Question

transformation- a function that moves or changes a figure in some way.

• original figure is called the preimage.
• new (transformed) figure is called the image.

a is read \a prime.\
use prime symbols
when naming
an image.
a→a
b→b
c→c
translation- a transformation in which a figure slides, but does not turn, every point of the figure moves the same distance and in the same direction. in a translation, the preimage & image are congruent.
ex. 1: write a rule for the translation of △abc to △abc.
(x, y) →
ex. 2: translate the figure 2 units left and 4 units down. what are the coordinates of the image?
a
b
c
d
ex. 3: the vertices of a square are a(1, -2), b(3, -2), c(3, -4), and d(1, -4). draw the figure and its image after a translation 4 units left and 6 units up.

vertices of abcd	(x, y)	vertices of abcd
b(3, -2)
c(3, -4)
d(1, -4)

Explanation:

EX.1

Step1: Identify coordinates of preimage and image

First, find the coordinates of a point in the preimage (e.g., \( A(-3, 1) \)) and its image \( A'(1, -2) \).

Step2: Calculate horizontal and vertical changes

Horizontal change: \( 1 - (-3) = 4 \) (right 4 units). Vertical change: \( -2 - 1 = -3 \) (down 3 units). So the translation rule is \( (x, y) \to (x + 4, y - 3) \).

Step1: Find original coordinates

From the graph, \( A(0, 3) \), \( B(3, 2) \), \( C(3, 0) \), \( D(0, -2) \).

Step2: Apply translation (left 2: \( x - 2 \), down 4: \( y - 4 \))

• \( A' \): \( (0 - 2, 3 - 4) = (-2, -1) \)
• \( B' \): \( (3 - 2, 2 - 4) = (1, -2) \)
• \( C' \): \( (3 - 2, 0 - 4) = (1, -4) \)
• \( D' \): \( (0 - 2, -2 - 4) = (-2, -6) \)

Step1: Apply translation rule (left 4: \( x - 4 \), up 6: \( y + 6 \))

• For \( A(1, -2) \): \( (1 - 4, -2 + 6) = (-3, 4) \)
• For \( B(3, -2) \): \( (3 - 4, -2 + 6) = (-1, 4) \)
• For \( C(3, -4) \): \( (3 - 4, -4 + 6) = (-1, 2) \)
• For \( D(1, -4) \): \( (1 - 4, -4 + 6) = (-3, 2) \)

Step2: Fill the table

Vertices of ABCD	(x, y)	Vertices of A'B'C'D'
\( B(3, -2) \)	\( (3, -2) \)	\( B'(-1, 4) \)
\( C(3, -4) \)	\( (3, -4) \)	\( C'(-1, 2) \)
\( D(1, -4) \)	\( (1, -4) \)	\( D'(-3, 2) \)

Answer:

\( (x + 4, y - 3) \)

EX.2