Identifying the Graphs of Polynomial
Many of the functions on the Math IIC are polynomial functions.
Although they can be difficult to sketch and identify, there are
a few tricks to make it easier. If you can find the roots of a function,
identify the degree, or understand the end behavior of a polynomial function,
you will usually be able to pick out the graph that matches the
function and vice versa.
The roots (or zeros) of a function are the x values
for which the function equals zero, or, graphically, the values
where the graph intersects the x-axis (x =
0). To solve for the roots of a function, set the function equal
to 0 and solve for x.
A question on the Math IIC that tests your knowledge of
roots and graphs will give you a function like f(x)
= x2 + x –
12 along with five graphs and ask you to determine which graph is
that of f(x). To approach a question
like this, you should start by identifying the general shape of
the graph of the function. For f(x) = x2 + x – 12,
you should recognize that the graph of the function in the paragraph
above is a parabola and that opens upward because of a positive
This basic analysis should immediately eliminate several
possibilities but might still leave two or three choices. Solving
for the roots of the function will usually get you to the one right
answer. To solve for the roots, factor the function:
The roots are –4 and 3, since those are the values at
which the function equals 0. Given this additional information,
you can choose the answer choice with the upward-opening parabola
that intersects the x-axis at –4 and 3.
The degree of a polynomial function is the highest exponent
to which the dependent variable is raised. For example, f(x)
= 4x5 – x2 +
5 is a fifth-degree polynomial, because its highest exponent is
A function’s degree can give you a good idea of its shape.
The graph produced by an n-degree function can
have as many as n – 1 “bumps” or “turns.” These
“bumps” or “turns” are technically called “extreme points.”
Once you know the degree of a function, you also know
the greatest number of extreme points a function can have. A fourth-degree
function can have at most three extreme points; a tenth-degree function
can have at most nine extreme points.
If you are given the graph of a function, you can simply
count the number of extreme points. Once you’ve counted the extreme
points, you can figure out the smallest degree that the function
can be. For example, if a graph has five extreme points, the function
that defines the graph must have at least degree six. If the function
has two extreme points, you know that it must be at least third
degree. The Math IIC will ask you questions about degrees and graphs
that may look like this:
||If the graph above represents a portion of
the function g(x), then which
of the following could be g(x)?
||ax2 + bx + c
||ax3 + bx2 + cx + d
||ax4 + bx3 + cx2 + dx + e
To answer this question, you need to use the graph to
learn something about the degree of the function. Since the graph
has three extreme points, you know the function must be at least
of the fourth degree. The only function that fits that description
is E. Note that the answer could have been any function
of degree four or higher; the Math IIC test will never present you
with more than one right answer, but you should know that even if
answer choice E had read ax7 + bx6 + cx5 + dx4 + ex3 + fx2 + gx + h it
still would have been the right answer.
Function Degree and Roots
The degree of a function is based on the largest exponent
found in that function. For instance, the function f(x)
= x2 + 3x +
2 is a second-degree function because its largest exponent is a
2, while the function g(x) = x4 +
2 is a fourth-degree function because
its largest exponent is a 4.
If you know the degree of a function,
you can tell how many roots that function will have. A second-degree
function will have two roots, a third-degree funtion will have three
roots, and a ninth-degree function will have nine roots. Easy, right?
Right, but with one complication.
In some cases, all the roots of a function will be distinct.
Take the function:
The factors of g(x)
are (x + 2) and
(x + 1),
which means that its roots occur when x equals –2 or –1. In contrast,
look at the function
While h(x) is a second-degree
function and has two roots, both roots occur when x equals
–2. In other words, the two roots of h(x)
are not distinct.
The Math IIC may occasionally present you with a function
and ask you how many distinct roots the function has. As long as
you are able to factor out the function and see how many of the
factors overlap, you can figure out the right answer. Whenever you
see a question that asks about the roots in a function, make sure
you determine whether the question is asking about roots or distinct
The end behavior of a function is a description of what
happens to the value of f(x) as x approaches
infinity and negative infinity. Think about what happens to a polynomial
containing x if you let x equal
a huge number, like 1,000,000,000. The polynomial is going to end
up being an enormous positive or negative number.
The point is that every polynomial function
either approaches infinity or negative infinity as x approaches
positive and negative infinity. Whether a function will
approach positive or negative infinity in relation to x is
called the function’s end behavior.
There are rules of end behavior that can allow you to
use a function’s end behavior to figure out its algebraic characteristics
or to figure out its end behavior based on its definition:
- If the degree of the polynomial is even,
the function behaves the same way as x approaches
both positive and negative infinity. If the coefficient of the term with
the greatest exponent is positive, f(x)
approaches positive infinity at both ends. If the leading coefficient
is negative, f(x) approaches negative
infinity at both ends.
- If the degree of the polynomial function is odd, the function
exhibits opposite behavior as x approaches positive
and negative infinity. If the leading coefficient is positive, the
function increases as x increases and decreases
as x decreases. If the leading coefficient is negative,
the function decreases as x increases and increases
as x decreases.
For the Math IIC, you should be able to determine a function’s
end behavior by simply looking at either its graph or definition.
Another type of question you might see on the Math IIC
involves identifying a function’s symmetry. Some functions have
no symmetry whatsoever. Others exhibit one of two types of symmetry
and are classified as either even functions or odd functions.
An even function is a function for which f(x)
= f(–x). Even functions are symmetrical
with respect to the y-axis. This means that a line
segment connecting f(x) and f(–x)
is a horizontal line. Some examples of even functions are f(x) = cos x, f(x) = x2,
and f(x) = |x|.
Here is a figure with an even function:
An odd function is a function for which f(x)
= –f(–x). Odd functions are symmetrical
with respect to the origin. This means that a line segment connecting f(x)
and f(–x) contains the origin.
Some examples of odd functions are f(x)
= sin x and f(x)
Here is a figure with an odd function:
Symmetry Across the x-Axis
No function can have symmetry across the x-axis,
but the Math IIC will occasionally include a graph that is symmetrical
across the x-axis to fool you. A quick check with
the vertical line test proves that the equations that produce such
lines are not functions: