The problem asks us to determine a series of transformations that map Figure M onto Figure N. The available transformations are reflection over the x-axis followed by a translation. We need to specify the direction and magnitude of the translation.

1. Problem Description

The problem asks us to determine a series of transformations that map Figure M onto Figure N. The available transformations are reflection over the x-axis followed by a translation. We need to specify the direction and magnitude of the translation.

2. Solution Steps

First, reflect Figure M over the x-axis. If a point on Figure M is $(x, y)$, the corresponding point on the reflected figure will be $(x, -y)$.

Next, we need to find the translation needed to map the reflected Figure M to Figure N. We can observe that Figure N is to the left of where Figure M would be after a reflection over the x-axis. Thus we will need to translate the reflected figure left. Let's pick a point on Figure M, such as $(8, 0)$ which is a vertex. The reflected point will be $(8, -0)$ or $(8,0)$. A corresponding vertex on Figure N is $(-6, -7)$.

Let's examine the top vertex of Figure M. This point appears to be approximately $(8,1)$. After reflection over the x-axis it becomes $(8,-1)$. A corresponding point in Figure N is near $(-6,-7)$. Thus, the $x$ coordinate must go from $8$ to $-6$, which is a shift of $8 - (-6) = 14$ units to the left.

Let's verify this with the rightmost vertex of M, which is near $(9,-2)$ after reflection over the x-axis. The corresponding vertex of N is approximately $(-5, -6)$. The x coordinate shift is $9 - (-5) = 14$ to the left. The y-coordinate shift from the reflected image of M after shifting down 4 is what maps figure M to figure N. Since the question only allows movement along the x-axis (right or left), we will reflect over the x axis and translate left 14 units.

3. Final Answer

A reflection over the x-axis followed by a translation left 14 unit(s).