Question

quadrilateral abcd is the image of quadrilateral abcd under a translation.
determine the translation.
use non-negative numbers.
a translation by units to the right/left and units up/down

Explanation:

Step1: Find coordinates of B and B'

Point B: $(0, 4)$ (assuming grid, x=0, y=4). Point B': $(3, -2)$? Wait, no, looking at graph: B is at (0,4)? Wait, x-axis: B is at (0,4)? Wait, the grid: B is at (0,4)? Wait, B' is at (3, -2)? No, wait, original B (blue) is at (0,4)? Wait, no, let's check x and y. Let's take point B: blue, x=0, y=4. Point B' (red) is at (3, -2)? Wait, no, the red B' is at (3, -2)? Wait, no, the translation: let's take point D. D is at (-6,1). D' is at (-4, -4)? Wait, no, D' is red, at (-4, -4)? Wait, no, maybe better to take B: B is (0,4)? Wait, no, the x-axis: from left, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7. So B (blue) is at (0,4)? Wait, no, the blue B is at (0,4)? Wait, the red B' is at (3, -2)? Wait, no, the red B' is at (3, -2)? Wait, no, let's count the horizontal and vertical distance. From B (blue) to B' (red): horizontal: from x=0 to x=3? No, wait, blue B is at (0,4)? Wait, no, looking at the graph, blue B is at (0,4)? Wait, the x-coordinate of B: the grid lines, B is at (0,4) (x=0, y=4). B' is at (3, -2)? Wait, no, the red B' is at (3, -2)? Wait, no, vertical distance: from y=4 to y=-2: that's 6 units down. Horizontal distance: from x=0 to x=3? No, wait, maybe I got B wrong. Wait, blue B is at (0,4)? Wait, no, the blue quadrilateral: A is at (-5,3), B at (0,4), C at (-6,-2), D at (-6,1). Red quadrilateral: A' at (-3,-2), B' at (3,-2)? Wait, no, B' is at (3,-2)? Wait, A is (-5,3), A' is (-3,-2). So horizontal change: -3 - (-5) = 2? No, wait, A' is (-3,-2), A is (-5,3). So x: -3 - (-5) = 2 (right 2). y: -2 - 3 = -5 (down 5). Wait, but B: B is (0,4), B' is (3,-2). x: 3 - 0 = 3? No, that's conflicting. Wait, maybe I misread the coordinates. Let's re-express:

Blue points:
- A: (-5, 3)
- B: (0, 4)
- C: (-6, -2)
- D: (-6, 1)

Red points (A', B', C', D'):
- A': (-3, -2)
- B': (3, -2)
- C': (-4, -7)
- D': (-4, -4)

Wait, no, C' is at (-4, -7)? No, the red C' is at (-4, -7)? Wait, maybe better to take B: B (blue) is (0,4), B' (red) is (3, -2). So horizontal change: 3 - 0 = 3? No, 3 units right? Vertical change: -2 - 4 = -6, so 6 units down. Wait, but A: A (-5,3) to A' (-3,-2): x change: -3 - (-5) = 2, y change: -2 - 3 = -5. That's inconsistent. Wait, maybe I picked the wrong points. Let's take D: D (blue) is (-6,1), D' (red) is (-4,-4). x change: -4 - (-6) = 2 (right 2). y change: -4 - 1 = -5 (down 5). Ah, that's consistent. So D to D': right 2, down 5. Let's check C: C (blue) is (-6,-2), C' (red) is (-4,-7). x: -4 - (-6) = 2 (right 2). y: -7 - (-2) = -5 (down 5). Yes, that's consistent. So translation: right 2 units? No, wait, D is (-6,1), D' is (-4,-4). So x: -4 - (-6) = 2 (right 2). y: -4 - 1 = -5 (down 5). Wait, but the problem says "use non-negative numbers" and "right/left" and "up/down". So horizontal: right 2? No, wait, D is (-6,1), D' is (-4,-4). So x increases by 2 (right 2). y decreases by 5 (down 5). Wait, but let's check B: B (blue) is (0,4), B' (red) is (2,-1)? Wait, no, maybe I misread B'. Wait, the red B' is at (3,-2)? No, the red quadrilateral: B' is at (3,-2)? Wait, no, the x-coordinate of B' is 3? Wait, the grid: x=3, y=-2. So B (blue) is at (0,4), B' at (3,-2). x: 3 - 0 = 3 (right 3). y: -2 - 4 = -6 (down 6). But D: D (-6,1) to D' (-4,-4): x 2, y -5. Conflict. So maybe I made a mistake. Wait, let's look again. The blue quadrilateral: A is at (-5,3), B at (0,4), C at (-6,-2), D at (-6,1). Red quadrilateral: A' at (-3,-2), B' at (3,-2), C' at (-4,-7), D' at (-4,-4). Wait, A' (-3,-2), A (-5,3): x: -3 - (-5) = 2 (right 2), y: -2 - 3 = -5 (down 5). B' (3,-2), B (0,4): x: 3 - 0 = 3 (right 3), y: -2 - 4 = -6 (down 6). No, that's not consistent. Wait, maybe the blue B is at (0,4), red B' is at (3,-2). So horizontal: 3 units right, vertical: 6 units down. But D: D (-6,1) to D' (-4,-4): x 2, y -5. Not matching. Wait, maybe the graph is different. Wait, the user's graph: blue D is at (-6,1), red D' is at (-4,-4). So x: -4 - (-6) = 2 (right 2), y: -4 - 1 = -5 (down 5). So translation is right 2, down 5? But let's check A: A (-5,3) to A' (-3,-2): x 2, y -5. Yes, that's correct. So A' is (-3,-2), A is (-5,3). So x: -3 - (-5) = 2 (right 2), y: -2 - 3 = -5 (down 5). So translation: 2 units right and 5 units down. Wait, but the problem says "use non-negative numbers" and "right/left" and "up/down". So horizontal: 2 units right, vertical: 5 units down. Wait, but let's confirm with C: C (-6,-2) to C' (-4,-7): x: -4 - (-6) = 2 (right 2), y: -7 - (-2) = -5 (down 5). Yes, that's consistent. So the translation is 2 units right and 5 units down? Wait, but the initial thought with B was wrong because I misread B's coordinates. So correct: from A (-5,3) to A' (-3,-2): right 2, down 5. So the translation is 2 units to the right and 5 units down.

Step2: Confirm translation

Using point A: original (x=-5, y=3), image (x=-3, y=-2). Horizontal change: -3 - (-5) = 2 (right 2). Vertical change: -2 - 3 = -5 (down 5). So translation is 2 units right and 5 units down.

Answer:

2 units to the right and 5 units down (so first box: 2, second dropdown: right, third box: 5, fourth dropdown: down)