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Transformation Sequences

Grade 8 · Geometry · Worksheet 3

  1. Mason is designing a digital logo for his school club. He starts with a triangle that has vertices at A(2, 7), B(7, 2), and C(7, 7). He first reflects the triangle across the y-axis. Then he rotates the reflected triangle 90° clockwise about the origin. Finally, he translates the rotated triangle 2 units left and 7 units down. What are the coordinates of vertex A after this sequence of transformations? Answer: ______________
  2. √(64) + 3² × (4 - 1) = ? Answer: ______________
  3. √(64) + 3² × (2³ - 4) = ? Answer: ______________
  4. Translate 6 units left and 1 unit down, then reflect over the y-axis, then rotate 90° clockwise about the origin. Answer: ______________
  5. Hana draws triangle ABC on a coordinate plane with vertices at A(12, 8), B(20, 8), and C(14, 16). The triangle undergoes the following sequence of transformations: first, it is reflected across the y-axis; then, it is rotated 180° about the origin; finally, it is translated 10 units left and 6 units down. What are the coordinates of the final image of vertex C? Answer: ______________
  6. Translate 9 units right and 7 units up, then reflect over the y-axis, then rotate 180° clockwise about the origin. Answer: ______________
  7. (4.2 × 10⁶) ÷ (2.1 × 10²) = ? Answer: ______________
  8. Kaia draws triangle ABC on a coordinate plane with vertices at A(3, 1), B(7, 1), and C(5, 9). She applies the following sequence of transformations: first, a translation 5 units left and 3 units down; then, a dilation by a scale factor of 3 with the origin as the center; finally, a reflection across the y-axis. What are the coordinates of the final image of vertex B? Answer: ______________
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Answer Key & Explanations

Transformation Sequences · Grade 8 · Worksheet 3

  1. Mason is designing a digital logo for his school club. He starts with a triangle that has vertices at A(2, 7), B(7, 2), and C(7, 7). He first reflects the triangle across the y-axis. Then he rotates the reflected triangle 90° clockwise about the origin. Finally, he translates the rotated triangle 2 units left and 7 units down. What are the coordinates of vertex A after this sequence of transformations? Answer: (-5, -9) Solution: Start with original point A(2, 7). Reflect across the y-axis. Rule: (x, y) → (-x, y).
    Full step-by-step solution

    Step 1: Start with original point A(2, 7). Step 2: Reflect across the y-axis. Rule: (x, y) → (-x, y). So (2, 7) → (-2, 7). Step 3: Rotate 90° clockwise about the origin. Rule: (x, y) → (y, -x). So (-2, 7) → (7, -(-2)) = (7, 2). Step 4: Translate 2 units left and 7 units down. Rule: (x, y) → (x - 2, y - 7). So (7, 2) → (7 - 2, 2 - 7) = (5, -5). The final coordinates of vertex A are (5, -5).

  2. √(64) + 3² × (4 - 1) = ? Answer: 35 Solution: Calculate inside the parentheses: (4 - 1) = 3 Calculate the square root: √(64) = 8 Calculate the exponent: 3² = 9 Multiply: 9 × 3 = 27 Add: 8 + 27 = 35 The answer is 35.
    Full step-by-step solution

    Step 1: Calculate inside the parentheses: (4 - 1) = 3 Step 2: Calculate the square root: √(64) = 8 Step 3: Calculate the exponent: 3² = 9 Step 4: Multiply: 9 × 3 = 27 Step 5: Add: 8 + 27 = 35 The answer is 35.

  3. √(64) + 3² × (2³ - 4) = ? Answer: 44 Solution: Calculate inside the parentheses: 2³ - 4 = 8 - 4 = 4 Calculate the square root: √(64) = 8 Calculate the exponent: 3² = 9 Multiply: 9 × 4 = 36 Add: 8 + 36 = 44 The answer is 44.
    Full step-by-step solution

    Step 1: Calculate inside the parentheses: 2³ - 4 = 8 - 4 = 4 Step 2: Calculate the square root: √(64) = 8 Step 3: Calculate the exponent: 3² = 9 Step 4: Multiply: 9 × 4 = 36 Step 5: Add: 8 + 36 = 44 The answer is 44.

  4. Translate 6 units left and 1 unit down, then reflect over the y-axis, then rotate 90° clockwise about the origin. Answer: Translate 1 unit up and 6 units right, then reflect over the x-axis Solution: Start with the original sequence: Translate 6 left and 1 down, then reflect over y-axis, then rotate 90° clockwise.
    Full step-by-step solution

    Step 1: Start with the original sequence: Translate 6 left and 1 down, then reflect over y-axis, then rotate 90° clockwise. Step 2: Apply the inverse of the last transformation first. The inverse of a 90° clockwise rotation is a 90° counterclockwise rotation, which maps (x, y) to (-y, x). Step 3: Apply this to our sequence: The rotation undoes itself, leaving: Translate 6 left and 1 down, then reflect over y-axis. Step 4: A reflection over the y-axis maps (x, y) to (-x, y). Step 5: Combining the translation and reflection: Translating 6 left and 1 down gives (x-6, y-1), then reflecting over y-axis gives (-(x-6), y-1) = (-x+6, y-1). Step 6: This is equivalent to reflecting over x-axis first: (x, -y), then translating 1 up and 6 right: (x+6, -y+1). Step 7: Therefore, the equivalent single transformation is: Reflect over x-axis, then translate 1 up and 6 right.

