Jan 13_Horizontal and vertical translation
.pdf
Horizontal & Vertical Translation
Compared to the graph y f ( x), the graph of y k f ( x h) results in vertical & horizontal translation of k and hunits
|
h 0 |
h 0 |
|||
|
|
|
|
|
|
k 0 |
Up |
↑ |
Up |
↑ |
|
Right |
→ |
Left |
← |
||
|
|||||
|
|
|
|
|
|
k 0 |
Down |
↓ |
Down |
↓ |
|
Right |
→ |
Left |
← |
||
|
|||||
|
|
|
|
|
|
y
y x2
x
y 3 ( x 3)2
Write the equation of the image of |
|
y f ( x) at lowest degree after each |
|
||||||||||||||||||||||||||||||||||||||||||||||
transformationy y |
|
|
|
|
|
|
|
|
y y |
|
|
|
|
|
|
|
|
|
|
|
|
|
y |
|
|
|
|
|
|
|
|
|
|
|
y y |
|
|
|
|
|
|||||||||
|
|
|
|
6 |
6 |
|
|
|
|
|
|
|
|
|
|
|
6 |
6 |
|
|
|
|
|
|
|
|
|
|
|
|
|
6 |
|
|
|
|
|
|
|
|
|
|
|
6 6 |
|
|
|
|
|
|
|
|
|
5 |
5 |
|
|
|
|
|
|
|
|
|
|
|
5 |
5 |
|
|
|
|
|
|
|
|
|
|
|
|
|
5 |
|
|
|
|
|
|
|
|
|
|
|
5 5 |
|
|
|
|
|
|
|
|
|
4 |
4 |
|
|
|
|
|
|
|
|
|
|
|
4 |
4 |
|
|
|
|
|
|
|
|
|
|
|
|
|
4 |
|
|
|
|
|
|
|
|
|
|
|
4 4 |
|
|
|
|
|
|
|
|
|
3 |
3 |
|
|
|
|
|
|
|
|
|
|
|
3 |
3 |
|
|
|
|
|
|
|
|
|
|
|
|
|
3 |
|
|
|
|
|
|
|
|
|
|
|
3 3 |
|
|
|
|
|
|
|
|
|
2 |
2 |
|
|
|
|
|
|
|
|
|
|
|
2 |
2 |
|
|
|
|
|
|
|
|
|
|
|
|
|
2 |
|
|
|
|
|
|
|
|
|
|
|
2 2 |
|
|
|
|
|
|
|
|
|
1 |
1 |
|
|
|
|
|
|
|
|
|
|
|
1 |
1 |
|
|
|
|
|
|
|
|
|
|
|
|
|
1 |
|
|
|
|
|
|
|
|
|
|
|
1 1 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
x |
x |
|
|
|
|
|
|
|
|
|
|
|
x |
x |
|
|
|
|
|
|
|
|
|
|
|
|
x |
|
|
|
|
|
|
|
|
|
|
x x |
6 6 |
5 5 |
4 4 |
3 3 |
2 2 1 1 |
1 |
1 2 |
2 3 |
3 4 |
4 5 |
5 6 |
6 |
|
6 6 |
5 5 |
4 4 |
3 3 |
2 2 1 1 |
1 |
1 2 |
2 3 |
3 4 |
4 5 |
5 6 |
6 |
|
6 |
5 |
4 |
3 |
2 |
1 |
1 |
2 |
3 |
4 |
5 |
6 |
|
66 |
55 |
4 4 |
3 3 |
2 2 |
1 1 |
1 1 2 2 3 |
3 4 |
4 5 |
5 6 |
6 |
|
|
|
|
1 |
1 |
|
|
|
|
|
|
|
|
|
|
|
1 |
1 |
|
|
|
|
|
|
|
|
|
|
|
|
|
1 |
|
|
|
|
|
|
|
|
|
|
|
1 1 |
|
|
|
|
|
|
|
|
|
2 |
2 |
|
|
|
|
|
|
|
|
|
|
|
2 |
2 |
|
|
|
|
|
|
|
|
|
|
|
|
|
2 |
|
|
|
|
|
|
|
|
|
|
|
2 2 |
|
|
|
