The graph of a function f is defined to be the graph of the corresponding equation . In other words, the graph is the set of all points in the Cartesian plane such that is in the domain of and .
For instance, if m and b are constants, then the graph of the function is the same as the graph of the equation , a line with slope and intercept b (as shown in the figure below). For this reason, a function of the form is called a linear function.
Graphs of functions that are not linear are often (but not always) smooth curves in the Cartesian plane.
The Vertical-Line Test
A set of points in the Cartesian plane is the graph of a function if and only if no vertical straight line intersects the set more than once.