  5. Hana draws triangle ABC on a coordinate plane with vertices at A(12, 8), B(20, 8), and C(14, 16). The triangle undergoes the following sequence of transformations: first, it is reflected across the y-axis; then, it is rotated 180° about the origin; finally, it is translated 10 units left and 6 units down. What are the coordinates of the final image of vertex C? Answer: (4, -22) Solution: Start with vertex C at (14, 16). Reflect across the y-axis. Rule: (x, y) → (-x, y).
    Full step-by-step solution

    Start with vertex C at (14, 16). Step 1: Reflect across the y-axis. Rule: (x, y) → (-x, y). So (14, 16) becomes (-14, 16). Step 2: Rotate 180° about the origin. Rule: (x, y) → (-x, -y). So (-14, 16) becomes (14, -16). Step 3: Translate 10 units left and 6 units down. Rule: (x, y) → (x - 10, y - 6). So (14, -16) becomes (14 - 10, -16 - 6) = (4, -22). The final coordinates of vertex C are (4, -22).

  6. Translate 9 units right and 7 units up, then reflect over the y-axis, then rotate 180° clockwise about the origin. Answer: Translate 9 units left and 7 units down, then reflect over the y-axis Solution: Start with the original sequence: Translate 9 units right and 7 units up, then reflect over the y-axis, then rotate 180° clockwise. Work backwards. The last transformation is a 180° clockwise rotation.
    Full step-by-step solution

    Step 1: Start with the original sequence: Translate 9 units right and 7 units up, then reflect over the y-axis, then rotate 180° clockwise. Step 2: Work backwards. The last transformation is a 180° clockwise rotation. Its inverse is a 180° counterclockwise rotation. A 180° rotation (either direction) maps (x, y) to (-x, -y). Step 3: Apply this inverse to undo the rotation: The sequence becomes: Translate 9 units right and 7 units up, then reflect over the y-axis. Step 4: The next transformation to undo is the reflection over the y-axis. Its inverse is itself: reflecting over the y-axis again. A reflection over the y-axis maps (x, y) to (-x, y). Step 5: Apply this inverse: The sequence becomes: Translate 9 units right and 7 units up. Step 6: The next transformation to undo is the translation. Its inverse is translating 9 units left and 7 units down. Step 7: Now, re-read the problem: The question asks for the sequence that would undo the original. So the answer is the inverse sequence we found: Translate 9 units left and 7 units down, then reflect over the y-axis. The answer is: Translate 9 units left and 7 units down, then reflect over the y-axis.

  7. (4.2 × 10⁶) ÷ (2.1 × 10²) = ? Answer: 20000 Solution: Divide the coefficients: 4.2 ÷ 2.1 = 2 Subtract the exponents: 6 - 2 = 4 Combine the results: 2 × 10⁴ Convert to standard form: 2 × 10,000 = 20,000 The answer is 20000.
    Full step-by-step solution

    Step 1: Divide the coefficients: 4.2 ÷ 2.1 = 2 Step 2: Subtract the exponents: 6 - 2 = 4 Step 3: Combine the results: 2 × 10⁴ Step 4: Convert to standard form: 2 × 10,000 = 20,000 The answer is 20000.

  8. Kaia draws triangle ABC on a coordinate plane with vertices at A(3, 1), B(7, 1), and C(5, 9). She applies the following sequence of transformations: first, a translation 5 units left and 3 units down; then, a dilation by a scale factor of 3 with the origin as the center; finally, a reflection across the y-axis. What are the coordinates of the final image of vertex B? Answer: (6, -6) Solution: Start with vertex B at (7, 1). Step 1: Translate 5 units left and 3 units down. This subtracts 5 from the x-coordinate and 3 from the y-coordinate: (7 - 5, 1 - 3) = (2, -2).
    Full step-by-step solution

    Start with vertex B at (7, 1). Step 1: Translate 5 units left and 3 units down. This subtracts 5 from the x-coordinate and 3 from the y-coordinate: (7 - 5, 1 - 3) = (2, -2). Step 2: Dilate by a scale factor of 3 with the origin as the center. Multiply both coordinates by 3: (2 * 3, -2 * 3) = (6, -6). Step 3: Reflect across the y-axis. This changes the sign of the x-coordinate: (-6, -6). The final coordinates of vertex B are (-6, -6).