Transformations of Functions and their Graphs

 

 

     Before we begin our discussion on transformations of function, it will be helpful to compile a list of some common functions and their graphs. Armed with those, we shall consider various transformations and what effect that has on the graph of the function.

 

               

Function: y = x             Function: y = |x|           Function: y = x2          Function: y = x3

Domain: (-, )         Domain: (-, )         Domain: (-, )        Domain: (-, )

Range: (-, )            Range: [0, )               Range: [0, )             Range: (-, )

 

               

Function:        Function:   Function: y = 1/x         Function: y = 1/x2

Domain: [0, )            Domain: [-1, 1]            Domain: x  0            Domain: x  0

Range: [0, )              Range: [0, 1]                Range: y  0               Range: (0, )

 

Figure 1: Eight Common Functions and their Graphs

 

 

     There are two types of transformations that shall discuss. First are translations. By a translation of a graph, we mean a shift in its location such that every point of the graph is moved the same distance and in the same direction. Essentially, think of lifting the graph out of the paper, moving it around, and then placing it down at a new location.

 

     There are four ways to move the graph: left, right, up and down. The effect this has on the graph is summarized in the following table:

 

 

Translations

Suppose that c is a positive constant.

 

Equation                      Effect on the Graph

1. y = f(x) + c               Translate c units upward

2. y = f(x)  c             Translate c units downward

3. y = f(x + c)               Translate c units to the left

4. y = f(x  c)             Translate c units to the right

 

 

 

Example 1:

 

     Write out the function of and graph the translation the graph of y = x2 to the left by 1.

 

Solution:

 

     From the table above, we see that a translation to the left by 1 can be accomplished by replacing x with x + 1. That is, our function is y = (x + 1)2. Its graph is the following:

 

Figure 2: The graph y = (x + 1)2

 

 

Example 2:

 

     Translate the graph of y = |x| to the right by 2 and down by 1.

 

Solution:

 

     First, we translate to the right by 2, and then we translate down by 1.

 

 

Figure 3: The graph of y = |x  2|  1

 

 

      The second type of transformation we are interested in is a reflection. There are two types of reflections that we will be concerned about. A reflection about the x-axis is where each point (x, y) is mapped to the point (x, -y). That is, we think of the x-axis as fixed and we spin our graph 180°.

 

     Similarly, a reflection about the y-axis is where each point (x, y) is mapped to the point (-x, y). This time we think of the y-axis as fixed and we spin our graph 180°. We record this in the following:

 

 

Reflections

Equation                      Effect on the Graph

1. y = f(x)                 Reflect about the x-axis

2. y = f(x)                 Reflect about the y-axis

 

 

 

Example 3:

 

     Graph the functions  and .

 

Solution:

 

     Notice that the first graph is a translation of 1/x2 to the right by 1. The second graph is a reflection of the first graph about the x-axis. Their graphs appear below.

 

    

Figure 4: The graphs of  and  

 

 

Example 4:

 

     Graph the functions  and .

Solution:

 

     Notice that the first graph is a reflection of  about the y-axis. The second graph is a translation of the first graph up by 1. Their graphs appear below.

 

 

 

Figure 5: The graphs of  and  

 

 

     What if instead we wanted to graph the function . There will be a reflection involved because of the x. But there will also be a translation, because of the +1. So, which do we do first?

 

     A common mistake would be to apply the translation rule y = f(x + c). The problem is that there is no “x” in the rule. Instead, we have to massage the function.

 

     Notice that . We read this to say “translate the graph to the right by 1, then reflect about the y-axis”.

 

     In general, when presented with both reflections and translations, factor out the negative signs first. Then perform the translations, and finally apply the reflections.