When a straight-line model is fitted to data, it is reliable only for the range of values it was built from. Interpolation means predicting inside that range. Extrapolation means predicting outside it, and that needs a judgement about whether the situation still behaves the same way.
This lesson finishes straight lines and linearisation. Questions that use it often add a sentence such as “comment on the validity of your answer”.
How do you judge a prediction?
Ask three questions in order.
First, is the value of x inside the data range? Second, is the predicted answer physically possible, for example not negative for a length?
Third, could the real process change shape, such as a tank filling up, a battery draining to zero or a cost that has a fixed cap?
A strong comment names the quantity and states the reason. “The model is not reliable” on its own earns little; “the model predicts more than the tank can hold” earns the mark.
Worked example
A tank is filled at a steady rate. Readings from 1 to 6 minutes suggest V = 12t + 30, where V is litres in the tank and t is minutes. The tank holds at most 300 litres.
Step 1, interpolate: at t = 3.5, V = 12(3.5) + 30 = 42 + 30 = 72 litres. This lies inside the data range of 1 to 6, so it is interpolation and reasonably reliable.
Step 2, extrapolate: at t = 30, V = 12(30) + 30 = 390 litres. This is outside the data and above the capacity of 300, so the model cannot be right there.
Step 3, find where it breaks: set 12t + 30 = 300, so 12t = 270 and t = 22.5 minutes. The model can be trusted only up to about that time.
Check: 12(22.5) + 30 = 270 + 30 = 300.
Comment: the model predicts 390 litres at 30 minutes, which is more than the tank holds, so it is not valid beyond 22.5 minutes.
The mistake to watch for
A common slip is to accept the calculated number as a fact about the world.
Mistaken working: “V = 390, so the tank holds 390 litres.”
The line continues forever, but the tank does not.
The correction is to compare every extrapolated answer with what you know about the situation, and say so in words. Also notice negative values: at t = −5 the model gives V = −30, which is impossible, so the model does not describe times before the filling started.
Check yourself
1. A model P = 4.5d + 18 was fitted using data for d between 2 and 8. For d = 5, 12 and 0, say whether each is interpolation or extrapolation, and find P.
Show answer
d = 5 is inside 2 to 8: interpolation, P = 4.5(5) + 18 = 22.5 + 18 = 40.5.
d = 12 is above 8: extrapolation, P = 54 + 18 = 72.
d = 0 is below 2: extrapolation, P = 18.
2. A battery level is modelled by B = 95 − 7.5t, where B is a percentage and t is hours. When does the model reach zero, and what does it give at t = 15?
Show answer
Zero when 7.5t = 95, so t = 95 ÷ 7.5 = 12 2/3 hours, about 12.67.
At t = 15, B = 95 − 112.5 = −17.5. A battery cannot be below 0%, so the model is not valid after about 12.67 hours.
3. A line y = 3x + 2 fits readings for x from 1 to 5. The real quantity cannot exceed 50. What is the largest x for which the model can be valid?
Show answer
3x + 2 ≤ 50, so 3x ≤ 48 and x ≤ 16. The largest value is x = 16. Check: 3(16) + 2 = 50. Even this is extrapolation, since the data stopped at 5.
Where this leads next
Bring this habit to every model question in the mixed practice set, where one question asks for a comment on validity. The non-calculator working trainer helps with the arithmetic so that your attention stays on the judgement.
Some students can calculate and then stop, leaving the final comment mark behind. Our teachers practise that short written step in online one-to-one Additional Mathematics tuition.