Let's analyze the translation function rule [tex]\( T_{-4,6}(x, y) \)[/tex]. This rule describes how to translate any point [tex]\((x, y)\)[/tex] on a coordinate plane.
When we apply [tex]\( T_{-4,6}(x, y) \)[/tex], we are transforming the point [tex]\((x, y)\)[/tex] according to:
[tex]\[
x \rightarrow x - 4
\][/tex]
[tex]\[
y \rightarrow y + 6
\][/tex]
### Here is a step-by-step breakdown of this transformation:
1. Horizontal Translation ([tex]\( x \)[/tex]-coordinate):
- The [tex]\( x \)[/tex]-coordinate is decreased by 4 units. This translates the point 4 units to the left.
2. Vertical Translation ([tex]\( y \)[/tex]-coordinate):
- The [tex]\( y \)[/tex]-coordinate is increased by 6 units. This translates the point 6 units up.
Given these translations, we need to identify which of the provided options matches this transformation.
### Considering each option:
1. A parallelogram on a coordinate plane that is translated 4 units down and 6 units to the right:
- This would require the transformation rule [tex]\( T_{4, -6}(x, y) \)[/tex], which means [tex]\( x \rightarrow x + 4 \)[/tex] and [tex]\( y \rightarrow y - 6 \)[/tex]. Thus, this option does not fit [tex]\( T_{-4,6}(x, y) \)[/tex].
2. A trapezoid on a coordinate plane that is translated 4 units to the left and 6 units up:
- This perfectly matches our translation rule [tex]\( T_{-4,6}(x, y) \)[/tex], where [tex]\( x \rightarrow x - 4 \)[/tex] and [tex]\( y \rightarrow y + 6 \)[/tex].
3. A rhombus on a coordinate plane that is translated 4 units down and 6 units to the left:
- This requires the transformation rule [tex]\( T_{-4, -6}(x, y) \)[/tex], which means [tex]\( x \rightarrow x - 4 \)[/tex] and [tex]\( y \rightarrow y - 6 \)[/tex]. Thus, this does not fit [tex]\( T_{-4,6}(x, y) \)[/tex].
4. A rectangle on a coordinate plane that is translated 4 units to the right and 6 units up:
- This requires the transformation rule [tex]\( T_{4, 6}(x, y) \)[/tex], which means [tex]\( x \rightarrow x + 4 \)[/tex] and [tex]\( y \rightarrow y + 6 \)[/tex]. Thus, this does not match [tex]\( T_{-4,6}(x, y) \)[/tex].
Given the analysis, the correct answer is:
[tex]\[ \text{a trapezoid on a coordinate plane that is translated 4 units to the left and 6 units up.} \][/tex]
Hence, the proper translation described by [tex]\( T_{-4,6}(x, y) \)[/tex] is:
[tex]\[ \boxed{2} \][/tex]
