Transforming Graphs
Vertical Translation
f (x)
y
3
2
1
x
0
-3
-2
-1
0
1
2
3
-1
-2
-3
Vertical Translation g ( x) = f ( x) + 1.5
f (x)
y
y
3
3
2
2
1
-3
-2
-1
0
1
2
3
-2
-1
0
-1
-2
-2
-3
-3
1
2
3
Horizontal Translation
Horizontal Translation
f (x)
g ( x ) = f ( x + 2)
y
g ( x ) = f ( x + 2)
y
y
y
3
3
3
3
2
2
2
2
1
1
1
x
0
-2
x
0
-3
-1
f (x)
-3
If g(x) = f(x) + c, then the graph of g(x) is the graph
of f(x) shifted up c units.
If g(x) = f(x) – c, then the graph of g(x) is the graph
of f(x) shifted down c units.
1
x
0
Vertical Translation
-1
0
1
2
3
x
0
-3
-2
-1
0
1
2
3
1
x
0
-3
-2
-1
0
1
2
3
x
0
-3
-2
-1
0
-1
-1
-1
-1
-2
-2
-2
-2
-3
-3
-3
-3
1
2
3
1
Horizontal Translation
Horizontal Translation
f (x)
g ( x ) = f ( x − 2)
y
If g(x) = f(x + c), then the graph of g(x) is the graph
of f(x) shifted left c units.
y
3
3
2
2
1
1
x
0
-3
Horizontal Translation
f (x)
y
3
3
2
2
1
-3
-2
-1
0
1
2
3
-2
-1
0
-1
-1
-2
-2
-3
-3
1
2
3
x
0
-3
-2
-1
0
-1
-1
-2
-2
-3
-3
1
2
3
x
0
-3
0
If g(x) = f(x + c), then the graph of g(x) is the graph
of f(x) shifted left c units.
If g(x) = f(x – c), then the graph of g(x) is the graph
of f(x) shifted right c units.
1
x
0
-1
Horizontal Translation
g ( x ) = f ( x − 2)
y
-2
1
2
3
2
Reflection about the x-axis
y
y
y
3
3
3
3
2
2
2
2
1
x
0
-2
g ( x) = − f ( x )
f (x)
y
1
-3
Reflection about the x-axis
g ( x) = − f ( x )
f (x)
-1
0
1
2
3
-3
-2
-1
1
1
x
0
0
1
2
3
x
0
-3
-2
-1
0
1
2
3
-2
-1
-1
-1
-2
-2
-2
-2
-3
-3
-3
-3
-3
-2
3
2
2
-1
Reflection about the y-axis
-2
0
1
2
3
-2
-1
-2
-2
-3
-3
3
2
2
2
2
1
1
1
2
3
x
0
-3
-2
-1
0
2
3
y
y
3
1
1
g ( x) = f (− x )
f (x)
3
0
3
x
0
-1
3
-1
2
0
-3
-1
y
x
1
1
x
Reflection about the y-axis
g ( x) = f (− x )
0
3
y
3
0
y
2
g ( x) = f (− x )
y
1
f (x)
1
Reflection about the y-axis
f (x)
If g(x) =– f(x) then the graph of g(x) is the graph of
f(x) reflected about the x-axis.
-3
0
-1
Reflection about the x-axis
x
0
-3
-1
1
2
3
1
x
0
-3
-2
-1
0
1
2
3
x
0
-3
-2
-1
0
-1
-1
-1
-1
-2
-2
-2
-2
-3
-3
-3
-3
1
Reflection about the y-axis
If g(x) = f(–x) then the graph of g(x) is the graph of
f(x) reflected about the y-axis.
