Graphs of Functions

The coordinate plane can be used for graphing functions. To graph a function in the
plane, you represent each input
and its corresponding output
f as a point
, where
. In other words, you use the
axis for the input and the
axis for the output. Below are several examples of graphs of elementary functions.
Ex. Consider the linear function defined by . Its graph in the plane is the line with the linear equation , as shown in the figure below.
Ex. Consider the quadratic function defined by The graph of is the parabola with the quadratic equation , as shown in the figure below.
Note that the graphs of and from the two examples above intersect at two points. These are the points at which . We can find these points algebraically by setting
and solving for x, using the quadratic formula, as follows.
We get , which represent the xcoordinates of the two solutions
and
With these input values, the corresponding coordinates can be found using either
or :
and
Thus, the two intersection points can be approximated by (0.78, 0.61) and (–1.28, 1.64).
Ex. Consider the absolute value function defined by . By using the definition of absolute value, h can be expressed as a piecewise defined function:
The graph of this function is Vshaped and consists of two linear pieces, and , joined at the origin, as shown in the figure below.
Ex. Consider the functions defined by and . These functions are related to the absolute value function and the quadratic function , respectively, in simple ways.
The graph of is the graph of shifted upward by 2 units, as shown in the figure below. Similarly, the graph of the function is the graph of shifted downward by 5 units (not shown).
The graph of is the graph of shifted to the left by 1 unit, as shown in the figure below. Similarly, the graph of the function is the graph of shifted to the right by 4 units (not shown). To doublecheck the direction of the shift, you can plot some corresponding values of the original function and the shifted function.
In general, for any function and any positive number, the following are true.

The graph of is the graph of shifted upward by units.

The graph of is the graph of shifted downward by c units.

The graph of is the graph of shifted to the left by units.

The graph