The Linear Function as a Tool

If a linear model is appropriate for the situation we are trying to analyze, there are several mathematical techniques that may be of interest.

Extrapolation
Interpolation
Regression or Fitting

Extrapolation

Extrapolation is the use of a linear model to make a prediction. Usually, our linear model is valid over a certain range. In an extrapolation, we assume the model is valid beyond that range and ask what our model would predict.

There are two approaches to extrapolation

doing a calculation based on the slope and an existing data point
using a ruler or other visual aid on a graph to estimate

Interpolation

Interpolation is the use of a linear model to make a prediction. If we have data on both sides of our region of interest, we can make predictions in between.

Linear Fits or Regression

If we have a bunch of data that is roughly linear, we can use a linear function as a model
The process of finding this linear model from the data is called regression or fitting

Linear Extrapolation

If we assume a relationship, we can predict its value in the future

Extrapolation Formula

It is helpful to be able to see how the pieces of the extrapolation formula correspond to a visual picture.

Find the slope of the known portion of the graph so you have the relationship between x and y.
You are calculating the distance on the x-axis between your last known data and the point you want to compute. This will be the run in a slope triangle.
Multiplying this x-distance by the slope will give you the rise in a slope triangle.
Adding this to the last known y-value will give you the corresponding value of y for the x you chose.