2
Vertical Stretching
Vertical Stretching
g ( x ) = 2 f ( x)
f (x)
y
y
-4
y
y
4
4
4
4
2
2
2
2
x
0
-6
g ( x ) = 2 f ( x)
f (x)
-2
0
2
4
x
0
-6
6
-4
-2
0
2
4
6
x
0
-6
-4
-2
0
2
4
-4
-2
0
-2
-2
-2
-2
-4
-4
-4
-4
Vertical Stretching
4
Vertical Crunching
g ( x ) = 2 f ( x)
f (x)
g ( x) =
f (x)
y
2
y
4
x
0
-6
6
2
4
6
4
6
1
f ( x)
3y
5
5
4
4
3
3
2
2
2
x
0
-6
-4
-2
0
2
4
6
x
0
-6
-4
-2
0
2
4
6
-2
1
-2
-4
-6
Vertical Crunching
g ( x) =
f (x)
1
x
0
-4
-2
0
2
4
6
x
0
-6
-4
-2
0
2
Vertical Stretching
1
f ( x)
3y
If g(x) = cf(x) then the graph of g(x) is the graph of
f(x) stretched by a factor of c if c is a constant
greater than 1.
If g(x) = cf(x) then the graph of g(x) is the graph of
f(x) crunched by a factor of c if c is a constant
between 0 and 1.
5
4
3
2
y
5
4
1
3
2
1
-4
-2
0
2
4
6
x
0
x
0
-6
-6
-4
-2
0
2
4
6
1
Transformation Summary
Using Transformations
g(x) = f(x) + c, Shift UP
g(x) = f(x) – c, Shift DOWN
g(x) = f(x + c), Shift LEFT
g(x) = f(x – c), Shift RIGHT
g(x) = – f(x) Reflect about the X-AXIS
g(x) = f(–x) Reflect about the Y-AXIS
g(x) = cf(x) Vertical Stretch if c > 1.
g(x) = cf(x) Vertical Crunch if 0 < c < 1.
How is the graph of g(x) = |x – 2| + 3 related to the
graph of f(x) = |x|?
What transformations would turn f(x) into g(x)
h(x)= f(x – 2)=|x – 2| (shift right 2)
g(x)= h(x) + 3= |x – 2|+ 3 (shift up 3.)
The graph of g(x) is the graph of f(x) shifted right 2
and up 3.
Using Transformations
f ( x) =| x |
Using Transformations
g ( x) =| x − 2 | +3
y
10
8
6
6
h( x) = 2 f ( x) ( vertical stretch)
4
4
g ( x) = − h( x) ( vertical reflection)
2
x
0
-4
What transformations would turn f ( x) into g ( x)?
8
2
-6
How is the graph of g ( x) = −2 x related to the graph of f ( x) = x?
y
10
-2
0
2
4
6
x
0
-6
-4
-2
0
-2
-2
-4
-4
2
4
6
Using Transformations
f ( x) = x
1
How is the graph of g ( x) = − ( x + 2) 2 + 3 related to
2
the graph of f ( x) = x 2?
y
6
6
4
4
2
2
x
0
0
Using Transformation
g ( x ) = −2 x
y
2
4
6
8
10
What transformations would turn f ( x) into g ( x)?
x
0
12
0
-2
-2
-4
-4
-6
-6
The graph of g ( x) is the graph of f ( x) stretched
and reflected vertically.
2
4
6
8
10
12
h( x) = f ( x + 2) = ( x + 2) 2 (left shift 2)
1
1
k ( x) = h( x) = ( x + 2) 2 ( vertical crunch)
2
2
1
p ( x) = − k ( x) = − ( x + 2) 2 ( vertical reflection)
2
1
g ( x) = p ( x) + 3 = − ( x + 2) 2 + 3 ( vertical shift 3)
2
1
Using Transformations
1
g ( x) = − ( x + 2) 2 + 3
2
y
f ( x) = x 2
y
6
6
4
4
2
2
x
0
-6
-4
-2
0
2
4
6
x
0
-6
-4
-2
0
-2
-2
-4
-4
-6
-6
2
4
6
2