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 lines or the coordinates for the corresponding vertices. Also, please include a sketch of the pre-image and the image.
It would help if you proofread your questions before you posted them.
"6 foot"?
Also we cannot post a sketch here.
Real-world Scenario: Park Renovation
Imagine a scenario where a city park is undergoing renovation. One of the features being added is a rectangular fountain. To ensure proper placement and design, the park architect needs to map the pre-image (the original blueprint) onto the coordinate plane and make transformations to create the final image (the renovated park).
Pre-image:
The original rectangular fountain is 20 feet wide and 30 feet long. We can place the bottom left corner of the fountain at the origin (0,0) on the coordinate plane. Therefore, the coordinates for the pre-image's vertices are:
A(0, 0)
B(0, 30)
C(20, 30)
D(20, 0)
Image:
To create the final image of the fountain, the park architect applies the following transformations:
1. Translation: The architect decides to move the fountain 10 feet to the right and 5 feet up. This means that all the coordinates of the pre-image will shift accordingly. The coordinates for the image's vertices become:
A'(10, 5)
B'(10, 35)
C'(30, 35)
D'(30, 5)
2. Scaling: The architect wants to increase the width of the fountain by 40% and the length by 20%. To achieve this, each x-coordinate (width) will be multiplied by 1.4, and each y-coordinate (length) will be multiplied by 1.2. The coordinates for the scaled image's vertices become:
A''(14, 6)
B''(14, 42)
C''(42, 42)
D''(42, 6)
Sketch:
Before transformation (pre-image):
A(0, 0)————————— D(20, 0)
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B(0, 30)———————C(20, 30)
After translation (image):
A'(10, 5)—————————D'(30, 5)
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B'(10, 35)————————C'(30, 35)
After scaling (final image):
A''(14, 6)—————————D''(42, 6)
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B''(14, 42)————————C''(42, 42)
Calculations:
1. Translation:
- Adding 10 to the x-coordinate of each vertex of the pre-image gives the new x-coordinate for the image.
- Adding 5 to the y-coordinate of each vertex of the pre-image gives the new y-coordinate for the image.
2. Scaling:
- Multiplying each x-coordinate of the image by 1.4 gives the new x-coordinate for the scaled image.
- Multiplying each y-coordinate of the image by 1.2 gives the new y-coordinate for the scaled image.
By applying these transformations and calculations, the park architect successfully maps the pre-image of the original rectangular fountain onto the coordinate plane and creates the final image for the renovated park.