Transformation Rules
Transformations
Transformation Rules
Translations applied to a graph mean move movements
Horizontally or Vertically or both
They can be described using a vector as shown below
5 Means translate(move) a graph 5 to the right
2 And translate (move it) 2 up
 
5 ^|
Y
4
3
2
1
-2
-1
0
-1
1
2
3
4
5
6
X->
7
Transformation Rules
The graph of
parabola
yx
2
forms a curve called a
y  x2
This point . . . is called the vertex
Transformation Rules
2
 Adding a constant translates y  x up the y-axis

y  x2
y  x2  3
e.g.
y  x2
y  x2  3

y  x2
The vertex is now ( 0, 3)
y x
2
 y x 3
2
has added 3 to the y-values
Transformation Rules
Adding 3 to x gives
y  x2
We get

y  ( x  3) 2
y  x2
y  ( x  3)
2
Adding 3 to
x moves the
curve 3 to
the left.
Transformation Rules
 Translating in both directions
2
2
y

x
y

(x

5
)
3
e.g.

y  (x  5) 2  3
y  x2
We can write this in vector form as:
translation 5
3
 
Transformation Rules
SUMMARY
 The curve y  ( x  p)  q
2
is a translation of y  x
2
by
 The vertex is given by ( p, q)
 p
 q 
 
Transformation Rules
SUMMARY
Memory Aid
HIVO –
HOVIS –
Horizontal Inside the Bracket
Vertical Outside the bracket
Horizontal Opposite of what it says
Vertical Is Same as what it says
Transformation Rules
Exercises: Sketch the following translations of y  x 2
2
2

y

x
y

(x

2
)
1
1.
y  x2
y  ( x  2)2  1
2. y  x
3.
2
 y  (x  3)  2
y  ( x  3)2  2
2
y  x2
y  x 2  y  (x  4) 2  3
y  x2
y  ( x  4)2  3
Transformation Rules
4 Sketch the curve found by translating
y  x by
2
 2
  3 .
 
What is its equation?
y  (x  2) 2  3
5 Sketch the curve found by translating
y x
2
 1
by   .
2
What is its equation?
y  (x  1) 2  2
Transformation Rules
e.g. The translation of the function y  x 3 by the
 2
vector  1  gives the function y  ( x  2) 3  1 .
 
The graph becomes
 2
 
1 
Means translate
horizontally by +2.
So put –2 in the
bracket.
HIvo
HOvis
yx
3
y  ( x  2) 3  1
Transformation Rules
e.g. The translation of the function y  x 3 by the
 2
vector  1  gives the function y  ( x  2) 3  1 .
 
The graph becomes
 2
 
1 
Means translate
vertically by +1.
So put +1 outside
the bracket.
HIVO
HOVIS
yx
3
y  ( x  2) 3  1
Transformation Rules
Vertical Stretches
e.g.1 Consider the following functions:
2
y

4x
y  x2
and
For
y  x2,
For
y  4x 2 ,
x2 
x2 
y 4
y  16
In transforming from y  x 2 to y  4 x 2
the y-value has been multiplied by 4
Transformation Rules
e.g.1 Consider the following functions:
2
y

4x
y  x2
and
For
y  x2,
For
y  4x 2 ,
x2 
y 4
x  2  y  16
In transforming from y  x 2 to y  4 x 2
the y-value has been multiplied by 4
Similarly, for every value of x, the y-value on
y  4x 2 is 4 times the y-value on y  x 2
y x
2

y  4x
2
is a stretch of scale factor
4 parallel to the y-axis
Transformation Rules
e.g.1 Consider the following functions:
2
y

4x
y  x2
and
HIVO –
HOVIS –
As the 4 is
Horizontal Inside the Bracket
Vertical Outside the bracket
Horizontal Opposite of what it says
Vertical Is Same as what it says
Outside the stretch is applied Vertically
Transformation Rules
The graphs of the functions are as follows:
y  4x2
y x
(1, 4)
2
(1, 1)
y  4x 2 is a stretch of y  x 2
by scale factor 4, parallel to the y-axis
Transformation Rules
y  4x 2 is a transformation of y  x 2 given by
a stretch of scale factor 4 parallel to the y-axis
y  4x 2
y  x2
4
Transformation Rules
Horizontal stretches
Now, for
and for
y  x2 ,
x  3  y9
y  (3x) , x  1  y  9
2
The x-value must be divided by 3 to give the same value
of y.
So the x coordinate is stretched by a factor of 
parallel to the x axis
2
y  (3x)
5 ^|
Y
4
3
yx
-2
2
2
-1
1
0
-1
1
2
3
4
5
6
Transformation Rules
e.g.1 Consider the following functions:
2
y

