Question

directions: graph and label each figure and its image under the sequence of transformations. give the coordinates of the image.

1. rectangle \\(defg\\) with vertices \\(d(-2, 7)\\), \\(e(2, 3)\\), \\(f(0, 1)\\), and \\(g(-4, 5)\\):

a) translation along the rule \\((x, y) \
ightarrow (x + 6, y - 8)\\)
b) reflection in the \\(y\\)-axis

\\(d(\underline{\quad}, \underline{\quad})\\)
\\(e(\underline{\quad}, \underline{\quad})\\)
\\(f(\underline{\quad}, \underline{\quad})\\)
\\(g(\underline{\quad}, \underline{\quad})\\)

Explanation:

Identify the given vertices and sequence

We are given rectangle \(DEFG\) with the following initial vertices:

• \(D(-2, 7)\)
• \(E(2, 3)\)
• \(F(0, 1)\)
• \(G(-4, 5)\)

We need to apply a sequence of two transformations to find the final coordinates of the image \(D'E'F'G'\):

1. Translation along the rule \((x, y)

ightarrow (x + 6, y - 8)\)

1. Reflection in the \(y\)-axis

Apply the translation rule

Let's apply the translation rule \((x, y)
ightarrow (x + 6, y - 8)\) to each vertex to find the intermediate coordinates, which we will call \(D_1, E_1, F_1, G_1\):

• For \(D(-2, 7)\):

$$ D_1 = (-2 + 6, 7 - 8) = (4, -1) $$

• For \(E(2, 3)\):

$$ E_1 = (2 + 6, 3 - 8) = (8, -5) $$

• For \(F(0, 1)\):

$$ F_1 = (0 + 6, 1 - 8) = (6, -7) $$

• For \(G(-4, 5)\):

$$ G_1 = (-4 + 6, 5 - 8) = (2, -3) $$

Apply the reflection rule

Next, we apply a reflection in the \(y\)-axis to the intermediate coordinates \(D_1, E_1, F_1, G_1\).
The rule for reflecting a point over the \(y\)-axis is:

$$ (x, y) ightarrow (-x, y) $$

Applying this rule gives the final coordinates \(D', E', F', G'\):

• For \(D_1(4, -1)\):

$$ D' = (-4, -1) $$

• For \(E_1(8, -5)\):

$$ E' = (-8, -5) $$

• For \(F_1(6, -7)\):

$$ F' = (-6, -7) $$

• For \(G_1(2, -3)\):

$$ G' = (-2, -3) $$

Answer:

Rectangle \(DEFG\) with vertices \(D(-2, 7)\), \(E(2, 3)\), \(F(0, 1)\), and \(G(-4, 5)\):
a) translation along the rule \((x, y)
ightarrow (x + 6, y - 8)\)
b) reflection in the \(y\)-axis

\(D'\) <blank>\((-4, -1)\)</blank>
\(E'\) <blank>\((-8, -5)\)</blank>
\(F'\) <blank>\((-6, -7)\)</blank>
\(G'\) <blank>\((-2, -3)\)</blank>