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.

Mauna Loa Data

Linear Scale and Music

This is a linear scale. Note that “so” falls at the midpoint (arithmetic mean) between do and do. Also, that “mi” falls at the midpoint between do and so and that re is at the midpoint between do and mi.

Because of these mathematical relationships, these notes are pleasing to our ears.

Linear Scale of Musical Notes

DBH Tape

We are placing two scales on our line:

  • circumference centimeter scale
    • this scale directly measures the circumference
  • diameter centimeter scale
    • this scale “calculates” the diameter

The circumference scale is a 1-to-1 scale so

1 \textrm{ circumference scale centimeter unit} = 1 \textrm{ paper cm}

Because C = \pi d

1 \textrm{ circumference scale centimeter unit} = \pi \cdot 1 \textrm{ diameter scale unit}

We can create unit conversion factors for each of these

To determine the distance on the page between the zero and the one on the diameter centimeter scale, we convert from

  • diameter centimeter scale to circumference centimeter scale
  • from circumference centimeter scale to paper centimeters

1 \textrm{ dia cm scale} \cdot \frac{1 \textrm{ circ cm scale}}{\pi \cdot 1 \textrm{ dia cm scale}} \cdot \frac{1 \textrm{ paper cm}}{1 \textrm{ circ cm scale}} = 0.318 \textrm{ cm}