|
|
|
|
|
|
3 |
3 |
|
|
|
|
|
|
|
|
|
|
|
3 |
3 |
|
|
|
|
|
|
|
|
|
|
|
|
|
3 |
|
|
|
|
|
|
|
|
|
|
|
3 3 |
|
|
|
|
|
|
|
|
|
4 |
4 |
|
|
|
|
|
|
|
|
|
|
|
4 |
4 |
|
|
|
|
|
|
|
|
|
|
|
|
|
4 |
|
|
|
|
|
|
|
|
|
|
|
4 4 |
|
|
|
|
|
|
|
|
|
5 |
5 |
|
|
|
|
|
|
|
|
|
|
|
5 |
5 |
|
|
|
|
|
|
|
|
|
|
|
|
|
5 |
|
|
|
|
|
|
|
|
|
|
|
5 5 |
|
|
|
|
|
|
|
|
|
6 |
6 |
|
|
|
|
|
|
|
|
|
|
|
6 |
6 |
|
|
|
|
|
|
|
|
|
|
|
|
|
6 |
|
|
|
|
|
|
|
|
|
|
|
6 6 |
|
|
|
|
|
y x3 |
|
|
|
|
|
y 2 x3 |
|
|
|
|
|
y ( x 2)3 |
y 2 ( x 2)3 |
||||||||||||||||||||||||||
|
|
|
|
|
|
y |
|
|
|
|
|
|
|
|
|
|
|
|
|
y |
|
|
|
|
|
|
|
|
|
|
|
|
y |
|
|
|
|
|
|
|
|
|
|
|
|
6 |
|
|
|
|
|
|
|
|
|
|
|
|
|
6 |
|
|
|
|
|
|
|
|
|
|
|
|
6 |
|
|
|
|
|
|
|
|
|
|
|
|
5 |
|
|
|
|
|
|
|
|
|
|
|
|
|
5 |
|
|
|
|
|
|
|
|
|
|
|
|
5 |
|
|
|
|
|
|
|
|
|
|
|
|
4 |
|
|
|
|
|
|
|
|
|
|
|
|
|
4 |
|
|
|
|
|
|
|
|
|
|
|
|
4 |
|
|
|
|
|
|
|
|
|
|
|
|
3 |
|
|
|
|
|
|
|
|
|
|
|
|
|
3 |
|
|
|
|
|
|
|
|
|
|
|
|
3 |
|
|
|
|
|
|
|
|
|
|
|
|
2 |
|
|
|
|
|
|
|
|
|
|
|
|
|
2 |
|
|
|
|
|
|
|
|
|
|
|
|
2 |
|
|
|
|
|
|
|
|
|
|
|
|
1 |
|
|
|
|
|
|
|
|
|
|
|
|
|
1 |
|
|
|
|
|
|
|
|
|
|
|
|
1 |
|
|
|
|
|
|
6 |
5 |
4 |
3 |
2 |
1 |
1 |
2 |
3 |
4 |
5 |
6 |
x |
|
6 |
5 |
4 |
3 |
2 |
1 |
1 |
2 |
3 |
4 |
5 |
6 |
x |
6 |
5 |
4 |
3 |
2 |
1 |
1 |
2 |
3 |
4 |
5 |
6 |
x |
|
|
|
|
||||||||||||||||||||||||||||||||||||
|
|
|
|
|
|
1 |
|
|
|
|
|
|
|
|
|
|
|
|
|
1 |
|
|
|
|
|
|
|
|
|
|
|
|
1 |
|
|
|
|
|
|
|
|
|
|
|
|
2 |
|
|
|
|
|
|
|
|
|
|
|
|
|
2 |
|
|
|
|
|
|
|
|
|
|
|
|
2 |
|
|
|
|
|
|
|
|
|
|
|
|
3 |
|
|
|
|
|
|
|
|
|
|
|
|
|
3 |
|
|
|
|
|
|
|
|
|
|
|
|
3 |
|
|
|
|
|
|
|
|
|
|
|
|
4 |
|
|
|
|
|
|
|
|
|
|
|
|
|
4 |
|
|
|
|
|
|
|
|
|
|
|
|
4 |
|
|
|
|
|
|
|
|
|
|
|
|
5 |
|
|
|
|
|
|
|
|
|
|
|
|
|
5 |
|
|
|
|
|
|
|
|
|
|
|
|
5 |
|
|
|
|
|
|
|
|
|
|
|
|
6 |
|
|
|
|
|
|
|
|
|
|
|
|
|
6 |
|
|
|
|
|
|
|
|
|
|
|
|
6 |
|
|
|
|
|
|
y 2 ( x 2)3 |
y 2 ( x 2)3 |
y |
2 ( x |
2)3 |
|||||||||||||||||||||||||||||||||||
