Section 1.6
Transformation of Functions
Graphs of Common Functions
Reciprocal Function
y


f ( x) 

1
x

x













Domain:  -,0    0,  
Range:  -,0    0,  
Decreasing on  -,0  and  0,  
Odd function
Vertical Shifts
Vertical Shifts
Let f be a function and c be a positive real number.
 The graph of y  f  x   c is the graph of y  f  x  shifted c units
vertically upward.
 The graph of y  f  x   c is the graph of y  f  x  shifted c units
vertically downward.
Vertical Shifts
Example
Use the graph of f(x)=|x| to obtain g(x)=|x|-2
y




x













Horizontal Shifts
Horizontal Shifts
Let f be a function and c a positive real number.
 The graph of y  f  x  c  is the graph of y  f  x  shifted
to the left c units.
 The graph of y  f  x  c  is the graph of y  f  x  shifted
to the right c units.
Horizontal Shifts
Example
Use the graph of f(x)=x2 to obtain g(x)=(x+1)2
y




x













Combining Horizontal and Vertical Shifts
Example
Use the graph of f(x)=x2 to obtain g(x)=(x+1)2+2
y




x













Reflections of Graphs
Refection about the x-Axis
The graph of y   f  x  is the graph of y  f  x  reflected
about the x-axis.
Reflections about the x-axis
Reflection about the y-Axis
The graph of y  f   x  is the graph of y  f  x  reflected
about y - axis.
Example
Use the graph of f(x)=x3 to obtain the graph
of g(x)= (-x)3.
y




x













Example
Use the graph of f(x)= x to graph g(x)=- x
y




x













Vertical Stretching and Shrinking
Vertically Shrinking
Vertically Stretching
Graph of
f(x)=x3
y

Graph of
g(x)=3x3
y







x
x
























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|
y




x













Horizontal Stretching and Shrinking
Horizontal Shrinking
Horizontal Stretching
Use the graph of f(x)= x to obtain the
Example
1
graph of g(x)= x
3
y




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
Starting graph.
Move the graph
to the left 3 units
Stretch the graph
vertically by 2.
Shift down 1 unit.
Example
1
Given the graph of f(x) below, graph f ( x  1).
2
y




x












Example
Given the graph of f(x) below, graph - f ( x  2)  1.
y




x












Example
Given the graph of f(x) below, graph 2f ( x)  1.

f ( x)  x




x






















Use the graph of f(x)= x to graph g(x)= -x.


The graph of g(x) will appear in which quadrant?


(a) Quadrant I
(b) Quadrant II
(c) Quadrant III
(d) Quadrant IV


g(x)



x























Write the equation of the given graph g(x).
The original function was f(x) =x2

2
(a) g ( x )  ( x  4)  3
2
g
(
x
)

(
x

4)
3
(b)
2
g
(
x
)

(
x

4)
3
(c)
2
(d) g ( x )  ( x  4)  3



x






















g(x)




Write the equation of the given graph g(x).
The original function was f(x) =|x|
(a) g ( x)   x  4
(b) g ( x)   x  4
(c) g ( x)   x  4
(d) g ( x)   x  4