Math Problem Statement

Solve

Solve transformation problems involving reflection over the x-axis, reflection over the y-axis, and translation on a coordinate plane.

Solution

The image you uploaded contains several transformation problems involving coordinates and graphs. Let's go through each problem step by step:

Problem 6 (Transformation: Reflection over x-axis)

  • Original coordinates of points A, B, C, D for quadrilateral ABCD are:
    • A (1, 4)
    • B (4, 4)
    • C (4, 2)
    • D (1, 2)

After reflecting over the x-axis, we change the y-coordinates' sign while keeping the x-coordinates the same. The new coordinates of A'B'C'D' are:

  • A' (1, -4)
  • B' (4, -4)
  • C' (4, -2)
  • D' (1, -2)

Problem 7 (Transformation: Reflection over y-axis)

  • Original coordinates of points A, B, C, D are:
    • A (2, 1)
    • B (5, 1)
    • C (5, 6)
    • D (2, 6)

After reflecting over the y-axis, we change the x-coordinates' sign while keeping the y-coordinates the same. The new coordinates of A'B'C'D' are:

  • A' (-2, 1)
  • B' (-5, 1)
  • C' (-5, 6)
  • D' (-2, 6)

Problem 8 (Transformation: Translation by the rule (x, y) → (x + 7, y + 6))

  • Original coordinates of points A, B, C, D are:
    • A (-8, 5)
    • B (-4, 2)
    • C (2, -1)
    • D (1, -4)

After translating the shape according to the rule, add 7 to the x-coordinate and 6 to the y-coordinate:

  • A' (-8 + 7, 5 + 6) = (-1, 11)
  • B' (-4 + 7, 2 + 6) = (3, 8)
  • C' (2 + 7, -1 + 6) = (9, 5)
  • D' (1 + 7, -4 + 6) = (8, 2)

Would you like to go over the details of the transformations or have any specific questions about the graphing?

Additional Questions:

  1. How does reflection across the x-axis change coordinates?
  2. How can you verify your translation results on the graph?
  3. What are other types of transformations besides reflection and translation?
  4. How do the rules of translation affect the orientation of the shape?
  5. What happens if you reflect over both axes (x and y) sequentially?

Tip: For reflections, always remember that x-axis affects y-coordinates, and y-axis affects x-coordinates when flipping across these lines!

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Math Problem Analysis

Mathematical Concepts

Geometry
Transformations
Reflections
Translations
Coordinate Plane

Formulas

Reflection over the x-axis: (x, y) → (x, -y)
Reflection over the y-axis: (x, y) → (-x, y)
Translation: (x, y) → (x + a, y + b)

Theorems

Reflection Theorem: Reflecting a point over the x-axis changes the sign of its y-coordinate.
Translation Theorem: Translating a point adds constant values to both x and y coordinates.

Suitable Grade Level

Grades 8-10