Section 2.5Transformation of Functions
Graphs of Common Functions
x
y
Reciprocal Function
Domain: - ,0 0,
Range: - ,0 0,
Decreasing on - ,0 0,
Odd function
and
1( )f x
x
Vertical Shifts
Vertical Shifts
Let be a function and be a positive real number.
The graph of is the graph of shifted units
vertically upward.
The graph of is the graph of shifted
f c
y f x c y f x c
y f x c y f x c
units
vertically downward.
Vertical Shifts
Example
Use the graph of f(x)=|x| to obtain g(x)=|x|-2
x
y
Horizontal Shifts
Horizontal Shifts
Let be a function and a positive real number.
The graph of is the graph of shifted
to the left units.
The graph of is the graph of shifted
to the
f c
y f x c y f x
c
y f x c y f x
right units.c
Horizontal Shifts
Example
Use the graph of f(x)=x2 to obtain g(x)=(x+1)2
x
y
Combining Horizontal and Vertical Shifts
Example
Use the graph of f(x)=x2 to obtain g(x)=(x+1)2+2
x
y
Reflections of Graphs
Refection about the -Axis
The graph of is the graph of reflected
about the -axis.
x
y f x y f x
x
Reflections about the x-axis
Reflection about the y-Axis
The graph of is the graph of reflected
about - axis.
y f x y f x
y
Example
Use the graph of f(x)=x3 to obtain the graph of g(x)= (-x)3.
x
y
Example
x
y
Use the graph of f(x)= x to graph g(x)=- x
Vertical Stretching and Shrinking
Vertically Shrinking
Vertically Stretching
x
y
x
yGraph of f(x)=x3
Graph of g(x)=3x3
This is vertical stretching – each y coordinate is multiplied by 3 to stretch the graph.
Example
Use the graph of f(x)=|x| to graph g(x)= 2|x|
x
y
Horizontal Stretching and Shrinking
Horizontal Shrinking
Horizontal Stretching
Example
x
y
Use the graph of f(x)= to obtain the
1graph of g(x)=
3
x
x
Sequences of Transformations
A function involving more than one transformation can be graphed by performing transformations in the following order:
1. Horizontal shifting
2. Stretching or shrinking
3. Reflecting
4. Vertical shifting
Summary of Transformations
A Sequence of Transformations
Move the graph to the left 3 units
Starting graph.
Stretch the graph vertically by 2.
Shift down 1 unit.
Example
x
y
1Given the graph of f(x) below, graph ( 1).
2f x
Example
x
y
Given the graph of f(x) below, graph - ( 2) 1.f x
Example
Given the graph of f(x) below, graph 2 ( ) 1.f x
(a)
(b)
(c)
(d)
x
y
Use the graph of f(x)= x to graph g(x)= -x.
The graph of g(x) will appear in which quadrant?
Quadrant I
Quadrant II
Quadrant III
Quadrant IV
( )f x x
(a)
(b)
(c)
(d)
x
y
Write the equation of the given graph g(x). The original function was f(x) =x2
g(x)
2
2
2
2
( ) ( 4) 3
( ) ( 4) 3
( ) ( 4) 3
( ) ( 4) 3
g x x
g x x
g x x
g x x
(a)
(b)
(c)
(d)
Write the equation of the given graph g(x). The original function was f(x) =|x|
g(x)
( ) 4
( ) 4
( ) 4
( ) 4
g x x
g x x
g x x
g x x
x
y