In this lesson you will be investigating the transformation of a function by examining the notations, the tables of values and the graphs of the function and its transformations. After working through problems 1 - 11, you will be able to explain the effects of the following transformations on the graph of the function
fx
f
x
if
c>0
c
0
.
fx+c
f
x
c
;
fx−c
f
x
c
;
fx+c
f
x
c
;
fx−c
f
x
c
;
cfx
c
f
x
if
0<c<1
0
c
1
;
cfx
c
f
x
if
c>1
c
1
;
fcx
f
c
x
if
0<c<1
0
c
1
;
fcx
f
c
x
if
c>1
c
1
;
−fx
f
x
;
f−x
f
x
.
In problems 1 - 11, you will examine the function
fx=x
f
x
x
and its transformations.
Given the function
fx=x
f
x
x
and the table of values for
fx
f
x
, graph
y=fx
y
f
x
on the grid (figure 1).
The function
y=x+2
y
x
2
is a transformation of
fx
f
x
. The notation
fx+2
f
x
2
can be used to show this transformation. Graph
fx
f
x
in red. Complete the table of values then use the table of values to help you graph the transformation (figure 3) . Explain how the transformation changed the graph of
fx
f
x
.
The graph of
f
f
has been translated 2 units to the left. (figure 4)
The function
y=x−1
y
x
1
is a transformation of
fx
f
x
. The notation
fx−1
f
x
1
can be used to show this transformation. Graph
fx
f
x
in red. Complete the table of values then use the table of values to help you graph the transformation (figure 5) . Explain how the transformation changed the graph of
fx
f
x
.
The graph of
f
f
has been translated 1 unit to the right. (figure 6)
The function
y=-3+x
y
-3
x
is a transformation of
fx
f
x
. The notation
fx−3
f
x
3
can be used to show this transformation. Graph
fx
f
x
in red. Complete the table of values then use the table of values to help you graph the transformation (figure 7) . Explain how the transformation changed the graph of
fx
f
x
.
The graph of
f
f
has been translated 3 units down. (figure 8)
The function
y=1+x
y
1
x
is a transformation of
fx
f
x
. The notation
fx+1
f
x
1
can be used to show this transformation. Graph
fx
f
x
in red. Complete the table of values then use the table of values to help you graph the transformation (figure 9) . Explain how the transformation changed the graph of
fx
f
x
.
The graph of
f
f
has been translated 1 unit up. (figure 10)
The function
y=2x
y
2
x
is a transformation of
fx
f
x
. The notation
2fx
2
f
x
can be used to show this transformation. Graph
fx
f
x
in red. Complete the table of values then use the table of values to help you graph the transformation (figure 11) . Explain how the transformation changed the graph of
fx
f
x
.
The graph of
f
f
has been vertically stretched by a scale factor of 2. (figure 12)
The function
y=.5x
y
.5
x
is a transformation of
fx
f
x
. The notation
.5fx
.5
f
x
can be used to show this transformation. Graph
fx
f
x
in red. Complete the table of values then use the table of values to help you graph the transformation (figure 13) . Explain how the transformation changed the graph of
fx
f
x
.
The graph of
f
f
has been vertically compressed by a scale factor of 0.5 . (figure 14)
The function
y=2x
y
2
x
is a transformation of
fx
f
x
. The notation
f2x
f
2
x
can be used to show this transformation. Graph
fx
f
x
in red. Complete the table of values then use the table of values to help you graph the transformation (figure 15) . Explain how the transformation changed the graph of
fx
f
x
.
The graph of
f
f
has been horizontally compressed by a scale factor of 0.5 . (figure 16)
The function
y=.5x
y
.5
x
is a transformation of
fx
f
x
. The notation
f.5x
f
.5
x
can be used to show this transformation. Graph
fx
f
x
in red. Complete the table of values then use the table of values to help you graph the transformation (figure 17) . Explain how the transformation changed the graph of
fx
f
x
.
The graph of
f
f
has been horizontally stretched by a scale factor of 2. (figure 18)
The function
y=−x
y
x
is a transformation of
fx
f
x
. The notation
−fx
f
x
can be used to show this transformation. Graph
fx
f
x
in red. Complete the table of values then use the table of values to help you graph the transformation (figure 19) . Explain how the transformation changed the graph of
fx
f
x
.
The graph of
f
f
has been reflected over the x - axis. (figure 20)
The function
y=−x
y
x
is a transformation of
fx
f
x
. The notation
f−x
f
x
can be used to show this transformation. Graph
fx
f
x
in red. Complete the table of values then use the table of values to help you graph the transformation (figure 21) . Explain how the transformation changed the graph of
fx
f
x
.
The graph of
f
f
has been reflected over the y - axis. (figure 22)
Answer questions 12 - 21 to test your knowledge of functions and their transformations.
Given the function
gx
g
x
,
c
c
is a real number , and
c>0
c
0
explain how the transformation
gx+c
g
x
c
changes the graph of
gx
g
x
.
The graph of
gx
g
x
will be translated
c
c
units up.
Given the function
gx
g
x
,
c
c
is a real number , and
c>0
c
0
explain how the transformation
gx−c
g
x
c
changes the graph of
gx
g
x
.
The graph of
gx
g
x
will be translated
c
c
units down.
Given the function
gx
g
x
,
c
c
is a real number , and
c>0
c
0
explain how the transformation
gx+c
g
x
c
changes the graph of
gx
g
x
.
The graph of
gx
g
x
will be translated
c
c
units to the left.
Given the function
gx
g
x
,
c
c
is a real number , and
c>0
c
0
explain how the transformation
gx−c
g
x
c
changes the graph of
gx
g
x
.
The graph of
gx
g
x
will be translated
c
c
units to the right.
Given the function
gx
g
x
,
c
c
is a real number , and
c>1
c
1
explain how the transformation
gcx
g
c
x
changes the graph of
gx
g
x
.
The graph of
gx
g
x
will be horizontally compressed by a scale factor of
1c
1
c
.
Given the function
gx
g
x
,
c
c
is a real number , and
0<c<1
0
c
1
explain how the transformation
gcx
g
c
x
changes the graph of
gx
g
x
.
The graph of
gx
g
x
will be horizontally stretched by a scale factor of
c
c
.
Given the function
gx
g
x
,
c
c
is a real number , and
c>1
c
1
explain how the transformation
cgx
c
g
x
changes the graph of
gx
g
x
.
The graph of
gx
g
x
will be vertically stretched by a scale factor of
c
c
.
Given the function
gx
g
x
,
c
c
is a real number , and
0<c<1
0
c
1
explain how the transformation
cgx
c
g
x
changes the graph of
gx
g
x
.
The graph of
gx
g
x
will be vertically compressed by a scale factor of
1c
1
c
.
Given the function
gx
g
x
explain how the transformation
−gx
g
x
changes the graph of
gx
g
x
.
The graph of
gx
g
x
will be reflected over the x - axis.
Given the function
gx
g
x
explain how the transformation
g−x
g
x
changes the graph of
gx
g
x
.
The graph of
gx
g
x
will be reflected over the y - axis.
