To determine the description of the translation given by the rule [tex]\((x, y) \rightarrow (x-2, y+7)\)[/tex], let's closely examine what happens to the coordinates of any point [tex]\((x, y)\)[/tex] under this transformation.
1. The new x-coordinate is obtained by subtracting 2 from the original x-coordinate:
[tex]\[
x' = x - 2
\][/tex]
This means we are moving the point 2 units to the left because subtracting from the x-coordinate shifts it to the left.
2. The new y-coordinate is obtained by adding 7 to the original y-coordinate:
[tex]\[
y' = y + 7
\][/tex]
This means we are moving the point 7 units up because adding to the y-coordinate shifts it upward.
Putting these together, the translation [tex]\((x, y) \rightarrow (x-2, y+7)\)[/tex] describes the following transformations:
- 2 units to the left (since [tex]\(x' = x - 2\)[/tex])
- 7 units up (since [tex]\(y' = y + 7\)[/tex])
Therefore, the correct description of this translation is:
- a translation of 2 units to the left and 7 units up
So, the correct answer is:
- a translation of 2 units to the left and 7 units up
