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In this section we'll focus on how to sketch the graph of any polynomial function, a really important skill.

There are basically three things to do in order to sketch the graph of a polynomial function. Remember that you have many methods at your disposal: GCF, grouping, idendification of special forms, and the rational root theorem.

In the previous section we discussed several ways of finding the roots of polynomial functions. When the independent variable increases in size in either direction ( A± ), the ends of a polynomial graph will eventially increase or decrease without bound (infinitely). End behavior is a clue about the shape of a polynomial graph that you just can't do without, so you should either memorize these possibilities or (better yet) understand where they come from. The table below also shows that a polynomial function of degree n can have at most n - 1 points where it changes direction from down-going to up-going. The end behavior is down on the left and up on the right, consistent with an odd-degree polynomial with a positive leading coefficient.

It takes a few tries to get the hang of this kind of curve sketching, but it will develop with practice. Likewise there are no other options, given the right-end behavior, for the part of f(x) between 0 and 3. What we don't know from such a sketch is just exactly how high the maxima rise and how low the minima dive.

The key to sketching a function like this quickly is seeing that it's just the parent function of all cubic functions, y = x3, shifted to the right by 2 units and inverted across the x-axis.

For these kinds of graphs, I like to lightly sketch in the parent function, then apply the transformations one at a time. Sometimes you won't find a GCF, grouping won't work, your function is not a sum or difference of cubes and it doesn't look like a quadratic, . That means graphing the function on a calculator and estimating x-axis crossings or using a numerical root-finding algorithm. Still, you should be aware of end behavior, the y-intercept and maybe a few other things in order to check that computer output is what you expect it to be. We learned what a Polynomial is here in the Introduction to Multiplying Polynomials section. As a review, here are some polynomials, their names, and their degrees. Remember that the degree of the polynomial is the highest exponent of one of the terms (add exponents if there are more than one variable in that term). We learned that a Quadratic Function is a special type of polynomial with degree 2; these have either a cup-up or cup-down shape, depending on whether the leading term (one with the biggest exponent) is positive or negative, respectively.

There are certain rules for sketching polynomial functions, like we had for graphing rational functions. Again, the degree of a polynomial is the highest exponent if you look at all the terms (you may have to add exponents, if you have a factored form). The leading coefficient of the polynomial is the number before the variable that has the highest exponent (the highest degree). So for, the degree is 4, and the leading coefficient is 5; for, the degree is 7 (add exponents since the polynomial isn’t multiplied out and don’t forget the x to the first power), and the leading coefficient is –10 (you can tell by the –5 in front and the 2x in the factor with the highest exponent). Each factor in a polynomial has what we call a multiplicity, which just means how many times it’s multiplied by itself in the polynomial (its exponent). Remember that x – 4 is a factor, while 4 is a root (zero, solution, x-intercept, or value). Now we can use the multiplicity of each factor to know what happens to the graph for that root – it tells us the shape of the graph at that root. Also note that you won’t be able to determine how low and high the curves are when you sketch the graph; you’ll just want to get the basic shape.

Here are a few problems where we use the Conjugate Zeroes Theorem and Complex Conjugate Zeroes Theorem (also called Conjugate Root Theorem or Conjugate Pair Theorem), which states that if is a root, then so is . The complex form of this theorem states that if is a root, then so is .

Also remember that when we factor to solve quadratics or any polynomials, we can never just divide by factors (with variables) on both sides to get rid of them. If we do this, we may be missing solutions! Many times we’re given a polynomial in Standard Form, and we need to find the zeros or roots. For higher level polynomials, the factoring can be a bit trickier, but it can be sort of fun — like a puzzle!

Remember that if we divide a polynomial by “x – c” and get a remainder of 0, then “x – c” is a factor of the polynomial and “c” is a root.

