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MCR 3U U3-4
Transformations of Functions
Date:
1. Vertical and Horizontal Translations of Functions


f(x)

 f(x) + 3

 f(x) – 1
y  f (x)  k represents a vertical translation of k units.
y  f (x  h) represents a horizontal translation of h units.
x2
x

Sketch
 f(x + 4)
 f(x – 2)
Sketch
 f(x – 2) + 1
 f(x + 3) – 2
Sketch
2. Reflections of Functions


f(x)
  f (x)

Sketch
 f (x)

Sketch
Reflection in xaxis: y   f (x)
Reflection in yaxis: y  f (x)


means the point (x, y) becomes (x, y)
means the point (x, y) becomes (x, y)
x2
x

3. Stretches of Functions
Vertical stretches are written in the form y  af (x) . The graph is vertically stretched or compressed by a
factor of a.
 if a > 1 then stretch the graph vertically.
 For y  af (x) , (x, y) becomes (x, ay)
 if 0 < a < 1 then compress the graph vertically.

Horizontal stretches are written in the form y  f (kx) . The graph is horizontally compressed or stretched
1

by a factor of .
k
 if k > 1 then compress the graph horizontally.
x 

 For y  f (kx) ,(x, y) becomes  , y 
 if 0 < k < 1 then stretch the graph horizontally.
k 
f(x)
 2 f (x)



1
f (x)
2

x



Sketch
f(x)
 f (2x)

x2
1 
 f  x 
2 
Sketch
x2
x

4. Combinations of Transformations
Combining the functions we have investigated we can now compare the graph of y  af [k ( x  p )]  q
to the graph of y  f (x ) .
What information does a tell us? __________________ & __________________
What information does k tell us? __________________ & __________________
What information does p tell us? __________________
What information does q tell us? __________________
The order of translations is important  perform them in the following order:
1) Expansions and Compressions
2) Reflections
3) Translations
* Notice that this means you are performing multiplications (expansions, compressions & reflections)
before additions and subtractions (translations)
For functions such as y  (2 x  6) 2 and y   x  5 , factor the coefficient of the xterm to identify
the characteristics of the function more easily:
y  (2 x  6) 2
y
y   x5
y
Description of Transformation:
Description of Transformation:
Graph:
Graph:
Complete the following:
f(x)
y  2 f ( x  3)  3
x
Equation:

Description of Transformations:
Mapping Equation
f(x)
y   f (0.5( x  1))  2
Input/Output Diagram
x2
Equation:
Description of Transformations:
Mapping Equation
Input/Output Diagram