Answer :
To determine which geometric transformation the function rule [tex]\( T_{-4,6}(x, y) \)[/tex] describes, we need to break down the transformation values provided:
1. The function rule [tex]\( T_{-4,6}(x, y) \)[/tex]:
- The [tex]\(-4\)[/tex] represents a transformation in the x-direction.
- The [tex]\(6\)[/tex] represents a transformation in the y-direction.
2. Translation in the x-direction:
- A negative value [tex]\(-4\)[/tex] in the x-direction means the figure is being translated 4 units to the left.
3. Translation in the y-direction:
- A positive value [tex]\(6\)[/tex] in the y-direction means the figure is being translated 6 units up.
Combining these two translations, the function rule [tex]\( T_{-4,6}(x, y) \)[/tex] translates any figure on the coordinate plane 4 units to the left and 6 units up.
Next, let's match this translation to the given options:
1. A parallelogram on a coordinate plane that is translated 4 units down and 6 units to the right.
- This option describes a translation 4 units down and 6 units right. This does not match our description.
2. A trapezoid on a coordinate plane that is translated 4 units to the left and 6 units up.
- This option exactly matches our description of translating 4 units to the left and 6 units up.
3. A rhombus on a coordinate plane that is translated 4 units down and 6 units to the left.
- This option describes a translation down and to the left, which does not match our description.
4. A rectangle on a coordinate plane that is translated 4 units to the right and 6 units up.
- This option describes a translation to the right and up, which does not match our description.
Therefore, the correct interpretation of the function rule [tex]\( T_{-4,6}(x, y) \)[/tex] is:
[tex]\[ \boxed{\text{a trapezoid on a coordinate plane that is translated 4 units to the left and 6 units up}} \][/tex]
1. The function rule [tex]\( T_{-4,6}(x, y) \)[/tex]:
- The [tex]\(-4\)[/tex] represents a transformation in the x-direction.
- The [tex]\(6\)[/tex] represents a transformation in the y-direction.
2. Translation in the x-direction:
- A negative value [tex]\(-4\)[/tex] in the x-direction means the figure is being translated 4 units to the left.
3. Translation in the y-direction:
- A positive value [tex]\(6\)[/tex] in the y-direction means the figure is being translated 6 units up.
Combining these two translations, the function rule [tex]\( T_{-4,6}(x, y) \)[/tex] translates any figure on the coordinate plane 4 units to the left and 6 units up.
Next, let's match this translation to the given options:
1. A parallelogram on a coordinate plane that is translated 4 units down and 6 units to the right.
- This option describes a translation 4 units down and 6 units right. This does not match our description.
2. A trapezoid on a coordinate plane that is translated 4 units to the left and 6 units up.
- This option exactly matches our description of translating 4 units to the left and 6 units up.
3. A rhombus on a coordinate plane that is translated 4 units down and 6 units to the left.
- This option describes a translation down and to the left, which does not match our description.
4. A rectangle on a coordinate plane that is translated 4 units to the right and 6 units up.
- This option describes a translation to the right and up, which does not match our description.
Therefore, the correct interpretation of the function rule [tex]\( T_{-4,6}(x, y) \)[/tex] is:
[tex]\[ \boxed{\text{a trapezoid on a coordinate plane that is translated 4 units to the left and 6 units up}} \][/tex]
