Question

consider the figure and its image. describe the transformation. use decimals, if necessary. (x, y) → ( , )

Explanation:

Step1: Identify a point and its image

Let's take point \( R \). From the graph, let's assume \( R \) has coordinates \( (-2, 3) \) and its image \( R' \) has coordinates \( (-3, 1) \)? Wait, no, maybe better to take \( S \) and \( S' \). Wait, maybe I misread. Wait, let's check the grid. Let's find coordinates of a vertex and its image. Let's take \( R \): Let's say \( R \) is at \( (-2, 3) \), \( R' \) is at \( (-3, 1) \)? No, maybe better to look at the horizontal and vertical shifts. Wait, the gray figure is the image. Let's take point \( U \) and \( U' \). Wait, \( U \) is at \( (-2, -2) \), \( U' \) is at \( (3, 1) \)? No, maybe I should look at the x and y changes. Wait, let's take point \( R \): Let's assume \( R \) is at \( (-2, 3) \), \( R' \) is at \( (-3, 1) \)? No, maybe the transformation is a translation. Let's check the x - coordinate change and y - coordinate change. Let's take point \( S \): Suppose \( S \) is at \( (1, 4) \), \( S' \) is at \( (-4, 2) \)? No, maybe I made a mistake. Wait, let's look at the horizontal (x) and vertical (y) shifts. Let's take point \( R \): Let's say \( R \) is at \( (-2, 3) \), \( R' \) is at \( (-3, 1) \). The change in x: \( -3 - (-2)=-1 \)? No, that doesn't seem right. Wait, maybe the correct approach is to find the vector from a point to its image. Let's take point \( U \): Let's assume \( U \) is at \( (-2, -2) \), \( U' \) is at \( (3, 1) \). No, that's not. Wait, maybe the original figure (black) and the image (gray). Let's take point \( R \): \( R \) is at \( (-2, 3) \), \( R' \) is at \( (-3, 1) \). Wait, the x - coordinate change: \( -3 - (-2)=-1 \)? No, maybe I should look at the horizontal shift first. Wait, looking at the horizontal line (y - coordinate) of \( R' \) and \( U' \), they are on the same horizontal line. Similarly, \( S' \) and \( T' \) are on the same horizontal line. Let's find the x - coordinate difference between \( R \) and \( R' \). Let's say \( R \) is at \( x=-2 \), \( R' \) is at \( x = - 3\)? No, maybe \( R \) is at \( (-2, 3) \), \( R' \) is at \( (-3, 1) \). Wait, the y - coordinate change: \( 1 - 3=-2 \), x - coordinate change: \( -3 - (-2)=-1 \)? No, that can't be. Wait, maybe the correct points: Let's take \( S \) at \( (1, 4) \), \( S' \) at \( (-4, 2) \). No, this is confusing. Wait, maybe the transformation is a translation. Let's look at the horizontal shift (Δx) and vertical shift (Δy). Let's take point \( R \): Let's assume \( R \) has coordinates \( (-2, 3) \), \( R' \) has coordinates \( (-3, 1) \). Then Δx = - 3 - (-2)=-1, Δy = 1 - 3=-2. But let's check another point. Take \( U \): \( U \) is at \( (-2, -2) \), \( U' \) is at \( (3, 1) \)? No, that's not. Wait, maybe I misidentified the points. Wait, the black figure: vertices \( R, S, T, U \). The gray figure: \( R', S', T', U' \). Let's find the coordinates properly. Let's set the origin \( O(0,0) \). Let's find \( R \): From the grid, \( R \) is at \( (-2, 3) \) (2 units left of origin on x, 3 up on y). \( R' \) is at \( (-3, 1) \) (3 left, 1 up). So the change in x: \( -3 - (-2)=-1 \), change in y: \( 1 - 3=-2 \). Wait, but let's check \( S \): \( S \) is at \( (1, 4) \), \( S' \) is at \( (0, 2) \). Then Δx = 0 - 1=-1, Δy = 2 - 4=-2. Ah, that's consistent. So the translation is (x, y) → (x - 1, y - 2). Wait, but let's check \( T \): \( T \) is at \( (1, -2) \), \( T' \) is at \( (0, -4) \). Δx = 0 - 1=-1, Δy = - 4 - (-2)=-2. Yes, that works. And \( U \): \( U \) is at \( (-2, -2) \), \( U' \) is at \( (-3, -4) \). Δx = - 3 - (-2)=-1, Δy = - 4 - (-2)=-2. Perfect. So the transformation is a translation where we subtract 1 from the x - coordinate and subtract 2 from the y - coordinate.

Step2: Formulate the transformation

The general form of a translation is \( (x,y)\to(x + a,y + b) \), where \( a \) is the horizontal shift and \( b \) is the vertical shift. From our calculations, \( a=-1 \) (since \( x\to x - 1=x+(-1) \)) and \( b = - 2 \) (since \( y\to y - 2=y+(-2) \)). So the transformation is \( (x,y)\to(x - 1,y - 2) \).

Answer:

\( (x - 1, y - 2) \)