In this section we are going to look at a method for getting
a rough sketch of a general polynomial.
The only real information that we’re going to need is a complete list of
all the zeroes (including multiplicity) for the polynomial.
In this section we are going to either be given the list of
zeroes or they will be easy to find. In
the next section we will go into a method for determining a large portion of
the list for most polynomials. We are
graphing first since the method for finding all the zeroes of a polynomial can
be a little long and we don’t want to obscure the details of this section in
the mess of finding the zeroes of the polynomial.
Let’s start off with the graph of couple of polynomials.
Do not worry about the equations for these polynomials. We are giving these only so we can use them
to illustrate some ideas about polynomials.
First, notice that the graphs are nice and smooth. There are no holes or breaks in the graph and
there are no sharp corners in the graph.
The graphs of polynomials will always be nice smooth curves.
Secondly, the “humps” where the graph changes direction from
increasing to decreasing or decreasing to increasing are often called turning points. If we know that the polynomial has degree n then we will know that there will be
at most turning points in the graph.
While this won’t help much with the actual graphing process
it will be a nice check. If we have a
fourth degree polynomial with 5 turning point then we will know that we’ve done
something wrong since a fourth degree polynomial will have no more than 3
Next, we need to explore the relationship between the x-intercepts of a graph of a polynomial
and the zeroes of the polynomial. Recall
that to find the x-intercepts of a function we need to solve the equation
Also, recall that is a zero of the polynomial, ,
provided . But this means that is also a solution to .
In other words, the zeroes of a polynomial are also the x-intercepts of the graph. Also, recall that x-intercepts can either cross the x-axis or they can just touch the x-axis without actually crossing the axis.
Notice as well from the graphs above that the x-intercepts can either flatten out as
they cross the x-axis or they can go
through the x-axis at an angle.
The following fact will relate all of these ideas to the
multiplicity of the zero.
Finally, notice that as we let x get large in both the positive or negative sense (i.e. at either end of the graph) then
the graph will either increase without bound or decrease without bound. This will always happen with every polynomial
and we can use the following test to determine just what will happen at the
endpoints of the graph.
Suppose that is a polynomial with degree n.
So we know that the polynomial must look like,
We don’t know if there are any other terms in the
polynomial, but we do know that the first term will have to be the one listed
since it has degree n. We now have the following facts about the
graph of at the ends of the graph.
- If and n
is even then the graph of will increase without bound positively at
both endpoints. A good example of
this is the graph of x2.
- If and n
is odd then the graph of will increase without bound positively at
the right end and decrease without bound at the left end. A good example of this is the graph of x3.
- If and n
is even then the graph of will decrease without bound positively at
both endpoints. A good example of
this is the graph of -x2.
- If and n
is odd then the graph of will decrease without bound positively at
the right end and increase without bound at the left end. A good example of this is the graph of -x3.
Okay, now that we’ve got all that out of the way we can
finally give a process for getting a rough sketch of the graph of a polynomial.
Process for Graphing
all the zeroes of the polynomial and their multiplicity. Use the fact above to determine the x-intercept that corresponds to
each zero will cross the x-axis
or just touch it and if the x-intercept
will flatten out or not.
the y-intercept, .
the leading coefficient test to determine the behavior of the polynomial
at the end of the graph.
a few more points. This is left
intentionally vague. The more
points that you plot the better the sketch. At the least you should plot at least
one at either end of the graph and at least one point between each pair
We should give a quick warning about this process before we
actually try to use it. This process
assumes that all the zeroes are real numbers.
If there are any complex zeroes then this process may miss some pretty
important features of the graph.
Let’s sketch a couple of polynomials.
Example 1 Sketch
the graph of .
We found the zeroes and multiplicities of this polynomial
in the previous section
so we’ll just write them back down here for reference purposes.
So, from the fact we know that will just touch the x-axis and not actually cross it and that will cross the x-axis and will be flat as it does this since the multiplicity is
greater than 1.
Next, the y-intercept
The coefficient of the 5th degree term is
positive and since the degree is odd we know that this polynomial will
increase without bound at the right end and decrease without bound at the
Finally, we just need to evaluate the polynomial at a
couple of points. The points that we
pick aren’t really all that important.
We just want to pick points according to the guidelines in the process
outlined above and points that will be fairly easy to evaluate. Here are some points. We will leave it to you to verify the
Now, to actually sketch the graph we’ll start on the left
end and work our way across to the right end.
First, we know that on the left end the graph decreases without bound
as we make x more and more negative and this agrees with the point that
we evaluated at .
