It’s Professor Dave, let’s graph higher-degree polynomials.
We now know how to graph a variety of functions, including parabolas, which are second-degree,
or quadratic functions.
But what about cubic functions, or quartic, or functions of higher degree still?
This can get very difficult very fast, as plotting points gets rather time consuming,
and the behavior of these functions is not as predictable as lines and parabolas.
But there are a few tricks we can use to get rough sketches of these functions that are
fairly accurate, and provide some key information about the function that is useful.
Let’s learn these tricks now.
First, we must realize that polynomial functions with a degree of two or higher are smooth
and continuous.
They can have hills and valleys in the middle, but on either side they tend to go towards
positive or negative infinity.
There are several possibilities for the end behavior of polynomial functions, which means
what it does as X goes to positive or negative infinity, so let’s start by learning how
to figure out what the end behavior of a function will be.
We can do this by using something called the leading coefficient test.
We look at the leading term in a polynomial, or the term with the largest exponent, and
we look at its exponent and coefficient.
Depending on whether the exponent is even or odd , and whether the coefficient is positive
or negative, we know what the end behavior will be.
If the exponent is odd, such as with a cubic function, or one that starts with X to the
fifth, X to the seventh, and so forth, if the leading coefficient is positive, the function
will fall downwards to the left, and rise upwards to the right.
If the leading coefficient is negative, the function will rise upwards to the left, and
fall downwards to the right.
We don’t yet know what will happen in the middle, but this end behavior is a great start
in terms of sketching the function.
Then, if the exponent is even, such as with quadratic or quartic functions, or ones beginning
with X to the sixth, X to the eighth, and so on, if the leading coefficient is positive,
the function will rise upwards on both sides.
If the leading coefficient is negative, the function will fall downwards on both sides.
So those are the four possibilities in terms of the end behavior of higher degree polynomials.
Now that we have the end behavior, the next thing to do is to find the zeroes of the function.
These will be the values of X that make the function equal to zero, which can be understood
graphically as X intercepts, because these are points where Y equals zero.
To get these, we set the function equal to zero, and solve for X.
This can be very tricky, but sometimes a little factoring will do it.
Other times, we have to use the rational roots test in conjunction with synthetic division
to find out what the zeroes are.
We learned this technique earlier in the series, so it would be a very good idea to check out
these tutorials right now if you need to learn how to graph higher order polynomials.
To see what to do once we find the zeroes, take something like X cubed plus three X squared
minus X minus three equals zero.
If we can factor this or perform synthetic division, we can eventually express this as
X plus three times X plus one times X minus one.
That means that the X intercepts are negative three, negative one, and one, since any of
these values will make one term equal to zero, and thus the whole function will be equal
to zero, since anything times zero is zero.
So we can plot these three points for the function, right on the X axis.
Then, we can get the end behavior by noticing that the leading term has an odd exponent
and a positive coefficient of one, meaning that the function falls to the left and rises
to the right.
This is already enough information to sketch this function, because going left to right,
it must rise up from negative infinity, cross the X axis at this point, curve back down
and cross again here, curve up and cross one more time, and then rise up towards positive infinity.
It’s true that we aren’t getting the exact maxima and minima when using such a simple
approach, that would require calculus, so this isn’t an exact representation, but
the function must indeed look roughly like this, and sometimes that’s all we need to know.
We must point out one other thing at this time.
If a zero is listed precisely once when a function is expressed as its factors, the
function will cross the X axis at that value.
But if a zero is listed twice, then the function will touch the X axis at that value and then
curve back in the direction it came from.
For example, take negative X to the fourth plus four X cubed minus four X squared.
We could do synthetic division, or just recognize that we can factor out a negative X squared.
Then this perfect square can be expressed as X minus two quantity squared.
We can either leave it like this, or understand that this also means negative X times X times X minus
two times X minus two.
This means that zero is one of the zeroes, but it is a zero twice.
Positive two is another zero, but it is a zero twice.
This makes sense because a quartic function should have four zeroes, and each these just
has a multiplicity of two.
Finding the end behavior, the leading exponent is even, and the coefficient is negative,
so this should fall towards negative infinity on either side.
Given what we just said about zeroes that occur twice, the function must rise up from
the left, touch the X axis at X equals zero but then curve back down, curve back up to
touch the X axis again at X equals two, and then fall back down towards negative infinity.
Again, this isn’t flawlessly precise, but for many purposes this is good enough.
So what we can say is that if a zero has an odd multiplicity, like one, or sometimes three
or greater, the function will cross the X axis at that point.
If a zero has an even multiplicity, usually two but sometimes more, the function will
touch the X axis at that point and then turn around.
So now we have a solid strategy for graphing higher degree polynomials.
Start with the leading coefficient test to find the end behavior of the function.
Then, find the zeroes of the function by whatever method necessary, whether factoring or the
rational roots test and synthetic division.
Then, assess the multiplicity of each zero to find out whether the function will cross
or just touch the X axis at each zero.
If we want one more additional point, we can easily get the Y intercept by plugging in
zero for X, which will take just a few seconds.
Then, obeying the end behavior of the function, we sketch the graph.
If you’re confused, watch this clip one more time, otherwise, let’s check comprehension.