Math Problem Statement

Reflect the figure over the line y = -1.

Solution

The task in the image is to reflect the given figure over the line y=1y = -1.

Steps to reflect a shape over the line y=1y = -1:

  1. Identify the coordinates of the vertices of the polygon in the image. Let's call these points (x1,y1),(x2,y2),(x3,y3),(x_1, y_1), (x_2, y_2), (x_3, y_3), \dots.
  2. Reflect each point across the line y=1y = -1. To reflect a point (x,y)(x, y) over this line, the new y-coordinate is determined by the formula: y=1(y(1))=2yy' = -1 - (y - (-1)) = -2 - y The x-coordinate remains unchanged.
  3. Plot the new points after reflection and connect them in the same order to form the reflected shape.

Example of the reflection formula:

  • If a point (x,y)(x, y) is above the line, for example, y=5y = 5, the reflected point would be: y=25=7y' = -2 - 5 = -7 So the reflected point would be (x,7)(x, -7).

Would you like me to walk through the reflection of each point in the figure, or would you like further clarification?

Here are some related questions for better understanding:

  1. What are the coordinates of the vertices of the original figure?
  2. How do you calculate reflections over other lines, like y=2y = 2?
  3. What is the general reflection rule for any horizontal line y=cy = c?
  4. How does the process differ for reflecting over a vertical line, like x=3x = 3?
  5. How would this reflection change if the line of reflection was diagonal, like y=xy = x?

Tip: When reflecting points over a horizontal line, only the y-coordinates change, but the x-coordinates stay the same!

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Math Problem Analysis

Mathematical Concepts

Geometry
Reflections
Coordinate Plane

Formulas

y' = -2 - y (for reflecting over y = -1)

Theorems

Reflection Theorem over a Horizontal Line

Suitable Grade Level

Grades 8-10