TRANSFORMATIONS OF FUNCTIONS

 

Variable Replace
With
   
y y - 2 Vertical translation Transfer the graph vertically 2 units up
y y + 2 Vertical translation Transfer the graph vertically 2 units down
       
x x - 3 Horizontal translation Transfer the graph horizontally 3 units right
x x + 3 Horizontal translation Transfer the graph horizontally 3 units left
       
x -x Reflection in the y-axis  
y -y Reflection in the x-axis  
       
y   Vertical expansion Expand the graph vertically by a factor 3
y 3y Vertical compression Compress the graph vertically by a factor 1/3
       
x Horizontal expansion Expand the graph horizontally by a factor 2
x 3x Horizontal compression Compress the graph horizontally by a factor 1/3

 

 

 

TRANSFORMATIONS OF EXPONENTIAL FUNCTIONS

 

Example 1:

Graph:

Red:  y = 2x

Blue: y = (1/2)x

 

 


 

Example 2:

 

 

Graph:  y = 2x-2 3

 

Steps are given below:

 

Step 1

y = 2x

red

 

 

Step 2

y = 2x-2

blue

 

Translate the graph 2 units to the right (replace x by (x-2))

 

 

Step 3

y = 2x-2 3

green

 

Translate the graph 3 units down ((replace y by (y+3)

 


 

Example 3:

Graph:  y = (-2)(1/3)x+4 + 2

 

Steps are given below:

 

 

Step 1

y = 3x

red

 

Step 2

y = (1/3)x

 

blue

Reflect the graph in y axis

 

Step 3

3 = (1/3)x+4

green

 

Translate the graph 4 units to the left (replace x by (x+4)).

 

Step 4

y = 2(1/3)x+4

black

 

Expand the graph vertically by a factor of 2

 

 

Step 5

y = -2(1/3)x+4

 

magenta

Reflect the graph in x axis

 

Step 6

y = (-2)(1/3)x+4 + 2

 

brown

Translate the graph 2 units up

((replace y by (y-2)

 

Equation of asymptote:

x=2

Domain:  all real values

Range:  y < 2