### Transformations

```TRANSFORMATIONS
Shifts
Stretches
And
Reflections
When a function y = f(x) is multiplied by –1, the
which results is f(x) reflected about the x- axis
y  f (x)
x
x
 x
0
0
0
1
1
-1
4
2
-2
9
3
-3
y  1( f ( x))   f ( x)
When the argument x of a function y = f(x) is multiplied by
–1, the which results is f(x) reflected about the y- axis
x x
x
x
0
0
0
0
1
1
-1
1
4
2
-2
2
9
3
-3
3
Vertical Shift
• If a positive real number k is added to a function
f(x), the graph of the new function f(x)+k is shifted
vertically k units
f ( x)  x 2
x2
x
x2 1
x2  3
-2
4
7
3
-1
1
4
0
0
0
3
-1
1
1
4
0
2
4
7
3
f ( x)  x 2  3
K=3
f ( x)  x 2  1
K= -1
Vertical shifts
f ( x)  x 2  3
f ( x) f (xx2) x4
2
f ( x)  x 2
f ( x)  x 2  1
Right Horizontal Shift
• If the argument x of a function y = f(x) is
replaced by x – h for h > 0, the graph of the
new function y = f(x – h) is the graph of f(x)
shifted horizontally right h units.
x
x2
( x  3) 2
-1
1
16
0
0
9
1
1
4
2
2
1
3
5
0
4
16
1
5
25
4
f ( x)  ( x  3) 2
Left Horizontal shift
• If the argument x of a function y = f(x) is
replaced by x + h for h > 0, the graph of
the new function y = f(x + h) is the graph
of f(x) shifted horizontally left h units.
f ( x)  ( x  3) 2
2
x
x
2
(x + 3)
-5
25
4
-3
9
0
-1
1
4
0
0
9
2
4
25
4
16
49
Combining Vertical and Horizontal Shift
f ( x)  x 2  3
f ( x)  ( x  3)2  3
f ( x)  x 2  3
Vertically Stretched and Vertically
Compressed Functions
When a function f(x) is multiplied by a positive number a, the graph
of the new function af(x) is obtained by multplying each y coordinate
on the graph of y =f(x) by a.
x
|x|
2|x|
½|x|
-1
1
2
½
-2
2
4
1
0
0
0
0
1
1
2
½
2
2
4
1
Vertical Stretch
When a function f(x) is multiplied by a positive number a
the graph of the new function af(x) is obtained by
multplying each y coordinate on the graph of y =f(x) by a.
The new graph is a vertically stretched version of a
f ( x)  2 x
f ( x)  x
Vertical Compressions
If the graph of a function f(x) is multiplied by a number
a so that 0 < a < 1, then the new function af(x) is a
vertical compression
f ( x)  x
f ( x) 
1
x
2
Shifts
Vertical
Name the function whose graph is given below
1.
2.
3.
4.
5.
6.
```