In this section we will explore the graphs of polynomials. We have already discussed
the limiting behavior of even and odd degree polynomials with positive
and negative leading coefficients. Also recall that an nth degree polynomial can have at
most n real roots (including multiplicities) and n−1 turning points. We will explore these ideas by looking
at the graphs of various polynomials.
Cubic functions can have at most 3 real roots (including multiplicities) and 2 turning points. We will look
at the graphs of cubic functions with various combinations of roots and turning
points as pictured below.
The multiplicity of a root affects the shape of the graph of a polynomial. Specifically,
- If a root of a polynomial has odd multiplicity, the graph will cross the x-axis
at the the root.
- If a root of a polynomial has even multiplicity, the graph will
touch the x-axis at the root but will not cross the x-axis.
As the multiplicity
of the root increases, the graph flattens out more and more near the root. In
the red graph above, there is one distinct real root, x = 0, having multiplicity
3. Since the multiplicity is odd, the graph does cross the x-axis at the root, but
the graph flattens out near this root because the root is not simple. In the green
graph above, there are two distinct real roots, x1 = −1 and x2 = 2. Notice that
the graph does not cross the x-axis at the root x2 = 2 (it simply touches the x-axis). This indicates that x2 = 2 is a root of even multiplicity (in fact, the
multiplicity is 2 because a cubic is only degree 3). Thus, the cubic pictured in
green has one simple root, x = −1, and one double root, x = 2 (for a total,
including multiplicities, of 3).
We can actually take a look at how increasing multiplicity affects the shape of
the graph near a root. The figure below shows polynomials with one distinct
real root, x = 0. As you can see, the graph touches the root without crossing
the x-axis when the multiplicity is even, and crosses the x-axis through the root
when the multiplicity is odd. Furthermore, the graph flattens out more and more
near the root as the multiplicity increases.
Now try some problems that will test your knowledge of polynomial functions.