Selecting Polynomial Models (College Board AP® Precalculus): Study Guide

Selecting polynomial models

How do I choose an appropriate function type to model data?

• Different function types have characteristic behaviors that can help you identify which type best fits a data set or contextual scenario

• The key is to look at how the output values change as the input values change

  • and match that behavior to a known function type

When is a linear model appropriate?

• A linear model is appropriate when the data demonstrates a roughly constant rate of change

  • i.e. the differences between consecutive output values (over equally spaced input values) are roughly constant

• In contextual scenarios, look for situations where one quantity changes by a fixed amount for every unit change in another quantity

  • E.g. a car travelling at a constant speed

    • distance increases by the same amount each hour

When is a quadratic model appropriate?

• A quadratic model is appropriate when the data demonstrates roughly linear rates of change

  • I.e. the first differences are not constant

  • but the second differences (differences of the first differences) are roughly constant

• A quadratic model may also be appropriate when the data set is

  • roughly symmetric

  • with a unique maximum or minimum value

• In contextual scenarios involving area or two dimensions, a quadratic model is often appropriate

When is a higher-degree polynomial model appropriate?

• A cubic model is appropriate when

  • the first and second differences of the output values are not constant

    • but the third differences of the output values are roughly constant

  • In contextual scenarios involving volume or three dimensions, a cubic model is often appropriate

• More generally, a polynomial model of degree is appropriate when the data demonstrates roughly constant nonzero th differences

  • 1st differences constant: degree 1 (linear model)

  • 2nd differences constant: degree 2 (quadratic model)

  • 3rd differences constant: degree 3 (cubic model)

  • And so on

• A polynomial model is also appropriate when the data shows

  • multiple real zeros

  • or multiple maxima or minima

How many data points do I need?

• A polynomial function of degree or less can be used to model a set of points with distinct input values

  • The set of points could be represented on a graph or in a table

• E.g. 2 points determine a linear function (degree 1)

  • 3 points determine a quadratic function (degree 2)

  • 4 points determine a cubic function (degree 3)

• This means that if you have a specific number of data points

  • a polynomial of at most one degree less than the number of points can pass through all of them

  • E.g. a cubic function (degree 3) can always be found that will go exactly through 4 points

    • But depending on what the points are, a linear or quadratic function might also exist that goes through the 4 points

How do I use successive differences to identify the function type?

• Given a table of values with equally spaced input values

  • Calculate the 1st differences (differences between consecutive output values)

    • If the 1st differences are roughly constant, then use a linear model

  • If not, calculate the 2nd differences (differences between consecutive 1st differences)

    • If the 2nd differences are roughly constant, then use a quadratic model

  • If not, continue to 3rd differences, and so on

• The function type is determined by the first level at which the differences become roughly constant

• E.g. consider this table:

• 1st differences: — not constant

  • 2nd differences: — constant

• The 2nd differences are constant

  • so is best modeled by a quadratic function