — interpret m and c
For the bicycle hire model C(d) = 60d + 10 (d = days, C = cost in $), interpret m = 60 and c = 10.
Step by step
- Identify the gradient m.
- Interpret m in context: for each additional day of hire, the cost increases by $60.
- Identify the y-intercept c.
- Interpret c in context: there is a fixed charge of $10 (e.g. helmet and repair kit hire) before any days are added.
Final answer
m = 60: cost increases by $60 per day. c = 10: there is a fixed charge of $10 (helmet and repair kit hire).
c may not be meaningful in context: If x = 0 is outside the realistic domain (e.g. 0 passengers in a balloon), the y-intercept c is a mathematical extrapolation, not a physically meaningful value.
Note this in your answer.
A model is only valid within its domain: The linear model V = 342.9p + 657.1 was built from data for 1 to 15 passengers.
Predicting V for p = 50 is extrapolation — the relationship might not stay linear beyond the data range.
✓ Interpolation (reliable)
- Predicting within the data range
- p = 8 passengers (data goes 1 to 15)
- Reliable — the linear pattern has been observed here
✗ Extrapolation (unreliable)
- Predicting beyond the data range
- p = 50 passengers (far outside 1 to 15)
- Unreliable — the model may not hold; volume cannot grow infinitely
Validity comment = easy marks: IB mark schemes award 1–2 marks for commenting on model validity.
Write: "The model may not be reliable for p > 15 as this is extrapolation beyond the data range." This mark is free if you remember to include it.