Polynomials are often classified by their degree, the highest exponent on the independent variable. For example, a quadratic function, like , is considered a 2nd-degree polynomial because the highest exponent on  is 2. Similarly, a linear function like is considered a 1st-degree polynomial. Earlier, we considered the function , which gives the volume, in cubic inches, of a box made by removing the squares of side length , in inches, from each corner of a rectangle of paper and then folding up the 4 sides. This is an example of a 3rd-degree polynomial, which is easier to see if we use the distributive property to rewrite the equation as .

Graphs of polynomials have a variety of appearances. Here are three graphs of different polynomials with degree 1, 3, and 6, respectively:

Graph of third degree polynomial on coordinate plane with no grid, origin O. Horizontal axis from negative 10 to 8, by 2's. Vertical axis from negative 30 to 20, by 10s. Starting in quadrant 3, the function moves upwards, crossing the x axis at negative 0, continuing upwards until about negative 4 comma 20, curves downwards, crosses the y axis, and curves upwards at 2 comma 0.  

Graph of sixth degree polynomial on coordinate plane with no grid, origin O. Horizontal axis from negative 10 to 8, by 2's. Vertical axis from negative 30 to 20, by 10s.  Starting in quadrant 2, the line moves downwards, crosses the x axis at negative 8, curves around negative 7 comma negative 30, moves upwards crossing the x axis at negative 4, curves arond negative 2 comma 8, moves downwards crossing the y axis, follows the x axis from about 1 to 3, continues downwards, curves upwards around 6 negative 10, crossing the x axis around 7.

Since graphs of polynomials can curve up and down multiple times, they can have points that are higher or lower than the rest of the points around them. These points are relative maximums and relative minimums. In the second graph, there is a relative maximum at about and a relative minimum at . The word relative is used because while these are maximums and minimums relative to surrounding points, there are other points that are higher or lower.