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]
