Question

answer a reflection over the y -axis a reflection over the x -axis a translation 1 unit left and 4 units down a translation 4 units left and 1 unit down

Explanation:

Step1: Recall transformation rules

For reflection over y - axis, $(x,y)\to(-x,y)$; for reflection over x - axis, $(x,y)\to(x, - y)$; for translation $a$ units left and $b$ units down, $(x,y)\to(x - a,y - b)$.

Step2: Check key - point coordinates

Let's assume a key - point of Figure J, say $(1,0)$. In Figure K, the corresponding point seems to be at $(- 3,-4)$. If we consider the transformation $(x,y)\to(x - 4,y - 4)$, for the point $(1,0)$ in Figure J, we get $(1-4,0 - 4)=(-3,-4)$.

Answer:

A translation 4 units left and 4 units down (Note: There is an error in the provided options as the correct translation is 4 units left and 4 units down, but among the given options, the closest conceptually is a translation. If we assume a typing error and we go by the closest match in terms of the nature of transformation, we can analyze the x - and y - coordinate changes. The x - coordinates of corresponding points in Figure J and Figure K have a difference of 4 (decrease) and the y - coordinates also have a difference of 4 (decrease). So, if we have to choose from the given options, the answer is A translation 4 units left and 1 unit down is incorrect, A translation 1 unit left and 4 units down is incorrect, A reflection over the y - axis is incorrect, A reflection over the x - axis is incorrect. The closest in terms of the type of transformation is a translation, but the correct values should be 4 units left and 4 units down). If we assume we have to pick the best out of the given ones, we analyze the x and y changes. The x - value of a point in Figure J to Figure K decreases by 4 and y - value decreases by 4. Among the translation options, we note the x - shift and y - shift. The correct answer considering the nature of transformation is a translation, and among the given translation options, we consider the magnitudes of shifts. The x - shift from points in Figure J to Figure K is 4 units left and y - shift is 4 units down. But since we have to choose from the given ones, if we assume a mis - typing or approximation, we note the x - shift of 4 units left in one of the translation options. So, if we have to pick: A translation 4 units left and 1 unit down is wrong as y - shift is 4 not 1, A translation 1 unit left and 4 units down is wrong as x - shift is 4 not 1, A reflection over the y - axis is wrong as the orientation and coordinate changes don't match, A reflection over the x - axis is wrong as the orientation and coordinate changes don't match. The best option considering the type of transformation is a translation, and among the translation options, the one with 4 units left is more relevant in terms of x - coordinate change. So, if we assume we ignore the wrong y - value in the option, the answer is A translation 4 units left and 1 unit down (but this is with the caveat that the actual y - shift should be 4 units down).)