Question

question 1-36
figure 1 is the preimage of figure 2.
complete the statement about the transformation of figure 1 onto figure 2.
the transformation rule that describes the transformation of figure 1 onto figure 2 is
$(x,y)\to(2x,2y)$
$(x,y)\to(0.5x,0.5y)$
$(x,y)\to(x - 2,y - 2)$
$(x,y)\to(x + 2,y + 2)$
.

Explanation:

Step1: Analyze Figure 1 and Figure 2

First, observe the coordinates of Figure 1 and Figure 2. Let's take a vertex of Figure 1, say (3, 3) (assuming the bottom - left vertex). A vertex of Figure 2, say (5, 5) (corresponding vertex). Wait, actually, let's check the size. Figure 1 seems to have side length 2 (from x = 2 to x = 4, y = 2 to y = 4), Figure 2 has side length 4 (from x = 4 to x = 8, y = 4 to y = 8). So the scale factor is 2. Also, let's check the translation? Wait, no, the transformation is a dilation? Wait, no, let's check the coordinates. Wait, maybe first check the dilation. If we take a point from Figure 1, say (3, 3). If we apply \((x,y)\to(2x,2y)\), we get (6, 6). But Figure 2's bottom - left vertex is (4, 4)? Wait, maybe I made a mistake. Wait, Figure 1: let's look at the grid. Figure 1 is from x = 2 to x = 4 (length 2) and y = 2 to y = 4 (length 2). Figure 2 is from x = 4 to x = 8 (length 4) and y = 4 to y = 8 (length 4). Wait, also, the position: Figure 1 is at (2,2) - (4,4), Figure 2 is at (4,4) - (8,8)? No, wait the Figure 2 is at (4,4) - (8,8)? Wait, no, the Figure 2 in the graph: the bottom - left corner is at (4,4)? Wait, no, looking at the graph, Figure 1 is a small square, Figure 2 is a larger square. Wait, maybe the transformation is a dilation with scale factor 2 and also a translation? Wait, no, let's check the options. The options are:

1. \((x,y)\to(2x,2y)\)
2. \((x,y)\to(0.5x,0.5y)\)
3. \((x,y)\to(x - 2,y - 2)\)
4. \((x,y)\to(x + 2,y + 2)\)

Wait, let's take a vertex of Figure 1. Let's say the bottom - left vertex of Figure 1 is (2, 2) (assuming the grid: x - axis and y - axis, each grid is 1 unit). Then, if we apply \((x,y)\to(2x,2y)\), we get (4, 4). But Figure 2's bottom - left vertex is (4, 4)? Wait, no, Figure 2 in the graph: the bottom - left corner is at (4, 4)? Wait, the Figure 2 is a larger square. Wait, Figure 1 has side length 2 (from x = 2 to x = 4, y = 2 to y = 4), Figure 2 has side length 4 (from x = 4 to x = 8, y = 4 to y = 8). Wait, no, maybe the coordinates are: Figure 1: let's take the bottom - left point as (2, 2) and top - right as (4, 4). Figure 2: bottom - left as (4, 4) and top - right as (8, 8)? No, that would be a translation and dilation. But the options: the first option is \((x,y)\to(2x,2y)\). Let's check: if (2, 2) is transformed by \((x,y)\to(2x,2y)\), we get (4, 4). If (4, 4) (top - right of Figure 1) is transformed, we get (8, 8). But Figure 2's top - right is (8, 8)? Wait, the Figure 2 in the graph: the x - axis goes up to 14, y - axis up to 14. Figure 2 is from x = 4 to x = 8, y = 4 to y = 8? Wait, no, the Figure 2 is a square with bottom - left at (4, 4) and top - right at (8, 8), and Figure 1 is a square with bottom - left at (2, 2) and top - right at (4, 4). So the transformation from Figure 1 to Figure 2: each coordinate (x,y) of Figure 1 is multiplied by 2. For example, (2, 2) becomes (4, 4), (4, 4) becomes (8, 8), (2, 4) becomes (4, 8), (4, 2) becomes (8, 4). Wait, that matches the Figure 2. So the transformation rule is \((x,y)\to(2x,2y)\).

Step2: Eliminate other options

- Option \((x,y)\to(0.5x,0.5y)\): This would shrink the figure, but Figure 2 is larger than Figure 1, so eliminate.
- Option \((x,y)\to(x - 2,y - 2)\): This would move the figure left and down, but Figure 2 is to the right and up and larger, so eliminate.
- Option \((x,y)\to(x + 2,y + 2)\): This is a translation, but Figure 2 is also larger, so this is just a translation, not a size change. But Figure 1 has side length 2, Figure 2 has side length 4, so it's a dilation with scale factor 2, so \((x,y)\to(2x,2y)\) is correct.

Answer:

\((x, y) \to (2x, 2y)\)