To determine which translation is described by the function rule [tex]\( T_{-4,6}(x, y) \)[/tex], let's break down what the notation means:
1. Translation Definition: A translation moves every point of a figure or a space by the same distance in a given direction. It is defined as [tex]\( T_{a,b}(x, y) = (x + a, y + b) \)[/tex]. This translation shifts a point [tex]\((x, y)\)[/tex] by [tex]\( a \)[/tex] units horizontally and [tex]\( b \)[/tex] units vertically.
2. Understanding [tex]\( T_{-4,6}(x, y) \)[/tex]:
- The translation [tex]\( T_{-4, 6}(x, y) \)[/tex] specifically means:
- [tex]\( x \)[/tex]-coordinate is changed by [tex]\(-4\)[/tex]
- [tex]\( y \)[/tex]-coordinate is changed by [tex]\(+6\)[/tex]
- Therefore, every point moves 4 units to the left (negative direction in the [tex]\( x \)[/tex]-axis) and 6 units up (positive direction in the [tex]\( y \)[/tex]-axis).
3. Analyzing Each Option:
- Option 1: A parallelogram on a coordinate plane that is translated 4 units down and 6 units to the right.
- This would be symbolized by [tex]\( T_{6,-4}(x, y) \)[/tex], where [tex]\( x \)[/tex]-coordinates increase by 6 and [tex]\( y \)[/tex]-coordinates decrease by 4. This does not match our translation.
- Option 2: A trapezoid on a coordinate plane that is translated 4 units to the left and 6 units up.
- This matches [tex]\( T_{-4,6}(x, y) \)[/tex], where [tex]\( x \)[/tex]-coordinates decrease by 4 and [tex]\( y \)[/tex]-coordinates increase by 6.
- Option 3: A rhombus on a coordinate plane that is translated 4 units down and 6 units to the left.
- This would be symbolized by [tex]\( T_{-6,-4}(x, y) \)[/tex], where [tex]\( x \)[/tex]-coordinates decrease by 6 and [tex]\( y \)[/tex]-coordinates decrease by 4. This does not match our translation.
- Option 4: A rectangle on a coordinate plane that is translated 4 units to the right and 6 units up.
- This would be symbolized by [tex]\( T_{4, 6}(x, y) \)[/tex], where [tex]\( x \)[/tex]-coordinates increase by 4 and [tex]\( y \)[/tex]-coordinates increase by 6. This does not match our translation.
Therefore, the function rule [tex]\( T_{-4,6}(x, y) \)[/tex] correctly describes:
A trapezoid on a coordinate plane that is translated 4 units to the left and 6 units up.
Hence, the correct option is:
[tex]\[
\boxed{\text{2}}
\][/tex]
