# Quantitative Aptitude > Graph of Functions

### 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 x-coordinates 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 V-shaped 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 double-check 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