Given a set of data relating two variables $x$x and $y$y, we can use a model to best estimate how the dependent variable $y$y changes in response to the independent variable $x$x. The model allows us to go one step further and make predictions about other possible ordered pairs that fit this relationship.
Exploration
Say we gathered several measurements on the population $P$P of a small town $t$t years after an earthquake. We can then plot the data on the $xy$xy-plane as shown below.
 |
| Population of a town measured at several instances. |
We can fit a model through the observed data to make predictions about the population at certain times after the earthquake. One plausible model might be the following exponential function.
 |
| Exponential curve modelling population of a town over time. |
To make a prediction on the population, say two years after the earthquake, we first identify the point on the curve when $t=2$t=2. Then we find the corresponding value of $P$P. As you can see below, the model predicts that two years after the earthquake, the population of the town was $250$250.
 |
| A predicted population of $250$250 when $t=2$t=2. |
A prediction which is made within the observed data set is called an interpolation. Roughly speaking, we've gathered data between $t=0.3$t=0.3 and $t=2.3$t=2.3 so a prediction at $t=2$t=2 would be classified as an interpolation.
If we predict the population six years after the earthquake, we find that the population is roughly $16$16. A prediction outside the observed data set such as this one is called an extrapolation.
 |
| A predicted population of $16$16 when $t=6$t=6. |
How reliable are these predictions? Well, any model that fits the observed data will make reliable predictions from interpolations since the model roughly passes through the centre of the data points. We can say that the model follows the trend of the observed data.
However extrapolations are generally unreliable since we make assumptions about how the relationship continues outside of collected data. Sometimes extrapolation can be made more reliable if we have additional information about the relationship.
Consider if we were to use the following polynomial model to fit the data. We can see that interpolating doesn't change much from the previous exponential model, but the predicted values from extrapolation are very different.
 |
| Polynomial curve modelling population of a town over time. |
With further information, say like government funding and support aid, the population of the town might increase after a certain point and so the polynomial curve may be an appropriate model to use.
A prediction made within the observed data is called an interpolation.
A prediction made outside the observed data is called an extrapolation.
Generally, extrapolation is less reliable than interpolation since the model makes assumptions about the relationship outside the observed data set.
Practice questions
question 1
Seven data points have been plotted below with a cubic curve of best fit.
Predict the $y$y-value of a point with an $x$x-value of $7$7.
Which of the following points would be predicted by the cubic curve of best fit?
$\left(3,6\right)$(3,6)
A$\left(6,11\right)$(6,11)
B$\left(9,3\right)$(9,3)
C$\left(7,8\right)$(7,8)
D
question 2
The following logarithmic curve of best fit was used to model the set of data points:
$\left(3,7.9\right),\left(4,10.2\right),\left(5,11.3\right),\left(6,12.9\right),\left(7,14.1\right)$(3,7.9),(4,10.2),(5,11.3),(6,12.9),(7,14.1)
Predict the $y$y-value of a point with an $x$x-value of $16$16.
Which of the following points would be predicted by the logarithmic curve of best fit?
$\left(8,15\right)$(8,15)
A$\left(12,19\right)$(12,19)
B$\left(14,20\right)$(14,20)
C$\left(10,18\right)$(10,18)
D
question 3
The height of a particular projectile $y$y in metres was measured at different times $t$t in seconds and the following curve of best fit was drawn.
Using the curve of best fit, what is the predicted height of the projectile after $1$1 second?
Using the curve of best fit, what is the predicted height of the projectile after $4$4 seconds?
The height at a given time has the following relationship $y=-5t^2+100$y=−5t2+100. Which of the following statements about interpolation and extrapolation is true?
Extrapolation is less reliable since we make assumptions about the relationship outside of the observed data.
AExtrapolation is less reliable since we should use a line of best fit instead.
B