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A polygon is translated 3 units to the right and 4 units down, then it is reflected across the x-axis, and finally, it is dilated by a factor of 4 with the origin as the center of dilation. Create a function modeling the transformation.

[tex]\[ f(x, y) = (\quad \quad) \][/tex]



Answer :

GinnyAnswer
To model the transformation of a point [tex]\((x, y)\)[/tex] through a series of geometric operations—translation, reflection, and dilation—we can break down each step systematically.

1. Translation:
- Translate the point [tex]\((x, y)\)[/tex] 3 units to the right and 4 units down.
- The new coordinates [tex]\((x', y')\)[/tex] after translation are:
[tex]\[ x' = x + 3 \][/tex]
[tex]\[ y' = y - 4 \][/tex]

2. Reflection:
- Reflect the translated point across the x-axis.
- The coordinates [tex]\((x'', y'')\)[/tex] after reflection are:
[tex]\[ x'' = x' \][/tex]
[tex]\[ y'' = -y' \][/tex]

3. Dilation:
- Dilate the reflected point by a factor of 4 with the origin as the center of dilation.
- The coordinates [tex]\((x''', y''')\)[/tex] after dilation are:
[tex]\[ x''' = 4x'' \][/tex]
[tex]\[ y''' = 4y'' \][/tex]

Now, let's combine all these steps into one function [tex]\( f(x, y) \)[/tex]:

1. Translation:
[tex]\[ x' = x + 3 \][/tex]
[tex]\[ y' = y - 4 \][/tex]

2. Reflection:
[tex]\[ x'' = x' \][/tex]
[tex]\[ y'' = -y' \][/tex]

3. Dilation:
[tex]\[ x''' = 4x'' \][/tex]
[tex]\[ y''' = 4y'' \][/tex]

Substituting the intermediate steps into the final expressions:

1. Reflecting the translated coordinates:
[tex]\[ x'' = x' = x + 3 \][/tex]
[tex]\[ y'' = -y' = -(y - 4) = -y + 4 \][/tex]

2. Applying dilation:
[tex]\[ x''' = 4(x + 3) = 4x + 12 \][/tex]
[tex]\[ y''' = 4(-y + 4) = -4y + 16 \][/tex]

Thus, the function modeling the transformation [tex]\( f(x, y) \)[/tex] is:

[tex]\[ f(x, y) = (4x + 12, -4y + 16) \][/tex]

So, the final answer is:
[tex]\[ f(x, y) = (4x + 12, -4y + 16) \][/tex]

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