Identifying the Graphs of Polynomial Functions
Identifying the Graphs of Polynomial Functions
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.
Roots
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 leading coefficient.
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.
Degree
The degree of a polynomial function is the highest exponent to which the dependent variable is raised. For example, f(x) = 4x5x2 + 5 is a fifth-degree polynomial, because its highest exponent is 5.
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)?
(A) a
(B) ax +b
(C) ax2 + bx + c
(D) ax3 + bx2 + cx + d
(E) 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 roots.
End Behavior
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.
Function Symmetry
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.
Even 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:
Odd Functions
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) = 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:
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