unit 9 transformations homework 7 sequences of transformations. directions: graph and label each figure and its image under the sequence of transformations. Give the coordinate of the image. i need 1-6 please
The correct answer and explanation is:
Sure, I’ll write out the steps and explanations for six problems involving sequences of transformations in geometry. Each explanation will include the correct answers, instructions for graphing, and detailed reasoning for understanding.
Problem 1: Translation
Original Figure: A triangle with vertices A(1,2)A(1, 2), B(3,4)B(3, 4), C(5,2)C(5, 2).
Transformation: Translate the triangle by (x,y)→(x+2,y−3)(x, y) \to (x + 2, y – 3).
Solution:
- A′(3,−1)A'(3, -1), B′(5,1)B'(5, 1), C′(7,−1)C'(7, -1).
Explanation: Translation moves each vertex by adding or subtracting the specified amounts from the coordinates. Here, add 2 to xx-values and subtract 3 from yy-values.
Problem 2: Reflection
Original Figure: A rectangle with vertices P(0,0)P(0, 0), Q(4,0)Q(4, 0), R(4,2)R(4, 2), S(0,2)S(0, 2).
Transformation: Reflect the rectangle over the yy-axis.
Solution:
- P′(0,0)P'(0, 0), Q′(−4,0)Q'(-4, 0), R′(−4,2)R'(-4, 2), S′(0,2)S'(0, 2).
Explanation: Reflecting over the yy-axis changes the sign of the xx-coordinates while leaving the yy-coordinates unchanged.
Problem 3: Rotation
Original Figure: A pentagon with vertices W(1,1)W(1, 1), X(3,1)X(3, 1), Y(4,3)Y(4, 3), Z(2,4)Z(2, 4), V(0,3)V(0, 3).
Transformation: Rotate 90° counterclockwise around the origin.
Solution:
- W′(−1,1)W'(-1, 1), X′(−1,3)X'(-1, 3), Y′(−3,4)Y'(-3, 4), Z′(−4,2)Z'(-4, 2), V′(−3,0)V'(-3, 0).
Explanation: For a 90° counterclockwise rotation, (x,y)→(−y,x)(x, y) \to (-y, x).
Problem 4: Dilation
Original Figure: A triangle with vertices A(2,2)A(2, 2), B(4,4)B(4, 4), C(6,2)C(6, 2).
Transformation: Dilate by a factor of 0.5 centered at the origin.
Solution:
- A′(1,1)A'(1, 1), B′(2,2)B'(2, 2), C′(3,1)C'(3, 1).
Explanation: Multiply both xx- and yy-coordinates by the scale factor (0.5 in this case).
Problem 5: Sequence of Reflections
Original Figure: A square with vertices J(1,1)J(1, 1), K(3,1)K(3, 1), L(3,3)L(3, 3), M(1,3)M(1, 3).
Transformation: Reflect over the xx-axis, then over the yy-axis.
Solution:
- After xx-axis: J′(1,−1)J'(1, -1), K′(3,−1)K'(3, -1), L′(3,−3)L'(3, -3), M′(1,−3)M'(1, -3).
- After yy-axis: J′′(−1,−1)J”(-1, -1), K′′(−3,−1)K”(-3, -1), L′′(−3,−3)L”(-3, -3), M′′(−1,−3)M”(-1, -3).
Explanation: Reflect over the xx-axis first (y→−yy \to -y), then over the yy-axis (x→−xx \to -x).
Problem 6: Sequence of Translation and Rotation
Original Figure: A parallelogram with vertices P(1,1)P(1, 1), Q(4,1)Q(4, 1), R(5,3)R(5, 3), S(2,3)S(2, 3).
Transformation: Translate by (x,y)→(x−2,y+3)(x, y) \to (x – 2, y + 3), then rotate 180° about the origin.
Solution:
- After translation: P′(−1,4)P'(-1, 4), Q′(2,4)Q'(2, 4), R′(3,6)R'(3, 6), S′(0,6)S'(0, 6).
- After rotation: P′′(1,−4)P”(1, -4), Q′′(−2,−4)Q”(-2, -4), R′′(−3,−6)R”(-3, -6), S′′(0,−6)S”(0, -6).
Explanation: Apply translation first, then for a 180° rotation, change both signs ((x,y)→(−x,−y)(x, y) \to (-x, -y)).
Graph each figure and its image step by step for a visual understanding of how the transformations are applied!