Math Problem Statement

Graph the image of the figure below on a reflection over the x-axis.

Solution

To solve this reflection problem, we need to reflect each point of the figure over the x-axis.

Step-by-Step Solution:

  1. Identify the coordinates of points Z, W, X, and Y from the original image:

    • ZZ is at (8,2)(-8, 2)
    • WW is at (4,1)(-4, 1)
    • XX is at (1,3)(1, 3)
    • YY is at (3,4)(-3, 4)
  2. Apply the reflection rule over the x-axis:

    • Reflecting a point (x,y)(x, y) over the x-axis changes the yy-coordinate’s sign, resulting in (x,y)(x, -y).
  3. Determine the coordinates of the reflected points:

    • The reflection of Z(8,2)Z(-8, 2) will be Z(8,2)Z'(-8, -2).
    • The reflection of W(4,1)W(-4, 1) will be W(4,1)W'(-4, -1).
    • The reflection of X(1,3)X(1, 3) will be X(1,3)X'(1, -3).
    • The reflection of Y(3,4)Y(-3, 4) will be Y(3,4)Y'(-3, -4).
  4. Plot these new points on the graph:

    • Place ZZ', WW', XX', and YY' at their respective reflected coordinates.
    • Connect the points ZZ', WW', XX', and YY' in the same order as the original shape.

This will create the reflection of the figure over the x-axis.

Would you like a visual representation or have any specific questions about this solution?

Further Related Questions:

  1. How do you find the reflection of a point over the y-axis?
  2. What is the rule for reflecting a point over the origin?
  3. How would you reflect a shape over a line like y=xy = x?
  4. Can you apply the same method to find a reflection over other lines?
  5. How do reflections affect the orientation of shapes?

Tip: When reflecting over the x-axis, only the y-coordinates change, flipping to their opposite values.

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Math Problem Analysis

Mathematical Concepts

Reflection
Coordinate Geometry

Formulas

Reflection over x-axis: (x, y) -> (x, -y)

Theorems

Reflection Theorem

Suitable Grade Level

Grades 8-10