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.
