Question

the figure below has a point marked with a large dot. first, translate the figure 6 units to the right and 4 units up. then, give the coordinates of the marked point in the original figure and the final figure. point in original figure: ( , ) point in final figure: ( , )

Explanation:

Step1: Find original coordinates

Looking at the grid, the marked point in the original figure is at \( x = -3 \), \( y = -6 \)? Wait, no, wait. Wait, the grid: let's check the axes. The x-axis: left is negative, right positive. Y-axis: up positive, down negative. The marked point (the dot on the parallelogram) – let's see, the parallelogram is at, let's count the grid. Let's see, the x-coordinate: from the origin (0,0), moving left 3 units? Wait, no, maybe I misread. Wait, the original point: let's look at the grid. Let's see, the point is at \( x = -3 \), \( y = -6 \)? No, wait, the y-axis: the bottom part, the point is at y = -6? Wait, no, maybe the original point is at (-3, -6)? Wait, no, let's check again. Wait, the grid lines: each square is 1 unit. Let's find the original point. Let's see, the marked dot: let's count the x-coordinate (horizontal) and y-coordinate (vertical). So, starting from the origin (0,0), moving left 3 units (so x = -3) and down 6 units? No, wait, the y-axis: the numbers. Wait, the y-axis has 0 at the center, then up is positive (1,2,3,...) and down is negative (-1,-2,-3,...). Wait, the parallelogram is at the bottom left. Let's see, the marked point: x-coordinate: from 0, move left 3 units (so x = -3), y-coordinate: move down 6 units? No, wait, the y-axis labels: let's see, the grid has y-axis with 0, then above 0: 1,2,3,4,5,6,7,8? Wait, no, the image shows y-axis with numbers: 0, then above 0: 1,2,3,4,5,6,7,8? Wait, no, the original figure's point: let's look at the grid. Wait, maybe the original point is at (-3, -6)? Wait, no, maybe I made a mistake. Wait, the problem says "translate the figure 6 units to the right and 4 units up". So first, find the original coordinates. Let's re-examine the grid. Let's see, the marked point (the dot) is on the parallelogram. Let's count the x and y. So, x-coordinate: from the origin (0,0), moving left 3 units (so x = -3), y-coordinate: moving down 6 units? No, wait, the y-axis: the bottom part, the point is at y = -6? Wait, no, maybe the original point is at (-3, -6)? Wait, no, maybe the original point is at (-3, -6)? Wait, no, let's check again. Wait, maybe the original point is at (-3, -6)? Wait, no, perhaps I messed up. Wait, let's do it properly. Let's find the original coordinates:

Original point: Let's count the horizontal (x) and vertical (y) distances from the origin. So, x: left 3 units (so x = -3), y: down 6 units (so y = -6)? Wait, no, the y-axis: the numbers. Wait, the y-axis has 0, then below 0: -1, -2, -3, -4, -5, -6? Wait, the point is at y = -6? Wait, no, the parallelogram is at the bottom, so the y-coordinate is -6? Wait, maybe. So original coordinates: (-3, -6). Then, translate 6 units right (so add 6 to x) and 4 units up (add 4 to y). So new x: -3 + 6 = 3, new y: -6 + 4 = -2. Wait, but that seems off. Wait, maybe the original point is at (-3, -6)? Wait, no, maybe I misread the y-coordinate. Wait, maybe the original point is at (-3, -6)? Wait, let's check again. Wait, the grid: the x-axis: each square is 1 unit. So from 0, moving left 3: x = -3. Y-axis: moving down 6: y = -6. Then translate 6 right: x becomes -3 + 6 = 3. Translate 4 up: y becomes -6 + 4 = -2. So original point: (-3, -6), final point: (3, -2). Wait, but maybe I made a mistake in the original coordinates. Wait, maybe the original point is at (-3, -6)? Let's confirm. Alternatively, maybe the original point is at (-3, -6). Let's proceed.

Step1: Determine original coordinates

Looking at the grid, the marked point (dot) in the original figure is at \( x = -3 \), \( y = -6 \). So original coordinates: \( (-3, -6) \).

Step2: Apply translation

Translation: 6 units right (add 6 to x-coordinate) and 4 units up (add 4 to y-coordinate).

New x-coordinate: \( -3 + 6 = 3 \)

New y-coordinate: \( -6 + 4 = -2 \)

So final coordinates: \( (3, -2) \)

Wait, but maybe I misread the original y-coordinate. Wait, maybe the original point is at (-3, -6)? Let's check the grid again. The parallelogram is at the bottom left, so the y-coordinate is negative, and x is negative. So yes, (-3, -6) for original, then after translation: (3, -2).

Answer:

Point in original figure: \((-3, -6)\)

Point in final figure: \((3, -2)\)