y f ( x) |
vs. y f ( x2 ax) |
f ( x) x2
Increasing the positive coefficient of x moves the graph down and to the left
Increasing the negative coefficient of x moves the graph down and to the right
Coefficient of x determines second x-intercept if coefficient of x² equals 1
13
y f ( x) |
vs. y k f ( x2 ax) |
the value of the constant is the point at which the graph crosses the y axis
14
Replacement notation vs. Mapping notation
Consider the function |
y f ( x)to be transformed to the graph |
y 2 f ( x 3) |
|
Replacement notation |
x x 3 and |
y y 2 |
|
Mapping notation ( x, y) ( x 3, y 2)
The coordinates (4,6) will be translated to coordinates (7,8)
15
Given the graph of the function |
y f ( x), sketch the graph of the |
|
|
y 2 f ( x 3) |
||||||||||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
y y 2 |
( 1,2) |
( 1 3,2 |
2) (2,0) |
||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
x x 3 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
y |
|
|
|
|
|
|
|
|
|
|
|
|
y |
|
|
|
|
|
|
|
|
|
|
|
|
|
6 |
|
|
|
|
|
|
|
|
|
|
|
|
6 |
|
|
|
|
|
|
|
|
|
|
|
|
|
5 |
|
|
|
|
|
|
|
|
|
|
|
|
5 |
|
|
|
|
|
|
|
|
|
|
|
|
|
4 |
|
|
|
|
|
|
|
|
|
|
|
|
4 |
|
|
|
|
|
|
|
|
|
|
|
|
|
3 |
|
|
|
|
|
|
|
|
|
|
|
|
3 |
|
|
|
|
|
|
|
|
|
|
|
|
|
2 |
|
|
|
|
|
|
|
|
|
|
|
|
2 |
|
|
|
|
|
|
|
|
|
|
|
|
|
1 |
|
|
|
|
|
|
|
|
|
|
|
|
1 |
|
|
|
|
|
|
|
6 |
5 |
4 |
3 |
2 |
1 |
1 |
2 |
3 |
4 |
5 |
6 |
x |
|
6 |
5 |
4 |
3 |
2 |
1 |
1 |
2 |
3 |
4 |
5 |
6 |
x |
|
|
|
||||||||||||||||||||||||
|
|
|
|
|
|
1 |
|
|
|
|
|
|
|
|
|
|
|
|
1 |
|
|
|
|
|
|
|
|
|
|
|
|
|
2 |
|
|
|
|
|
|
|
|
|
|
|
|
2 |
|
|
|
|
|
|
|
|
|
|
|
|
|
3 |
|
|
|
|
|
|
|
|
|
|
|
|
3 |
|
|
|
|
|
|
|
|
|
|
|
|
|
4 |
|
|
|
|
|
|
|
|
|
|
|
|
4 |
|
|
|
|
|
|
|
|
|
|
|
|
|
5 |
|
|
|
|
|
|
|
|
|
|
|
|
5 |
|
|
|
|
|
|
|
|
|
|
|
|
|
6 |
|
|
|
|
|
|
|
|
|
|
|
|
6 |
|
|
|
|
|
|
|
16
HW is due Jan 14th 2014
•iWrite Math. Pre-Calculus Mathematics Book 12
•Read p. 75-79
•Assignment p.80 (#1-10)