Question

congruence transformations
find the composition of transformations that map abcd to ehgf.
reflect over the x-axis, then translate (x+?, y+ ).

Explanation:

Step1: Analyze reflection over x - axis

Let's take a point from ABCD, say point A (assuming A is at (-5, 2)). Reflecting over the x - axis changes the y - coordinate's sign. So the reflected point \(A'\) will be \((-5,-2)\).

Step2: Determine translation for x - coordinate

We want to map \(A'\) to point E. Looking at the graph, point E seems to be at (-2,-2). The change in x - coordinate is \(-2-(-5)=3\). So the x - component of the translation is \(+ 3\) (i.e., \(x + 3\)).

Step3: Determine translation for y - coordinate

The y - coordinate of \(A'\) is -2 and the y - coordinate of E is -2. So the change in y - coordinate is \(-2-(-2) = 0\)? Wait, maybe we take another point. Let's take point B (say B is at (-3,4)). Reflecting over x - axis gives \(B'(-3,-4)\). Point H (corresponding to B in EHGF) seems to be at (0,-4). The change in x - coordinate: \(0 - (-3)=3\), change in y - coordinate: \(-4-(-4) = 0\)? Wait, maybe my initial point assumption is wrong. Wait, maybe the figure: Let's re - examine. After reflecting over x - axis, let's take a vertex of ABCD, say D (let's assume D is at (-1,2)). Reflecting over x - axis: \(D'(-1,-2)\). Now, the corresponding vertex in EHGF, say F, let's assume F is at (2,-2). Then the translation in x: \(2-(-1)=3\), translation in y: \(-2-(-2)=0\)? Wait, no, maybe the y - translation. Wait, maybe the lower figure: EHGF has vertices at lower y. Wait, maybe the reflection over x - axis, then translation. Let's take a point from ABCD, say A (-5,2). Reflect over x - axis: (-5,-2). Now, E is at (-2,-2)? Wait, no, maybe E is at (let's see the graph) E is at (-2,-2)? Wait, no, the lower figure: E is at (-2,-2)? Wait, maybe the x - translation is 3 (from -5 to -2: -5 + 3=-2) and y - translation: let's see the y - coordinate. After reflection, the y - coordinate is -2, and in EHGF, the y - coordinate is -2? Wait, no, maybe the y - translation is - 0? Wait, maybe I made a mistake. Wait, let's check the x - translation. Let's take point A (-5,2), reflect over x - axis: (-5,-2). Now, we need to get to E. If E is at (-2,-2), then the translation is (x + 3,y+0). Let's check another point: B (-3,4), reflect over x - axis: (-3,-4). If H is at (0,-4), then -3+3 = 0, which matches. So the x - translation is 3 and y - translation is 0.

Answer:

The translation is \((x + 3,y+0)\), so the values are 3 and 0.