Functions and translations of graphs
The two graphs shown opposite are
a) the graph of the function:
f ( x)  x 2
b) the graph of this function after a
translation of :
 2
 1 
 
i.e. “minus” 2 in the x-direction and
“plus” 1 in the y-direction.
To understand what is happening here and the care required with the “plus and minus”
AND the (brackets in the) notation……….read on and follow these instructions.
Once UNDERSTOOD you will be able to apply these skills to any functions you care to ‘define’
Open a NEW PAGE on ‘Autograph’ then…


Equation
then
Function Definitions…
f ( x)  x 2

Enter Equation… y  f (x)
the graph of f ( x)  x 2 now appears in

(insert colour)
Enter Equation… y  f ( x  2)
the graph of f ( x)  ( x  2) 2 now appears in
(insert colour)
In effect the “x” has been replaced by “x+2” in the function
 2
The original function has undergone a translation of  
 0 
Now, on the same Graph Page, enter the equation
the graph of y  x 2  1 shown in
y  f ( x)  1 and you should see
(again insert the colour!)
 0
The original function has now been translated through  
 1
The next page will COMBINE the two translations covered on this page…………
Combining the two translations of the previous page
On a New Graph Page (using Autograph)………….

Equation…

Function Definitions…

f ( x)  x 2

Enter Equation…

y  f ( x  2)  1
You should now see that the original function has been………..
…….translated through
 2
 1 
 
Hopefully you can see why great care is needed with the “+” and “-“ signs!!!!!!
If you understand this far then try repeating the whole thing with:
a) different functions for f(x)
b) different translations
Here are some suggestions…………
ACTION - use Autograph and your Algebra skills to complete this table:
Original
Function
Function Notation
(to define the translation)
f ( x)  x 3
y  f ( x  2)  7
f ( x)  1 / x
y  f ( x  1)  3
f ( x)  1 / x 2
y  f ( x  2)  1
f ( x)  x 2  2 x
y  f ( x  2)  5
EQUATION
(of the translated
graph)
y
3x  5
x 1
Translation


















 2 
 5 