(3x)
y  x2
and
HIVO –
HOVIS –
Horizontal Inside the Bracket
Vertical Outside the bracket
Horizontal Opposite of what it says
Vertical Is Same as what it says
Inside the stretch is applied Horizontally
and is the Opposite. So instead of being 3 times as
As the 3 is
wide it is a  as wide.
Transformation Rules
SUMMARY
The transformation of y  x 2 to y  4 x 2
y x
2

y  4x
is a stretch of scale factor
4 parallel to the y-axis
2
or
y x
2

y  (2 x )
2
is a stretch of scale factor
1 parallel to the x-axis
2
Transformation Rules
SUMMARY
 The function
y  kf ( x ) is obtained from y  f ( x )
by a stretch of scale factor ( s.f. ) k,
parallel to the y-axis.
 The function
y  f (kx) is obtained from y  f ( x )
by a stretch of scale factor ( s.f. ) 1 ,
k
parallel to the x-axis.
Transformation Rules
1
e.g. 2 Describe the transformation of y 
that
x
3
gives y 
.
x
Using the same axes, sketch both functions.
3
Solution: y 
x
can be written as
1
y  3
x
so it is a stretch of s.f. 3, parallel to the y-axis
1
y
x
3
y
3
x
We always stretch
from an axis.
Transformation Rules
Exercises
1. (a) Describe a transformation of y  x 2 that
2
gives y  9x .
(b) Sketch the graphs of both functions to
illustrate your answer.
Solution:
(a) A stretch of s.f. 9 parallel to the y-axis.
(b)
y  9x 2
y  x2
Transformation Rules
Transforming the exponential graph y = ex
5
4
3
2
1
-4
-3
-2
-1
0
-1
-2
-3
-4
-5
1
2
3
4
5
Transformation Rules
y = ex
-4
-3
-2
-1
y = e(2x)
5
5
4
3
4
3
2
1
2
1
0
1
2
3
4
5
-4
-3
-2
-1
0
-1
-2
-1
-2
-3
-4
-3
-4
-5
-5
1
2
3
4
5
Horizontal stretch scale factor 
All the x coordinates are  as wide
Transformation Rules
y = ex
-4
-3
-2
-1
y = 2e(2x)
5
5
4
3
4
3
2
1
2
1
0
1
2
3
4
5
-4
-3
-2
-1
0
-1
-2
-1
-2
-3
-4
-3
-4
-5
-5
1
2
3
4
5
Horizontal stretch scale factor 
Vertical stretch scale factor 2
All the y coordinates are 2x
as high
Transformation Rules
y = 2e(2x)–4
y = ex
-4
-3
-2
-1
5
5
4
3
4
3
2
1
2
1
0
1
2
3
4
5
-4
-3
-2
-1
0
-1
-2
-1
-2
-3
-4
-3
-4
-5
1
2
3
4
5
Horizontal stretch-5scale factor 
Vertical stretch scale factor 2
 0 
Vertical translation  
 4 
Transformation Rules
Two more Transformations
 Reflection in the x-axis
Every y-value changes sign when we reflect in
the x-axis e.g.
y=x2
5 ^|
Y
4
5 ^|
Y
4
3
3
2
1
-4
-3
-2
-1
0
-1
-2
-3
So,
1
2
3
4
X->
5

2
1
-4
-3
-2
y=–(x)2
-1
0
1
2
3
-1
-2
-3
-4
-4
-5
-5
y  x 2  y  ( x )2
In general, a reflection in the x-axis is given by
y  f ( x)  y   f ( x)
4
X->
5
Transformation Rules
 Reflection in the y-axis
Every x-value changes sign when we reflect in
the y-axis e.g.
5 ^|
Y
4
y  x3
3
2
1
-4
-3
-2
-1
0
-1
1
2
3
4
X->
5

5 ^|
Y
4
y  (  x )3
3
2
1
-4
-3
-2
-1
0
1
2
3
4
-1
-2
-2
-3
-3
-4
-4
-5
-5
y  x 3  y  ( x )3
In general, a reflection in the y-axis is given by
y  f ( x )  y  f ( x )
X->
5
Transformation Rules
SUMMARY
 Reflections in the axes
•
Reflecting in the x-axis changes the sign of y
y  f ( x)  y   f ( x)
•
Reflecting in the y-axis changes the sign of x
y  f ( x )  y  f ( x )