THE GRAPH OF A POLYNOMIAL FUNCTION

Review: The Degree of a Polynomial  The degree of a polynomial is equal to the power of its highest power of x.  For example, the degree of x 4 + 3x 3 – 2x is four.

The Graph of a Polynomial Function  Graphs of polynomial functions have several important properties.  Graphs of polynomial functions are continuous for all x – they never jump from one point to another without crossing the distance between the two points.  Graphs of polynomial functions are smooth – they don’t have any sharp corners.

Degree and the Graph  The degree of a polynomial influences its graph in several ways.  First, it determines how many zeroes the graph could have. A function can have a number of zeroes equal to its degree (although not all will have that many)  Second, it determines how many turning points (points where the function switches between increasing and decreasing) the graph can have. A function can have a number of turning points equal to its degree minus one (again, not all will have that many).  Finally, along with the leading coefficient, the degree of a polynomial determines its end behavior – what it does as x goes to infinity or negative infinity.

Example  What can we say about the graph of y = x 3 - 4x without doing any algebra?  x 3 - 4x is a polynomial, so we know the graph will be continuous and smooth.  The polynomial is third degree, so we know it has at most two turning points.  In addition, the graph has at most 3 zeroes.  It’s degree is odd and its leading coefficient is positive, so by the leading coefficient test it increases without bound as x goes to infinity and decreases without bound as x goes to negative infinity.

Sample Graphs This is the graph of: x 3 - 4x Note the three roots and two turning points.

Sample Graphs This is a graph of: x 3 +2x 2 +x Note the two turning points and two roots.

Sample Graphs This is a graph of x + 36x 2 + x 3 - 6x 4 + x 5 Note the five roots and four turning points.