Question

give the coordinates for rectangle defg with vertices d(-2, 7), e(2, 3), f(0, 1), and g(-4, 5):
a) translation along the rule (x, y) → (x + 6, y - 8)
b) reflection in the y-axis
enter the numerical value for your answer (ie if your answer is 2, type \2\ not \two\)
d ( type your answer... type your answer... )
e ( type your answer... type your answer... )
f ( type your answer... type your answer... )

Explanation:

Part a) Translation along the rule \((x, y) \to (x + 6, y - 8)\)

For point \(D(-2, 7)\):

Step 1: Apply translation to \(x\)-coordinate

Add 6 to the \(x\)-coordinate: \(-2 + 6 = 4\)

Step 2: Apply translation to \(y\)-coordinate

Subtract 8 from the \(y\)-coordinate: \(7 - 8 = -1\)
So, \(D'' = (4, -1)\)

For point \(E(2, 3)\):

Step 1: Apply translation to \(x\)-coordinate

Add 6 to the \(x\)-coordinate: \(2 + 6 = 8\)

Step 2: Apply translation to \(y\)-coordinate

Subtract 8 from the \(y\)-coordinate: \(3 - 8 = -5\)
So, \(E'' = (8, -5)\)

For point \(F(0, 1)\):

Step 1: Apply translation to \(x\)-coordinate

Add 6 to the \(x\)-coordinate: \(0 + 6 = 6\)

Step 2: Apply translation to \(y\)-coordinate

Subtract 8 from the \(y\)-coordinate: \(1 - 8 = -7\)
So, \(F'' = (6, -7)\)

For point \(G(-4, 5)\):

Step 1: Apply translation to \(x\)-coordinate

Add 6 to the \(x\)-coordinate: \(-4 + 6 = 2\)

Step 2: Apply translation to \(y\)-coordinate

Subtract 8 from the \(y\)-coordinate: \(5 - 8 = -3\)
So, \(G'' = (2, -3)\)

Part b) Reflection in the \(y\)-axis (rule: \((x, y) \to (-x, y)\))

For point \(D(-2, 7)\):

Step 1: Apply reflection to \(x\)-coordinate

Negate the \(x\)-coordinate: \(-(-2) = 2\)

Step 2: \(y\)-coordinate remains the same

\(y = 7\)
So, \(D' = (2, 7)\)

For point \(E(2, 3)\):

Step 1: Apply reflection to \(x\)-coordinate

Negate the \(x\)-coordinate: \(-2\)

Step 2: \(y\)-coordinate remains the same

\(y = 3\)
So, \(E' = (-2, 3)\)

For point \(F(0, 1)\):

Step 1: Apply reflection to \(x\)-coordinate

Negate the \(x\)-coordinate: \(-0 = 0\)

Step 2: \(y\)-coordinate remains the same

\(y = 1\)
So, \(F' = (0, 1)\)

For point \(G(-4, 5)\):

Step 1: Apply reflection to \(x\)-coordinate

Negate the \(x\)-coordinate: \(-(-4) = 4\)

Step 2: \(y\)-coordinate remains the same

\(y = 5\)
So, \(G' = (4, 5)\)

Final Answers (for part a)

• \(D''\): \(4, -1\)
• \(E''\): \(8, -5\)
• \(F''\): \(6, -7\)
• \(G''\): \(2, -3\)

Final Answers (for part b)

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

Answer:

Part a) Translation along the rule \((x, y) \to (x + 6, y - 8)\)

For point \(D(-2, 7)\):

Step 1: Apply translation to \(x\)-coordinate

Add 6 to the \(x\)-coordinate: \(-2 + 6 = 4\)

Step 2: Apply translation to \(y\)-coordinate

Subtract 8 from the \(y\)-coordinate: \(7 - 8 = -1\)
So, \(D'' = (4, -1)\)

For point \(E(2, 3)\):

Step 1: Apply translation to \(x\)-coordinate

Add 6 to the \(x\)-coordinate: \(2 + 6 = 8\)

Step 2: Apply translation to \(y\)-coordinate

Subtract 8 from the \(y\)-coordinate: \(3 - 8 = -5\)
So, \(E'' = (8, -5)\)

For point \(F(0, 1)\):

Step 1: Apply translation to \(x\)-coordinate

Add 6 to the \(x\)-coordinate: \(0 + 6 = 6\)

Step 2: Apply translation to \(y\)-coordinate

Subtract 8 from the \(y\)-coordinate: \(1 - 8 = -7\)
So, \(F'' = (6, -7)\)

For point \(G(-4, 5)\):

Step 1: Apply translation to \(x\)-coordinate

Add 6 to the \(x\)-coordinate: \(-4 + 6 = 2\)

Step 2: Apply translation to \(y\)-coordinate

Subtract 8 from the \(y\)-coordinate: \(5 - 8 = -3\)
So, \(G'' = (2, -3)\)

Part b) Reflection in the \(y\)-axis (rule: \((x, y) \to (-x, y)\))

For point \(D(-2, 7)\):

Step 1: Apply reflection to \(x\)-coordinate

Negate the \(x\)-coordinate: \(-(-2) = 2\)

Step 2: \(y\)-coordinate remains the same

\(y = 7\)
So, \(D' = (2, 7)\)

For point \(E(2, 3)\):

Step 1: Apply reflection to \(x\)-coordinate

Negate the \(x\)-coordinate: \(-2\)

Step 2: \(y\)-coordinate remains the same

\(y = 3\)
So, \(E' = (-2, 3)\)

For point \(F(0, 1)\):

Step 1: Apply reflection to \(x\)-coordinate

Negate the \(x\)-coordinate: \(-0 = 0\)

Step 2: \(y\)-coordinate remains the same

\(y = 1\)
So, \(F' = (0, 1)\)

For point \(G(-4, 5)\):

Step 1: Apply reflection to \(x\)-coordinate

Negate the \(x\)-coordinate: \(-(-4) = 4\)

Step 2: \(y\)-coordinate remains the same

\(y = 5\)
So, \(G' = (4, 5)\)

Final Answers (for part a)

• \(D''\): \(4, -1\)
• \(E''\): \(8, -5\)
• \(F''\): \(6, -7\)
• \(G''\): \(2, -3\)

Final Answers (for part b)

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