So, as we move to the right the function will actually be
increasing at and we will continue to increase until we
hit the first x-intercept at . At this point we know that the graph just
touches the x-axis without actually
crossing it. This means that at the graph must be a turning point.
The graph is now decreasing as we move to the right. Again, this agrees with the next point that
we’ll run across, the y-intercept.
Now, according to the next point that we’ve got, ,
the graph must have another turning point somewhere between and since the graph is higher at than at . Just where this turning point will occur is
very difficult to determine at this level so we won’t worry about trying to
find it. In fact, determining this
point usually requires some Calculus.
So, we are moving to the right and the function is
increasing. The next point that we hit
is the x-intercept at and this one crosses the x-axis so we know that there won’t be
a turning point here as there was at the first x-intercept. Therefore,
the graph will continue to increase through this point until we hit the final
point that we evaluated the function at, .
At this point we’ve hit all the x-intercepts and we know that the graph will increase without
bound at the right end and so it looks like all we need to do is sketch in an
Here is a sketch of the polynomial.
Note that one of the reasons for plotting points at the
ends is to see just how fast the graph is increasing or decreasing. We can see from the evaluations that the
graph is decreasing on the left end much faster than it’s increasing on the
Okay, let’s take a look at another polynomial. This time we’ll go all the way through the
process of finding the zeroes.
Example 2 Sketch
the graph of .
First, we’ll need to factor this polynomial as much as
possible so we can identify the zeroes and get their multiplicities.
Here is a list of the zeroes and their multiplicities.
So, the zeroes at and will correspond to x-intercepts that cross the x-axis
since their multiplicity is odd and will do so at an angle since their
multiplicity is NOT at least 2. The
zero at will not cross the x-axis since its multiplicity is even.
is and notice that this is also an x-intercept.
The coefficient of the 4th degree term is
positive and so since the degree is even we know that the polynomial will
increase without bound at both ends of the graph.
Finally, here are some function evaluations.
Now, starting at the left end we know that as we make x more and more negative the function
must increase without bound. That
means that as we move to the right the graph will actually be
At the graph will be decreasing and will
continue to decrease when we hit the first x-intercept at since we know that this x-intercept will cross the x-axis.
Next, since the next x-intercept
is at we will have to have a turning point
somewhere so that the graph can increase back up to this x-intercept. Again, we
won’t worry about where this turning point actually is.
Once we hit the x-intercept
at we know that we’ve got to have a turning
point since this x-intercept
doesn’t cross the x-axis. Therefore to the right of the graph will now be decreasing.
It will continue to decrease until it hits another turning
point (at some unknown point) so that the graph can get back up to the x-axis for the next x-intercept at . This is the final x-intercept and since the graph is increasing at this point and
must increase without bound at this end we are done.
Here is a sketch of the graph.
Example 3 Sketch
the graph of .
As with the previous example we’ll first need to factor
this as much as possible.
Notice that we first factored out a minus sign to make the
rest of the factoring a little easier.
Here is a list of all the zeroes and their multiplicities.
So, all three zeroes correspond to x-intercepts that actually cross the x-axis since all their multiplicities are odd, however, only the x-intercept at will cross the x-axis flattened out.
is and as with the previous example this is
also an x-intercept.
In this case the coefficient of the 5th degree
term is negative and so since the degree is odd the graph will increase
without bound on the left side and decrease without bound on the right side.
Here are some function evaluations.
Alright, this graph will start out much as the previous
graph did. At the left end the graph
will be decreasing as we move to the right and will decrease through the
first x-intercept at since know that this x-intercept crosses the x-axis.
Now at some point we’ll get a turning point so the graph
can get back up to the next x-intercept
at and the graph will continue to increase
through this point since it also crosses the x-axis. Note as well that
the graph should be flat at this point as well since the multiplicity is
greater than one.
Finally, the graph will reach another turning point and
start decreasing so it can get back down to the final x-intercept at . Since we know that the graph will decrease
without bound at this end we are done.
Here is the sketch of this polynomial.
The process that we’ve used in these examples can be a
difficult process to learn. It takes
time to learn how to correctly interpret the results.
Also, as pointed out at various spots there are several
situations that we won’t be able to deal with here. To find the majority of the turning points we
would need some Calculus, which we clearly don’t have. Also, the process does require that we have
all the zeroes and that they all be real numbers.
Even with these drawbacks however, the process can at least
give us an idea of what the graph of a polynomial will look like.