A graph is simply a drawing of the coordinate plane with points plotted on it. These points all have coordinates (x, y). In the graph of a function, the y-coordinate has the value f (x), meaning the coordinates of the graph of a function are (x, f (x)). The possible values of x are elements of the domain of the function, and the possible values for f (x), or y, are the elements of the range of the function. Viewing the graph, a given x-coordinate can be selected, and the value of y at that point is the value of the function at that x-coordinate. Below are the graphs for some simple functions.
Because a function can produce only one output for a given input, it is easy to determine whether a given graph is a graph of a function or not. If a vertical line can be placed somewhere in the coordinate plane such that it intersects with the graph twice, the graph is not a function. Two intersections with a vertical line signify that for a given value of x, there are two possible values of f (x), which by definition is impossible for a function. This method for testing a potential function is called the vertical line test.
The graphs of the trigonometric functions are plots of points whose coordinates are (x, f (x)), with x being an angle measure in radians. The y-coordinate, f (x), is a real number--specifically, it is the ratio that defines the value of a given trigonometric function. In the next section we'll see what the graphs of the six trigonometric functions look like.