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(xc) 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(x4) ⇒ y = x4 (red graph)
b. 3 units to the left ⇒ y = f(x+3) ⇒ y = x+3 (yellow graph)
Try these problems
QUESTIONS

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

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

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

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

Use the graph of f(x)=x3 to obtain the graph of g(x)=(x2)3.

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

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

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.

To graph g(x)=(x2)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)=(x2)3 is the red graph.

To graph g(x)=x2+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) = x2+4 is the red graph.

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.