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Classifying Polynomials by Their Graphs

When presented with the graph of a polynomial function, there are several pieces of information we can get from the graph of the polynomial, without ever actually seeing the equation.

First, though, we must establish some terminology.  When given a polynomial in the form:
 

, ,

We refer to the degree of the polynomial as n, because n is the highest non-zero power of x in the polynomial.  We need to be clear that all exponents on the variable x must be nonnegative integers in order for f(x) above to be a polynomial. 

We call  the leading coefficient, since it accompanies the x with the highest power. 

For example, in the polynomial , the degree is 4, and the leading coefficient is -13.

The first piece of information we can extract is whether the degree of a polynomial is odd or even.  In order to do this, we look at both “sides” of the graph: if they both go up or both go down, the degree is even.  If they go opposite directions, that is, one goes up and one goes down, then the degree is odd.


For instance, since both ends go up in this graph, we know the degree is even.  Think about it:  as x gets bigger, both positively and negatively, the associated y-values get bigger.  This could only happen if the degree is even, because, for example, (-100) raised to an even power, gives a large positive value.

 


And in this graph, since they go opposite directions, the degree must be odd.  Think about this:  odd powers of negative numbers are negative, while odd powers of positive numbers are positive.  So for large positive numbers, odd powers would still be positive, while odd powers of large negative numbers would be negative.  This graph seems to behave opposite from this analysis, but the leading coefficient is at work here.  The leading coefficient happens to be negative for this graph.  We’ll explain more about that soon.

 

 

In addition to determining whether the degree is odd or even, we can in many cases also determine the minimum value of the degree by looking at the graph.  To do this, we count the number of places where the graph changes direction from decreasing to increasing or vice versa; the x coordinates of these places are in a special set of x values called critical points.  The number of critical points is always less than the degree of the polynomial. 

Let’s think of a few simple polynomials.  y=2x+5 is a straight line, and there are no places where it changes direction and the degree is 1.  On a parabola, such as y = x2+1, there is one turning point, and notice that the degree is 2.  With polynomials, each time you add a degree, then, you can add a turning point.


Here are the same two graphs, with the critical points indicated.

 

In the graph above, since there are 5 critical points, and we know the degree is even, we know that the degree must be an even number greater than or equal to 6.

In the graph above, there are 4 critical points, and the degree is odd, so the degree is an odd number greater than or equal to 5. Of course, the degree could be much bigger.  What this provides, though, is a lowest bound for the degree.

Finally, let us address the effect of the leading coefficient as mentioned earlier.  All the graph tells us about the leading coefficient is whether it is positive or negative.  If the graph goes down to the right, then the leading coefficient is negative.  If the graph goes up to the right, then the leading coefficient is positive.

 

 

In the function above, we can see that the leading coefficient must be positive since the right-hand side of the function is rising upward as you look from left to right.

In the function above, the leading coefficient must be negative since the graph falls to the right.


Example 1

Determine if the degree is odd or even, the lowest bound for the degree, and the sign of the leading coefficient of the polynomial with this graph

Solution 1

Since there are two critical points, and the degree is odd, the lowest bound for the degree is 3.  Also, the leading coefficient is positive.


 Example 2

Determine if the degree is odd or even, the lowest bound for the degree, and the sign of the leading coefficient of the polynomial of this graph:

Solution 2

The degree is even, and there are three critical points, so the lowest bound for the degree is 4.  Also, the leading coefficient is negative.