Translations of Functions

Key Concepts

• Graph transformations of functions.
• Analyze vertical translations of the functions.
• Analyze horizontal translations of the functions.
• Combine both vertical and horizontal translations of the functions.

Parent functions

1. Match the given functions with their parent functions:  

Solution:   

Comparison of graphs of the functions

Example 1: Sketch the graph of the functions √x and √x+2 and compare them.  

When compared, the graph √x+2 has moved 2 units up to the graph √x.  

Example 2: Sketch the graph of the functions ∛ and ∛x−1 and compare them. 

When compared, the graph ∛x−1 has moved 2 units up to the graph ∛ .  

Vertical translations:   

Example:  

Compare the graph of the function:

p(x) = |x|,  q(x) = |x|−3 and g(x) = x+2

Solution:  

When compared, the graph |x|−3 has moved 3 units down to the graph |x|.

When compared, the graph |x|+2 has moved 2 units up to the graph |x|.  

Any function of the form g(x)=f(x)+k, where the value of x, g takes the output of f and adds the constant k.

Vertical translation:  

For a given g(x)=f(x)+k, the graph of the function g is the function f translates k units vertically.  

k>0 : shifts |k| units up  

k<0: shifts |k| units down  

Example: How the function f(x) = ∛ translated to obtain the graph of g(x) = ∛−5?  

Solution:

g(x) = f(x)−5

k=−5k ; shifts

|−5| units down 

So, the graph of the function g is the function f translates 5 units down vertically. 

Horizontal translations: 

Example:  

Compare the graph of the functions x², (x−2)² and (x+1)².

Solution:   

When compared, the graph (x−2)²  has moved 2 units right to the graph x².  

When compared, the graph (x+2)²  has moved 1 units right to the graph x².  .  

Any function g(x)=f(x−h) means that g takes the input of f and subtracts the constant h before applying function f.  

Horizontal translation 

For a given g(x)=f(x-h), the graph of the function g is the function f translates k units vertically.  

h>0 : shifts |k| units right   

h<0: shifts |h| units lefts  

Example: How the function f(x)=√x translated to obtain the graph of g(x) = √x−3

Solution:

g(x) = f(x−3)

h = 3; shifts

|3| units right 

So, the graph of the function g is the function f translates 3 units right horizontally. 

Combine translations:   

Example:  

Compare the graph of the functions √x and √x+1 – 2

Solution:   

Example:  

Compare the graph of the functions 2^x and 2^x−2+3.  

Solution:   

Any function in the form g(x) = f(x−h)+k

The combined horizontal and vertical translation are independent of each other.

Given: g(x) = f(x−h)+k the graph of the function g is the graph of function f translated h units horizontally, then translated k units vertically. 

Example: Graph

g(x) = f(x+4)−1 for the function f(x).  

Solution: Compare g(x) = f(x+4)−1 with g(x) = f(x−h)+k

We get h=−4 and k=−1

So, all the points of graph f are translated left 4 units and down 1 unit.

Exercise

1. For the function gx=fx-6, how does the value -6 affect the graph of the function ?
2. For the function gx=fx+4, how does the value  affect the graph of the function ?
3. How is the function fx=2 translated to obtain the graph of ?

Concept Summary   

 

What we have learned

• Graph transformations of functions.
• Analyze vertical translations of the functions.
• Analyze horizontal translations of the functions.
• Combine both vertical and horizontal translations of the functions.