THE GRAPH – TRANSLATION
THEOREM
Section 3.2
DEFINITIONS
Translation: In the coordinate plane, a
translation maps each point (x,y) onto
(x + h, y + k)
(In other words, it moves the graph h places to the
right/left, and k places up/down).

Pre-Image : before the move
 Translation image: after the move

GRAPH TRANSLATION THEOREM




In a relation described by the sentence in x and y,
the following two processes yield the same graph:
1. Replacing x by x – h and y – k in the sentence
2. Applying the translation (x,y)  (x + h, y + k)
to the graph of the original relation.
In other words, under the translation, T(x,y) = (x
+ h, y + k), an equation of the image of y = f(x) is
y – k = f(x – h)
EXAMPLES!

Under a translation, the image of (0,0) is (7,8).
 Find a rule for this translation:
T(x,y)  (x + 7, y + 8)

Find the image of (6,-10) under this
translation:
T(6,-10)  (13, -2)
EXAMPLE 2

If the graph of y = x2 is translated 2 units up and
3 units to the left, what is an equation for its
image?
(y – 2) = (x + 3)2
GRAPH SCALE-CHANGE THEOREM

A scale change centered at the origin with a horizontal
factor a ≠ 0 and a vertical scale factor b ≠ 0 is a
transformation that maps (x,y) to (ax, by)

S(x,y)  (ax, by)

In the equation, it should be

Just like Graph Translations!
EXAMPLE 2:

Sketch the graph of
 Vertical scale of
magnitude 4, horizontal
change of 1/6
 Some points are:
 (0,0) , (1,24) , (-1,24)
EXAMPLE 3

Sketch the image of y = x3 under S(x,y) = (-2x, y)
y = (-x/2)3
 Vertical scale of
magnitude 0, horizontal
change of 2
 Some points are:
 (0,0) , (-2,1) , (4, -8)
PUTTING IT ALL TOGETHER!

Describe the graph of
Ellipse, centered at (3, -4)
 Semi-major axis length of 5
 Semi-minor axis length of 2

HOMEWORK
Pages 182 – 183
3- 11