Question

Given the table of data, determine the function that would most appropriately model the data. x 0 1 2 3 f(x) 1.5 2 2.5 3 linear quadratic cubic square root

Answer 1

linear

Explanation:

The x-values are evenly spaced (1 apart), so it is helpful to look at the differences of the f(x) values.

Successive f(x) values all differ by the constant 0.5, a characteristic of a linear function.

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Comment on differences

The differences of successive y-values are called "first differences". The differences of those are called "second differences". And the differences of those are "third differences." The "degree" of the differences that are constant is the degree of the function describing the sequence.

That is, a sequence (like this one) with constant first differences can be described by a first-degree polynomial, a linear function. If third differences are constant, the sequence is described by a third-degree polynomial, a cubic function.

A square root function will have first differences that decrease by a decreasing amount. Successive differences of differences will continue to decrease, never becoming constant.

An exponential function will have first differences with a common ratio.