When we want to factor and get the roots of a higher degree polynomial using synthetic division, it can be difficult to know where to start! In the examples so far, we’ve had a root to start, and then gone from there. For a polynomial function with integers as coefficients (no fractions or decimals), if p = the factors of the constant (in our case, d), and q = the factors of the highest degree coefficient (in our case, a), then the possible rational zeros or roots are where p are all the factors of d above, and q are all the factors of a above. Remember that factors are numbers that divide perfectly into the larger number; for example, the factors of 12 are 1, 2, 3, 4, 6, and 12. The rational root test help us find initial roots to test with synthetic division, or even by evaluating the polynomial to see if we get 0. However, it doesn’t make a lot of sense to use this test unless there are just a few to try, like in the first case above.

Now let’s try to find roots of polynomial functions without having a first root to try. Also remember that you may end up with imaginary numbers as roots, like we did with quadratics.

Take out any Greatest Common Factors (GCFs) of the polynomial, and you’ll have to set those to 0 too, to get any extra roots. For example, if you take an x out, you’ll add a root of “0”. If you have access to a graphing calculator, graph the function and determine if there are any rational zeros with which you can use synthetic division. If you don’t have a calculator, guess a possible rational zero using the method above. There are a couple of theorems that you’ll learn about that will help you evaluate polynomials (for a given x, find the y) and also be able to quickly tell if a given number is a root. There’s another really neat trick out there that you may not talk about in High School, but it’s good to talk about and pretty easy to understand. Yes, and it was named after a French guy!

The DesCartes’ Rule of Signs will tell you the number of positive and negative real roots of a polynomial by looking at the sign changes of the terms of that polynomial. We talked a little bit about the Complex Conjugate Zeros Theorem here when we talked about all the steps required to find all the factors and roots of a polynomial.

End Behavior refers to the behavior of a graph as it approaches either negative infinity, or positive infinity. Anybody can plot a function on a computer, but you'll need to know what you're doing to see if the computer has done what you think you asked it to. The fourth item in this list is just a reminder that the graphs of polynomial functions are continuous with no breaks or sharp points of any kind. We can use a computer to draw a more precise and more accurate graph, but we need a sketch to make sure that graph looks right - that we asked the computer to do the right thing.

Think of it as xA·xA·x = 0, and that if either of the three x's are zero, then the whole function has a zero value.

This is a double root, which means that the graph of this function just touches the x-axis at x = -4. In the previous section we showed that the end behavior depends on the sign of the leading coefficient and on the degree of the polynomial. Notice that all three roots are single roots, so the function graph has to pass right through the x-axis at those points (and no others).

You can see that it has all of the essential features of our sketch, but that the details are filled in. The ends of this function both go in the same direction because its degree is even, and that direction is upward because the coefficient of the leading term, x4, is positive. You can see that all of the essential features of our sketch were correct; we just have to blow up the region in green to see the other 3 roots (1 double, 1 single). They work the same way every time, and knowing how they affect a known function will really help you visualize the transformed function.

If we can identify the function as just a series of transformations of some parent function that we know, the graph is pretty easy to visualize.

Jeff Cruzan is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Notice that the negative part of the graph is more of a “cup down” and the positive is more of a “cup up”. If there is no exponent for that factor, the multiplicity is 1 (which is actually its exponent!) And remember that if you sum up all the multiplicities of the polynomial, you will get the degree!

A triple root at x = 0 means that there is an inflection point there, a point where the curvature of the function changes between concave-upward and concave-downward.

This function doesn't have an inflection point on the x-axis (it may have one or more elsewhere, but we won't be able to find those until we can use calculus).

The curve has to smoothly pass right through both points on the x-axis and go to -∞ on the left.

Because we've already sketched the graph, we can be confident that the computer output is reliable. Here it is in one sketch with some explanations, but the process goes like this: Draw in the roots, then the end behavior. All text and images on this website not specifically attributed to another source were created by me and I reserve all rights as to their use. Well, you're stuck, and you'll have to resort to numerical methods to find the roots of your function.

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