Graphs

This section contains the graphs of six functions that you should be able to graph without using a calculator or spending a great deal of time plotting points. Knowing these functions will make it easier for you in the next section on translations of functions. These functions are reviewed in the first example.

Example 1

Graph a constant function such as y = 3, graph the identity function y = x, graph , graph , graph the squaring function , and graph the basic cubic

Example 2

In the previous section piecewise-defined functions were introduced. In this example we will work with the function that equals when x is less than or equal to one and 2x when x is larger than one.

Make a table for f with x between -2 and 3 in steps of 0.25. (Note the use of scientific notation on a couple of the function's values.)

 x             funct(x)   

----------------------------

-2.           4.            

-1.750000000  3.062500000   

-1.500000000  2.250000000   

-1.250000000  1.562500000   

-1.000000000  1.000000000   

-.750000000   .5625000000   

-.500000000   .2500000000   

-.250000000   .6250000000E-1

0             0             

.250000000    .6250000000E-1

.500000000    .2500000000   

.750000000    .5625000000   

1.000000000   1.000000000   

1.250000000   2.500000000   

1.500000000   3.000000000   

1.750000000   3.500000000   

2.000000000   4.000000000   

2.250000000   4.500000000   

2.500000000   5.000000000   

2.750000000   5.500000000   

3.000000000   6.000000000   

We can plot these points.

We can use even more points by using an increment of 0.1 between x values that are used.

The graph consists of two parts, a portion of a parabola and a part of a line. If you ask the computer or calculator to graph such a function it will usually incorrectly connect the two parts as in the following graph.

To correct this you must ask for the graph in dot mode. Below is such a graph.

Example 3

The greatest integer function is denoted by [[ x ]] which stands for the greatest integer less than or equal to x. For instance, [[ 0 ]] = 0, [[ 0.3 ]] = 0, [[ 0.9 ]] = 0, [[ 1 ]] = 1, and [[ 1.2 ]] =1. The table below shows values for [[ x ]] for x between -2 and 2 in steps of 0.4

 x             funct(x)   

----------------------------

-2.           -2.           

-1.600000000  -2.           

-1.200000000  -2.           

-.800000000   -1.           

-.400000000   -1.           

0             0             

.400000000    0             

.800000000    0             

1.200000000   1.            

1.600000000   1.            

2.000000000   2.            

Below is a graph of [[ x ]] for x between -2 and 2 with an increment of 0.05 between x values.

Note that each level in the graph should be a horizontal line segment which includes the left end point and excludes the right end point.

Definition

A function is called even if f(-x) = f(x) for all x in the domain of f.

Definition

A function is called odd if f(-x) = - f(x) for all x in the domain of f.

Example 4

is an example of an even function since which is that same as

is an exmaple of an odd function since which equals or

Below are the graphs. Note the symmetry involved with each.

Note :

Even functions are symmetric about the x-axis.

Odd functions are symmetric about the origin.

Example 5

Is even, odd, or neither ?

Solution .

Since , this function is even. The symmetry about the y-axis in the next graph serves as check on this.

Example 6

Is even, odd, or neither?

Solution .

Since , we do have . The function is odd. In the following table note that if the pair (x, y) appears so does the pair (-x,-y). This says that the function is symmetric about the origin.

 x             funct(x)   

----------------------------

-3.           21.           

-2.000000000  4.000000000   

-1.000000000  -1.000000000  

0             0             

1.000000000   1.000000000   

2.000000000   -4.000000000  

3.000000000   -21.00000000  

Here is its graph.