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
(note: the image contains a coordinate grid with figure 1 (a small square) and figure 2 (a larger square), but the specific grid coordinates and figure details are as shown in the image.)

Explanation:

Step1: Identify coordinates of Figure 1

Let's take a vertex of Figure 1, say the bottom - left vertex. From the grid, its coordinates are \((2,2)\) (assuming the grid has x - axis and y - axis with units as grid squares). Another vertex, the bottom - right vertex of Figure 1, is \((4,2)\), the top - left is \((2,4)\) and the top - right is \((4,4)\).

Step2: Identify coordinates of Figure 2

For Figure 2, the bottom - left vertex is \((4,4)\), the bottom - right vertex is \((8,4)\), the top - left is \((4,8)\) and the top - right is \((8,8)\).

Step3: Analyze the transformation in x and y directions

- **X - direction**: For the x - coordinate of a point \((x,y)\) in Figure 1, the x - coordinate in Figure 2 is \(x + 2\) (e.g., \(2+2 = 4\), \(4 + 2=6\)? Wait, no, wait the bottom - left of Figure 1 is \((2,2)\), bottom - left of Figure 2 is \((4,4)\). Wait, maybe scaling and translation? Wait, the side length of Figure 1: from \(x = 2\) to \(x = 4\), length is \(4 - 2=2\). The side length of Figure 2: from \(x = 4\) to \(x = 8\), length is \(8 - 4 = 4\). So the scale factor in x and y is \(k=\frac{4}{2}=2\). Then, the translation: Let's take the center of Figure 1. Center of Figure 1: \(x=\frac{2 + 4}{2}=3\), \(y=\frac{2+4}{2}=3\). Center of Figure 2: \(x=\frac{4 + 8}{2}=6\), \(y=\frac{4 + 8}{2}=6\). So first, scale the figure by a factor of 2 about the origin (or about its center), then translate? Wait, another approach: Let's take a point \((x,y)\) in Figure 1. Let's see the transformation of \((2,2)\) to \((4,4)\), \((4,2)\) to \((8,4)\), \((2,4)\) to \((4,8)\), \((4,4)\) to \((8,8)\).

Looking at the x - coordinate: For \((2,2)\) to \((4,4)\): \(x\) goes from 2 to 4 (multiplied by 2), \(y\) goes from 2 to 4 (multiplied by 2). For \((4,2)\) to \((8,4)\): \(x\) goes from 4 to 8 (multiplied by 2), \(y\) goes from 2 to 4 (multiplied by 2). For \((2,4)\) to \((4,8)\): \(x\) goes from 2 to 4 (multiplied by 2), \(y\) goes from 4 to 8 (multiplied by 2). For \((4,4)\) to \((8,8)\): \(x\) goes from 4 to 8 (multiplied by 2), \(y\) goes from 4 to 8 (multiplied by 2). Wait, but also, is there a translation? Wait, no, if we scale the figure with side length 2 (from \(x = 2\) to \(x = 4\) and \(y = 2\) to \(y = 4\)) by a factor of 2, the new side length is 4. The center of the original figure (Figure 1) is at \((3,3)\). After scaling by 2, the center would be at \((6,6)\), which is the center of Figure 2. So the transformation is a dilation (scaling) with a scale factor of 2 centered at the origin? Wait, no, the center of dilation: Let's check the distance from the center of Figure 1 \((3,3)\) to the center of Figure 2 \((6,6)\). The vector is \((6 - 3,6 - 3)=(3,3)\)? No, wait, if we scale the figure with center at \((3,3)\) by a factor of 2, then a point \((x,y)\) in Figure 1 is transformed as follows: The vector from the center \((3,3)\) to \((x,y)\) is \((x - 3,y - 3)\). After scaling by 2, the vector becomes \((2(x - 3),2(y - 3))\), and the new point is \((3+2(x - 3),3 + 2(y - 3))=(2x-3,2y - 3)\). Let's test with \((2,2)\): \(2x-3=4 - 3 = 1\)? No, that's not matching. Wait, maybe the figure is first translated and then scaled? Wait, the bottom - left of Figure 1 is \((2,2)\), bottom - left of Figure 2 is \((4,4)\). If we scale Figure 1 by a factor of 2, the bottom - left would be \((4,4)\) (since \(2\times2 = 4\) for x and \(2\times2=4\) for y). Oh! Wait, maybe the transformation is a dilation (scale) with a scale factor of 2 and then no translation? Wait, the coordinates of Figure 1: Let's list all vertices of Figure 1: \((2,2)\), \((4,2)\), \((2,4)\), \((4,4)\). Vertices of Figure 2: \((4,4)\), \((8,4)\), \((4,8)\), \((8,8)\). Notice that if we multiply the x - coordinate of each vertex of Figure 1 by 2 and the y - coordinate by 2, we get the vertices of Figure 2: \(2\times2 = 4\), \(2\times2 = 4\) (for \((2,2)\) to \((4,4)\)); \(4\times2=8\), \(2\times2 = 4\) (for \((4,2)\) to \((8,4)\)); \(2\times2 = 4\), \(4\times2=8\) (for \((2,4)\) to \((4,8)\)); \(4\times2=8\), \(4\times2 = 8\) (for \((4,4)\) to \((8,8)\)). So the transformation rule is \((x,y)\to(2x,2y)\), which is a dilation with a scale factor of 2 centered at the origin.

Wait, but also, is there a translation? Wait, the original figure is at \((2,2)\) to \((4,4)\), the new figure is at \((4,4)\) to \((8,8)\). So the x - coordinates are doubled and y - coordinates are doubled. So the transformation rule is a dilation (scaling) with a scale factor of 2, so the rule is \((x,y)\to(2x,2y)\) (or we can also think of it as a combination of translation and scaling, but in this case, scaling by 2 about the origin gives the correct coordinates).

Answer:

The transformation rule is a dilation with a scale factor of 2, so the rule is \((x,y)\to(2x,2y)\) (or in words: "Multiply each coordinate of the vertices of Figure 1 by 2 to get the vertices of Figure 2", which is a dilation with scale factor 2).