TRANSFORMATION OF FUNCTIONS: TRANSLATION

A translation is transformation which shifts a graph vertically, horizontally, or both.

Vertical Translation  Let f be a function and c a positive real number. The graph of y=f(x)+c is the graph of y=f(x) shifted c units upward.          The graph of y=f(x)-c is the graph of y=f(x) shifted c units downward.

Examples

Example 1

For each vertical translation of y=x2, write the resulting function and graph it using the graph of y=x2

   a. 3 units upward     b. 5 units upward    c. 4 units downward

Solution :

a. 3 units upward ⇒ y = f(x) +3 ⇒ y = x2 + 3

b. 5 units upward ⇒ y = f(x) + 5 ⇒ y = x2 + 5

c. 4 units downward ⇒ y= f(x) – 4 ⇒ y = x2 – 4

The graphs of y=x2 and its three vertical translations are shown below.

Horizontal Translation    Let f be a function and c a positive real number. The graph of y=f(x+c) is the graph of y=f(x) shifted to the left c units. The graph of y=f(x-c) is the graph of y=f(x) shifted to the right c units.

Example 2

For each horizontal translation of y=|x|, write the resulting function and graph it using the graph of y=|x|:

a. 4 units to the right    b. 3 units to the left

Solution:

Graph y=|x| (blue graph) and then do the horizontal translations.

a. 4 units to the right ⇒ y = f(x-4) ⇒ y = |x-4| (red graph)

b. 3 units to the left ⇒ y = f(x+3) ⇒ y = |x+3| (yellow graph)

Try these problems

QUESTIONS

1. Use the graph of f(x) = |x| to obtain the graph of g(x)= -|x|.

2. Use the graph of f(x)=x3 to obtain the graph of g(x)= - x3.

3. Use the graph of f(x)=√x to obtain the graph of g(x)= - √x.

4. Use the graph of f(x)=√x to obtain the graph of g(x)= - √x..

ANSWERS

1. Use the graph of f(x)=√x to obtain the graph of g(x)=√x +5.

2. Use the graph of f(x)=x3 to obtain the graph of g(x)=(x-2)3.

3. Use the graph of f(x) = |x| to obtain the graph of g(x)=|x-2|+4 .

4. Use the graph of f(x)=x2 to obtain the graph of g(x)= (x+3)2 -2.

1. To graph g(x)=√x +5, we first graph f(x)=√x (blue graph) and then do a vertical translation of 5 units upward. The graph of g(x)=√x +5 is the red graph.

   Note that the domain of both functions is x ³ 0.

   
   

   




2. To graph g(x)=(x-2)3, we first graph f(x)=x3 (blue graph) and then do a horizontal translation of 2 units
   to the right. The graph of g(x)=(x-2)3 is the red graph.

   
   

   




3. To graph g(x)=|x-2|+4 , we first graph f(x)=|x| (blue graph). Next, we do a horizontal translation of 2 units
   to the right followed by a vertical translation of 4 units upward. The graph of g(x) = |x-2|+4 is the red graph.

   
   

   




4. To graph g(x)= (x+3)2 -2, we first graph f(x)=x2 (blue graph). Next, we do a horizontal translation of 3 units
   to the left followed by a vertical translation of 2 units downward. The graph of g(x)= (x+3)2 -2 is the red graph.