Follow the detailed proof below.

Follow the detailed proof below for an explanation.

To begin this proof we first must let there exist the triangle , where the length of  and  is a shared side between the two triangles.

Follow the detailed proof below.

To determine the type of transformation that is occurring in this particular situation, first recall the different types of transformations.

Translation: To move an object from its original position a certain distance without changing the object in any other way.

Transformation: Refers to any of the four changes that can be done to an object geographically. Transformations include, reflections, translations, rotations, and resizing the object.

Reflection: To flip the orientation of an object over a specific line or function.

Rotation: To rotate an object either clockwise or counter clockwise around a center point.

Since each of the triangle's coordinates is moved to the left and down, it is seen that the size and shape of the triangle remains the same but its location is different. Therefore, the transformation the triangle has undergone is a translation.

"If a rectangle has the coordinate values, , , , and  and after a transformation results in the coordinates , , , and  identify the transformation."

A transformation that changes the  values by multiplying them by negative one is known as a reflection across the -axis or the line  

Therefore, this particular rectangle is being reflected across the -axis because the opposite of all the  values have been taken.

To identify the transformation that is occurring in this particular problem, recall the different transformations.

Translation: To move an object from its original position a certain distance without changing the object in any other way.

Transformation: Refers to any of the four changes that can be done to an object geographically. Transformations include, reflections, translations, rotations, and resizing the object.

Reflection: To flip the orientation of an object over a specific line or function.

Rotation: To rotate an object either clockwise or counter clockwise around a center point.

Looking at the starting and ending coordinates of the rectangle,

 , , , and   to  , , , and 

Since all the  coordinates are increasing by two this is known as a translation.

"If a rectangle has the coordinate values, , , , and  and after a transformation results in the coordinates  , , , and  identify the transformation."

A transformation that changes the  values by multiplying them by negative one is known as a reflection across the -axis or the line  

Therefore, this particular rectangle is being reflected across the -axis because the opposite of all the  values have been taken.

The above described transformation is a dilation. Notice that one point, (a,b), stays the same before and after the transformation. The point (c,b) retains the same y value of b, but c is dilated into 2c, extending the base of the rectangle. The point (a,d) is similar in that the x value of a stays the same, but the y value of d is extended or dilated to 2d. The final point (c,d) is extended in both length and width to become (2c,2d). The below graph shows the original figure in blue and the dilated larger figure in pink. 

The correct answer is (-a,-b), (-c,-d), and (-e,-f). In other words, you'd just take the opposite value of each x and y value of each vertex of the triangle. The following diagram shows one set of vertexes rotated 180^o about the origin to help demonstrate this. 

Please note that if one of our original points had any negative values, such as the point (2,-2), and we rotated it 180o about the origin, the signs of both the x and y values would change, and this point's image after translation would be (-2,2).