Graphing Higher Degree Polynomial Functions

Before you start this section, make sure you know how to factor higher degree polynomial functions. This part of the lesson assumes that you have already factored what you need to graph (otherwise the help page is 17 miles long!) Please review the section on factoring if you are unsure how to factor a cubic (or higher degree) polynomial.

Before you start graphing, there are some important rules that you teacher should have shown you. They are repeated here, as you need them to properly graph a polynomial function.

Rule 1

Graphs of polynomial functions of degree 2 or higher are all smooth curves.

Rule 2

The number of 'changes of direction' of a polynomial graph is one less than the degree of the polynomial (ie. A quartic polynomial (of degree 4) has AT MOST 3 changes in direction - it could have 0, 1, 2 or 3 changes of direction.)

Rule 3

For polynomials with odd degrees (3, 5, 7, etc.):

• If the leading coefficient is negative the graph falls to the left and rises to the right.
• If the leading coefficient is positive the graph rises to the left and falls to the right.

Rule 4

For polynomials with even degrees (2, 4, 6, etc):

• If the leading coefficient is negative the graph falls to the left and right.
• If the leading coefficient is positive the graph rises to the left and right.

Rule 5

The graph will pass through the x-axis at any single root (a root is a zero or a solution). At a double root, the graph will 'bounce' off the x-axis. A triple root will see the graph pass through the x-axis (with a bit of flattening), and a quadruple root will see a 'bounce'. This pattern repeats indefinitely. You will see examples of a 'bounce' below.

• f(x) = (x -2)(x - 1)(x + 3) has only SINGLE roots at 2, 1 and -3.
• f(x) = (x - 3)(x - 3)(x + 3) has a DOUBLE root at 3 (since it appears twice) and a SINGLE root at -3.

You must remember these rules! They will not be given to you on a test or exam. You are expected to know them as you will be expected to explore these types of graphs.

Example 1

Sketch the graph of the function f(x) = (x + 4)(x + 1)(x - 3).

Solution

This is the graph of a cubic polynomial. (You can tell this since multiplying the first term in each of the brackets results in x³). The leading coefficient is positive. Therefore, we know that this graph falls to the left and rises to the right (rule 3) AND that this graph has, at most, 2 changes in direction.

To graph ANY polynomial function, start with the zeroes. The zeroes of this function are -4, -1 and 3. They are all single roots so you know that the graph will pass through the x-axis at these points. Draw a grid with the zeroes marked:

You also know that the graph falls to the left and rises to the right. You can draw in these 'arms' on the graph:

The question now becomes "What happens between these 'legs'?" Well, you need to connect the graph using a smooth curve (it is all one piece, unlike the graph of f(x) = 1/x). You also know that the graph passes through the x-axis at -1 (and the other zeroes), you could rough out a graph that would meet these criteria that would look like so,

but how do you know how high and how low the actual 'humps' go? A better way to sketch is to get a couple of test points. A good method is to select x-values that are directly between the existing roots and find the value of the function (the y-value) at those points. The points we will test are x = -2.5 (halfway between -4 and -1) and x = 1 (halfway between -1 and 3). Our results are shown on the table below:

As you can see from the table of values, we need to extend our graph a bit! Even though these points (-2.5, 12.275) and (1, -20) may not be the highest or lowest points on the graph, they should be close. Extend your y-axis (or change your scale on the y-axis) and put these points on the graph:

Now draw your graph as a smooth curve passing through these points, and you are done! The final graph is shown below. It has 2 changes in direction, 3 single roots, and it falls to the left and rises to the right. All the conditions are met.

Notice that your two 'test points' are NOT the highest and lowest points on the graph. You may have drawn them as such on your own paper, and that is OK. You are just drawing a sketch of the graph, and the graph you see above was drawn using a computer (it is as close to correct as the monitor will allow). You can ask your teacher what his or her tolerance is for accuracy of graphs of this kind.

Example 2

Sketch the graph of f(x) = -(x - 4)(x - 4)(x + 3)(x + 6).

Solution

This example is a polynomial of degree 4 with a negative leading coefficient. This means that the graph falls to the left AND right (rule 4) and has at MOST 3 changes of direction (rule 2). It has SINGLE roots at x = -3 and -6, and a DOUBLE root at x = 4. This means the graph will pass through the x-axis at -3 and -6, and will bounce off the x-axis at x = 4. As before, let's start by putting the roots on a graph:

We know that this graph will fall to the left and right, so let's put those little 'legs' on the graph.

Now to see what happens between the two ends of the graph. We know that the graph passes through the x-axis at x = -6 and x = -3. It 'bounces' off the graph at x = 4. We can sketch a rough idea of what will happen:

But again, we don't know that this is exactly what happens. Let's make another table of values for values in-between the roots on the graph, and see how 'high' and 'low' this graph may travel.

Yikes! Those are BIG y-values! Don't panic - just change the scale on your y-axis and plot those points:

Now connect these points with a smooth curve remembering the behaviour at the roots. This will give you your final graph:

Remember, your graph may not look exactly like this one. Check with your teacher for acceptable tolerances.

Example 3

Sketch the graph of f(x) = x(x + 3)(x + 3)(x + 3).

Solution

This is a polynomial of degree 4 with a positive leading coefficient. It will have at most 3 changes of direction. The positive leading coefficient means that this graph will rise on the left and the right. This graph has a single root at x = 0 as well as a triple root at x = -3. This means that the graph will pass through the x-axis at x = -3, but will 'flatten out' around that point.

Let's begin (as usual) by plotting the roots on a graph:

We know that this graph rises on the left and right, so we can fill in those 'legs':

Now we have to see what happens between the 'legs'. We know that the graph passes through the x-axis at x = -3, and it has to pass through again at x = 0. We could sketch what happens:

But again we will use a 'test point' to determine how 'low' this graph goes. Let's use an x-value of -1.5 (halfway between -3 and 0). Plugging x = -1.5 into the function gives us f(-1.5) = -5.0625. Plot a point at this y-value and draw the graph passing through this point:

Notice that the actual lowest point of this graph occurs around x = -1. The graph that you drew may have turned back 'upward' at the point
(-1.5,-5.0625). This is okay, as you are just drawing a sketch of the graph: You are not expected to come up with a computer-perfect version of the graph.

Again, the examples on this page are a brief reminder of how to graph higher-degree polynomial functions. You should make sure you have ALL of the notes on this topic from your class. There are many examples to try in your booklet. You can now head back to the Higher Degree Polynomial Functions main page.