Graphing Functions
Graphing Functions
While most of the function questions on the Math IC will involve analysis and manipulation of the functions themselves, you will sometimes be asked a question about the graph of a function. A common question of this type asks you to match a function’s graph to its definition. Understanding the next few topics will help prepare you for questions relating to functions and their graphs.
Identifying Whether a Graph is a Function
For the Math IC, it’s important to be able to determine if a given graph is indeed a function. A foolproof way to do this is to use the vertical line test: if a vertical line intersects a graph more than once, then the graph is not a function.
The vertical line test makes sense because the definition of a function requires that any x-value have only one y-value. A vertical line has the same x-value along the entire line; if it intersects the graph more than once, then the graph has more than one y-value associated with that x-value.
Using the vertical line test, check to see that the three graphs below are functions.
The next three graphs are not functions. In each graph, a strategically placed vertical line (depicted by the dashed line) will intersect the graph more than once.
Range and Domain in Graphing
The range and domain of a function are easy enough to see in their graphs. The domain is the set of all x-values for which the function is defined. The range is the set of all y-values for which the function is defined. To find the domain and range of a graph, just look at which x- and y-values the graph includes.
Certain kinds of graphs have specific ranges and domains that are visible in their graphs. A line whose slope is not 0 (a horizontal line) or undefined (a vertical line) has the set of real numbers as its domain and range. Since a line, by definition, extends infinitely in both directions, it passes through all possible values of x and y:
An odd-degree polynomial, which is a polynomial whose highest degree of power is an odd number, also has the set of real numbers as its domain and range:
An even-degree polynomial, which is a polynomial whose highest degree of power is an even number, has the set of real numbers as its domain, but it has a restricted range. The range is usually bounded at one end and unbounded at the other. The following parabola has range {–∞, 2}:
Trigonometric functions have various domains and ranges depending on the function. Sine, for example, has the real numbers for its domain and {–1, 1} for its range. A more detailed breakdown of the domains and ranges for the various trigonometric functions can be found in the Trigonometry chapter.
Some functions have limited domains and ranges that cannot be categorized simply, but are still obvious to see. By looking at the graph, you can see that the function below has domain {3, ∞} and range {–∞, –1}.
Asymptotes and Holes
There are two types of abnormalities that can further limit the domain and range of a function: asymptotes and holes. Being able to identify these abnormalities will help you to match the domain and range of a graph to its function.
An asymptote is a line that a graph approaches but never intersects. In graphs, asymptotes are represented as dotted lines. You’ll probably only see vertical and horizontal asymptotes on the Math IC, though they can have other slopes as well. A function is undefined at the x value of a vertical asymptote, thus restricting the domain of the function graphed. A function’s range does not include the y value of a horizontal asymptote, since the whole point of an asymptote is that the function never actually takes on that value.
In this graph, there is a vertical asymptote at x = 1, and a horizontal asymptote at y = 1. Because of these asymptotes, the domain of the graphed function is the set of real numbers except 1 (x ≠ 1), and the range of the function graphed is also the set of real numbers except 1 (f(x) ≠ 1).
A hole is a point at which a function is undefined. You’ll recognize it in a graph as an open circle at the point where the hole occurs. Find it in the following figure:
The hole in the graph above is the point (–4, 3). This means that the domain of the function is the set of real numbers except 4 (x ≠ –4), and the range is the set of real numbers except 3 (f(x) ≠ 3).
The Roots of a Function
The roots (or zeroes) of a function are the x values for which the function equals zero. Graphically, the roots are the values where the graph intersects the x-axis (y = 0). To solve for the roots of a function, set the function equal to 0 and solve for x.
A question on the Math IC that tests your knowledge of roots and graphs will give you a function such as f(x) = x2 + x – 12 along with five graphs and ask you to determine which graph shows that function. 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 it opens upward because it has a positive leading coefficient.
This basic analysis should immediately eliminate several possibilities, but it might still leave two or three choices. Solving for the roots of the function will usually get the 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.
Function Symmetry
Another type of question you might find on the Math IC involves identifying a function’s symmetry. There are only two significant types of symmetry that come up on the Math IC: the symmetry of even functions and of 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 IC will occasionally include a graph that is symmetrical across the x-axis to fool you. A quick check with the vertical line test would prove that the equations that produce such lines are not functions:
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