Question

write the coordinates of the vertices after a translation 7 units left and 7 units down.

Explanation:

Step1: Find original coordinates

From the graph:

• \( S(3, -3) \)
• \( T(3, 7) \) (Wait, no, looking at the grid, T is at (3,7)? Wait, no, the y-axis: T is at y=7? Wait, no, the grid lines: let's check again. The point T: x=3, y=7? Wait, no, the vertical line: S is at (3, -3), T is at (3, 7)? Wait, no, the y-coordinate for T: looking at the grid, the y-axis has 8 at the top, so T is at (3, 7)? Wait, no, the point T is at (3, 7)? Wait, no, the graph: S is at (3, -3), T is at (3, 7)? Wait, no, the vertical line from S (3, -3) up to T: the distance is 10? Wait, no, maybe I misread. Wait, the grid: each square is 1 unit. So S is at (3, -3), T is at (3, 7)? Wait, no, the y-coordinate for T: let's see, the y-axis has 10 at the top, 8, 6, 4, 2, 0, -2, -4, etc. So T is at (3, 7)? Wait, no, the point T is at (3, 7)? Wait, no, the graph shows T at (3, 7)? Wait, no, the coordinates: S is (3, -3), T is (3, 7), U is (10, 7), V is (10, -3). Wait, that makes a rectangle. So original coordinates:
• \( S(3, -3) \)
• \( T(3, 7) \)
• \( U(10, 7) \)
• \( V(10, -3) \)

Step2: Apply translation (7 left, 7 down)

Translation rule: \((x, y) \to (x - 7, y - 7)\)

For \( S(3, -3) \):
\( x' = 3 - 7 = -4 \)
\( y' = -3 - 7 = -10 \)
So \( S'(-4, -10) \)

For \( T(3, 7) \):
\( x' = 3 - 7 = -4 \)
\( y' = 7 - 7 = 0 \)
So \( T'(-4, 0) \)

For \( U(10, 7) \):
\( x' = 10 - 7 = 3 \)
\( y' = 7 - 7 = 0 \)
So \( U'(3, 0) \)

For \( V(10, -3) \):
\( x' = 10 - 7 = 3 \)
\( y' = -3 - 7 = -10 \)
So \( V'(3, -10) \)

Wait, wait, maybe I misread the original coordinates. Let's check again. The graph: S is at (3, -3)? Wait, no, the x-axis: the grid lines, S is at x=3, y=-3? Wait, the y-axis: below the x-axis is negative. So S is (3, -3), T is (3, 7)? Wait, no, the vertical line from S (3, -3) up to T: the y-coordinate of T is 7? Wait, the grid has y=8 at the top, so T is at (3, 7)? Wait, no, the point T is at (3, 7)? Wait, the graph shows T at (3, 7), U at (10, 7), V at (10, -3), S at (3, -3). Yes, that's a rectangle. So translation 7 left (subtract 7 from x) and 7 down (subtract 7 from y).

So recalculating:

S: (3, -3) → (3-7, -3-7) = (-4, -10)

T: (3, 7) → (3-7, 7-7) = (-4, 0)

U: (10, 7) → (10-7, 7-7) = (3, 0)

V: (10, -3) → (10-7, -3-7) = (3, -10)

Answer:

• \( S'(-4, -10) \)
• \( T'(-4, 0) \)
• \( U'(3, 0) \)
• \( V'(3, -10) \)