Design your real-world scenario involving a geometric figure and its transformations. Model the pre-image and image on a coordinate plane. In your final answer, include the real-world scenario, written in complete sentences, the transformations that map the pre-image onto its image, and all calculations involved for the equations of the corresponding linesthe corresponding lines or the coordinates for the corresponding vertices. Also, please include a sketch of the pre-image and the image.
Real-World Scenario: Transforming a House Floor Plan
Imagine you are an architect designing a floor plan for a new house. You start with a simple rectangular pre-image that represents the initial layout of the house. Throughout the design process, you make several transformations to refine the plan and create the final image.
Pre-Image:
The pre-image is a rectangle on a coordinate plane, representing the original house floor plan.
To illustrate, assume the pre-image rectangle has its lower left vertex at (-3, -2) and its dimensions are 8 units in length and 6 units in width. The remaining vertices can be calculated as follows:
Vertex 1: (-3, -2)
Vertex 2: (-3 + 8, -2) = (5, -2)
Vertex 3: (-3 + 8, -2 + 6) = (5, 4)
Vertex 4: (-3, -2 + 6) = (-3, 4)
Transformation 1: Translation
To make the house larger, you decide to translate the pre-image 2 units to the right and 3 units upward.
Translation equation for each vertex: (x, y) → (x + 2, y + 3)
Vertex 1: (-3 + 2, -2 + 3) = (-1, 1)
Vertex 2: (5 + 2, -2 + 3) = (7, 1)
Vertex 3: (5 + 2, 4 + 3) = (7, 7)
Vertex 4: (-3 + 2, 4 + 3) = (-1, 7)
Image after Translation:
The new rectangle, obtained by translating the pre-image, is now the image.
Now, to enhance the design, you decide to rotate the house 90 degrees counterclockwise around the origin.
Transformation 2: Rotation
To rotate the image counterclockwise, each coordinate (x, y) changes to (-y, x).
Vertex 1: (1, -1)
Vertex 2: (7, -1)
Vertex 3: (7, 7)
Vertex 4: (1, 7)
Finally, to add symmetry to the design, you reflect the image across the y-axis.
Transformation 3: Reflection
To reflect a shape across the y-axis, each x-coordinate becomes its opposite (-x), while the y-coordinate remains the same.
Vertex 1: (-1, -1)
Vertex 2: (-7, -1)
Vertex 3: (-7, 7)
Vertex 4: (-1, 7)
Now, the resulting image represents the final modified house floor plan.