1.2.3 Transformations in the Coordinate Plane (continued) Example 2 Drawing and Identifying Transformations
In this example, the coordinates of the preimage’s vertices, as well as the image’s vertices, are given. Since three points are given for each figure, the preimage and image must be triangles. Begin by plotting the points to identify the vertices. Then, draw lines to connect the vertices. These are the sides of the figures. Now determine if the transformation of △ ABC → △ A ′ B ′ C ′ is a reflection, translation, or rotation. Notice that the horizontal distance from the y -axis to A is equal to the horizontal distance from the y -axis to A ′. Furthermore, the horizontal distance is the same from the y -axis to B and B ′, and the same is true for C and C ′. So, each point and its image is the same distance from the y -axis (which is a line). The transformation where each point and its image is the same distance from a line is a reflection. Therefore, the transformation △ ABC → △ A ′ B ′ C ′ is a reflection and the line of reflection is the y -axis.
Example 3 Translations in the Coordinate Plane All of the points on the preimage move the same distance in the same direction to form the image in a translation. Given a series of points, such as the vertices of a figure, the result of adding some number a to each of the x -coordinates and adding some number b to each of the y -coordinates is that all of the points will be moved the same distance in the same direction, which is a translation. So, a translation can be generalized as ( x , y ) → ( x + a , y + b ), which means that for any preimage point ( x , y ), the corresponding image point is ( x + a , y + b ).
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