Graphs

Example 1

Graph (in blue) and ( in red) on the same coordinate system.

Solution .

Example 2

Point out similarities and differences between the two curves in the last graph.

Solution .

Both functions are increasing. There are some differences to note in their domains, ranges, intercepts, and asymptotes.

For

The domain is ( )

The range is ( )

The y-intercept is (0, 1)

As x approaches , y approaches 0

For

The domain is ( ) (which is the range of )

The range is ( ) (which is the domain of )

The x-intercept is (1, 0)

As x approaches 0 from the right, y approaches

i.e. x = 0 is a vertical asypmtote.

Example 3

Graph (in blue) and (in red) on the same coordinate system.

Solution .

Note that both of these functions are decreasing

Summary

If a > 0 and , then has

domain of (0, ), range of ( ), x-intercept at (1, 0), and vertical asymptote given by x = 0.

When a > 1, the graph is increasing.

When 0 < a < 1, the graph is decreasing.

Example 4

Determine the domain, range, intercepts, and asymptotes for

Solution .

Since where e > 1, the graph is increasing. It has domain (0, ), range ( ), x-intercept (1, 0), and vertical asymptote given by x = 0.

This is confirmed in the graph given below.

Example 5

How is the graph of related to the graph of ?

Solution .

The vertical asymptote of is located where x + 2 = 0. Since this gives x = -2, we see that the graph of is the graph of shifted two units to the left. The domain of is x > -2, its x-intercept is (-1, 0). Its range is the same as the range for .

In the graph below, is in blue while is in red.

Example 6

Graph (in blue) and its reflection about the x-axis, , (in red)

on the same coordinate system.

Solution

The picture of seen in the above graph looks like the graph of in example 3. They are indeed the same as the following table suggests.

 x            funct1(x)        funct2(x)  

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

0            undef            undef           

.2500000000  2.000000000      2.000000000     

.5000000000  1.000000000      1.000000000     

.7500000000  .4150374994      .4150374994     

1.000000000  0                0               

1.250000000  -.3219280949     -.3219280949    

1.500000000  -.5849625008     -.5849625008    

1.750000000  -.8073549221     -.8073549221    

2.000000000  -1.000000000     -1.000000000    

2.250000000  -1.169925002     -1.169925002    

2.500000000  -1.321928095     -1.321928095    

2.750000000  -1.459431619     -1.459431619    

3.000000000  -1.584962501     -1.584962501    

It can be shown in general that if (i.e. r is the reciprocal of a ), then