# Polynomial Functions and Modeling

## Overview

### Description

Polynomial functions are functions made up of one or more terms, each term having the form $ax^n$, where $a$ is a real number and $n$ is a positive integer. A polynomial with one term is a power function. Power functions of the form $f(x)=x^n$ can be transformed as parent functions. Graphs of polynomial functions can be analyzed to determine intercepts, maxima and minima, and end behavior. The graph of a polynomial can be used to model trends in data and make predictions. Modeling data is used in a wide range of applications, such as tracking the spread of a disease or the busiest times for a restaurant. A scatterplot can be used to determine which type of polynomial is the best model, and then technology can be used to generate the polynomial of best fit with the given degree.

### At A Glance

• A power function has the form $f(x) = ax^n$, where $a$ is a nonzero real number and $n$ is a real number.
• The graph of a power function can be transformed by performing operations on the function rule.
• A polynomial function can be written as a sum or difference of terms with a variable raised to a nonnegative integer exponent.
• A polynomial function can be even, odd, or neither.
• The degree of a polynomial and the sign of the leading coefficient determine the end behavior of the polynomial.
• Turning points of a graph may be local or global minima or maxima of the function.
• The features of the graph of a polynomial function, including the intercepts and number of turning points, can be used to write the rule of the function.
• Given the graph of a polynomial, solutions of the related equation can be identified by estimating the $x$-intercepts.
• 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.
• 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.