A **polynomial function** is a function that involves variables raised to non-negative integer powers.

The **standard form** of a polynomial function is given by f\left(x\right)=a_{n}x^{n}+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+\ldots + a_{2} x^{2}+a_{1}x+a_{0} where n is a positive integer and a_{n},\,a_{n-1},\,a_{n-2},\,\ldots,\,a_{2},\,a_{1},\,a_{0} are constant coefficients.

The **domain** of every polynomial function is \left(-\infty,\,\infty\right).

The **degree**, n, of a polynomial function can be found by identifying the highest exponent on the independent variable. The degree tells us information about the key features, such as the **range**, **end behavior**, number of **x-intercepts, ** and turning points.

A polynomial of degree n can have at most n x-intercepts, with those of odd degree having at least one.

The rate of change of a polynomial function is variable, meaning it changes over the course of the domain.

Given this polynomial function:

a

Determine the end behavior of the function.

Worked Solution

b

State the domain and range of the function.

Worked Solution

c

Determine if the degree of the function is even or odd.

Worked Solution

This graph shows f\left(x\right)=x^{4}+2x^{3}-7x^{2}-8x+12.

a

Determine f\left(2\right).

Worked Solution

b

Name the type of extrema seen on the graph.

Worked Solution

c

Determine the range, writing your answer in set notation.

Worked Solution

The graphs of two functions are shown:

a

Identify and compare the intervals where each function is increasing, decreasing, or constant.

Worked Solution

b

Identify and compare the x and y-intercepts of each function.

Worked Solution

Idea summary

The **degree**, n, of a polynomial function is the same as the highest exponent on the variable. It tells us information about the key features, such as the:

range

number of x-intercepts

end behavior

number of turning points

The domain of every polynomial function is (-\infty,\,\infty).