The problem asks us to determine a series of transformations that would map Figure M onto Figure N. We are provided with a graph of the two figures and options to select a reflection over the x-axis or y-axis, followed by a translation.

1. Problem Description

The problem asks us to determine a series of transformations that would map Figure M onto Figure N. We are provided with a graph of the two figures and options to select a reflection over the x-axis or y-axis, followed by a translation.

2. Solution Steps

First, consider a reflection over the x-axis. If we reflect Figure M over the x-axis, the coordinates $(x, y)$ become $(x, -y)$. Let's consider the vertices of Figure M. Approximating the vertices of Figure M, we have $(7, -2)$, $(9, -1)$, and $(8, -5)$. Reflecting over the x-axis, these become $(7, 2)$, $(9, 1)$, and $(8, 5)$. Figure N's vertices can be approximated as $(2, -7)$, $(4, -6)$, and $(3, -9)$. Thus, a reflection over the x-axis does not bring it any closer to Figure N.

Next, consider a reflection over the y-axis. If we reflect Figure M over the y-axis, the coordinates $(x, y)$ become $(-x, y)$. Let's consider the vertices of Figure M again. Approximating the vertices of Figure M, we have $(7, -2)$, $(9, -1)$, and $(8, -5)$. Reflecting over the y-axis, these become $(-7, -2)$, $(-9, -1)$, and $(-8, -5)$. Figure N's vertices can be approximated as $(2, -7)$, $(4, -6)$, and $(3, -9)$. Reflecting over the y-axis does not map M to N.

Instead, let's try a reflection over the x-axis followed by a translation. After reflection over the x-axis, the vertices are approximately $(7, 2)$, $(9, 1)$, and $(8, 5)$. To map this to the approximated vertices of Figure N $(2, -7)$, $(4, -6)$, and $(3, -9)$, we must translate by the vector $(-5, -9)$. Thus, we translate 5 units to the left and 9 units down.

Let's verify: $(7, 2) \to (7-5, 2-9) = (2, -7)$

$(9, 1) \to (9-5, 1-9) = (4, -8)$. It appears the translation should be $(-5, -7)$.

Let's try again:

Approximate vertices of figure M: (7, -2), (9, -1), (8, -5).

Reflection over x-axis: (7, 2), (9, 1), (8, 5).

Approximate vertices of figure N: (2, -7), (4, -6), (3, -9).

Translation vector (2-7, -7-2) = (-5, -9).

Translation vector (4-9, -6-1) = (-5, -7).

Translation vector (3-8, -9-5) = (-5, -14).

The question allows us to select A reflection over the x-axis or y-axis, followed by a translation. Let's consider reflecting figure M over the y-axis.

Figure M has points (7, -2), (9, -1), (8, -5).

After reflecting, these are (-7, -2), (-9, -1), (-8, -5).

The target points are approximately (2, -7), (4, -6), (3, -9).

The required translation vectors are:

(2 - (-7), -7 - (-2)) = (9, -5)

(4 - (-9), -6 - (-1)) = (13, -5)

(3 - (-8), -9 - (-5)) = (11, -4)

The transformation is a reflection over the x-axis followed by a translation.

3. Final Answer

A reflection over the x-axis followed by a translation. The translation would be by (-5, -9).