MA 15800 Lesson 10 Notes
Polynomial Functions of Degree Greater than Two Summer 2016
1
We have already studied polynomial functions of degree 1 (lines of the form
๐(๐)=๐๐ + ๐). We have also studied polynomial functions of degree 2 (quadratic
functions) of the form ๐(๐)= ๐๐๐+๐๐ + ๐ whose graphs are parabolas.
A polynomial function with degree greater than 2 has a graph that is some type of curve.
The graph of a polynomial function will usually partially lie above the x-axis and partially
lie below the x-axis. The locations (values of x) where it crosses or touches the x-axis are
called zeros.
We say when a function is positive, the graph lies above the x-axis. When a function is
negative, the graph lies below the x-axis. Below is a possible graph for a polynomial function
with zeros at ๐ฅ = ๐, ๐, ๐๐๐ ๐ . (The points that are locations of the zeros are shown.) You will
notice that the function is negative in the intervals (โโ, ๐) and (๐, โ). The function is positive
on the intervals (๐, ๐) and (๐, ๐ ). Notice that these are open intervals because the zeros are on
the x-axis, not above or below it (neither positive or negative y-values).
To sketch a polynomial function, follow these steps.
1. Find the zeros (x-intercepts) and the y-intercept.
2. Use zeros to find where the function is positive and where it is negative by
making a โsign chartโ.
3. Make a table of (x, f(x)) values and find at least one point in each interval of the
sign chart.
4. Include some points close to the origin.
5. If possible, determine the increasing/decreasing nature of the function.
