### Polynomial Trends in Data

Data in a scatterplot can show a relationship that can be modeled by a polynomial. The degree of the polynomial can affect how closely the curve fits the data.

Linear and quadratic functions can be used to model data. For data that does not show a linear or quadratic trend, polynomials of higher degree may be used. In general, the higher the degree of the polynomial, the closer the curve can be made to fit the points. However, the coefficients of the terms may be harder to interpret, and the function may not be a good predictor of values outside the given data.
In general, look for the overall shape of the data, and try to choose the function with the lowest degree that will fit the trend. Most often, this will be a linear or quadratic function, but there may be data that fit the pattern of a higher-degree polynomial.

### Polynomials of Best Fit

Technology can be used to generate a polynomial of best fit that approximates the closest relationship between two variables for a polynomial of a given degree.

Just as technology can be used to find a line or quadratic of best fit, it is possible to find other polynomials to fit a given set of data. Graphing calculators will commonly have regression functions up to degree 3 (cubic), but online tools can be used to fit polynomials of any degree.

Step-By-Step Example

Modeling Data Trends with Polynomial Functions

The table shows data values from an experiment. Identify the polynomial of best fit for the data.

$x$ | $y$ |
---|---|

0.18 | 4.24 |

1.64 | 8.33 |

3.23 | 6.43 |

4.26 | 3.44 |

6.69 | 7.62 |

6.80 | 10.64 |

3.49 | 4.92 |

0.79 | 6.72 |

5.74 | 3.52 |

6.09 | 5.18 |

5.92 | 4.51 |

0.85 | 6.40 |

Step 1

Plot the data points.

Step 2

Look for a pattern.

The pattern displayed by the data points resembles a cubic polynomial.

Step 3

Use a graphing utility to identify the polynomial that best models the pattern of the data.

Follow the instructions for the graphing utility to identify the best fit curve. Use a polynomial curve of degree 3, based on the data plot.

Solution

The graphing utility will plot the curve and identify its algebraic rule.