We will go over some different methods we can use to identify the correct model for a given data set. We will discuss how to use scatter plots and what patterns to look for in each case.

Using a Scatter Plot

When we have a data set and we are not sure which model can we use, it can be very helpful to start by drawing a scatter plot. This, can help us visualize the behavior of the data set and possible help us to determine a good model.

If the data looks close enough to an exponential, polynomial or logarithmic function, we can use the corresponding regression feature in a graphic calculator to find the best fit model.

Looking For Patterns in the Data

In the case that we do not have a graphic calculator but the data has equally-spaced inputs we can still identify polynomial and exponential data by look for patterns in the outputs. Recall that we can know if the data set fits a polynomial by finding the finite differences.

If the $n th$ differences of function's values for equally-spaced data are nonzero and constant, then the data can be represented by a polynomial function of degree $n .$

For example, by looking at the finite differences of the data set below we can determine that it corresponds to a cubic function since the third differences are constant.

On the other hand, for exponential functions, equally-spaced inputs generate outputs with common ratios. This is because an exponential function grows or decreases by a determined constant factor. For example, consider the following table of values for the function $y = 2^{x} .$

Notice that the constant factor is not necessarily equal to the base. This depends on the spacing between the inputs. Nevertheless once we know that the data fits an exponential function we can find the corresponding function using two of these points into the general form of exponential function.

Exercise

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