The vertex of a figure is located at (2,4). The figure is transformed and the image of the vertex is located at (4,-2). Which of these describes the transformation?
a. reflection over the x-axis
b. reflection over the y-axis
c. rotation of 180º counter-clockwise
d. rotation of 270º counter-clockwise.
c. rotation of 180º counter-clockwise
To determine the transformation, we need to analyze the changes in the x-coordinate and y-coordinate of the vertex.
The x-coordinate of the original vertex is 2, and in the image, it becomes 4. There is an increase of 2 units, which indicates a horizontal translation or shifting to the right.
The y-coordinate of the original vertex is 4, and in the image, it becomes -2. There is a decrease of 6 units, which indicates a vertical translation or shifting downwards.
Therefore, the transformation can be described as a translation 2 units to the right and 6 units downward.
The correct answer is not provided in the options, as the given options describe rotation and reflection transformations, which do not match the given coordinates of the vertex in the question.
To determine the transformation, we need to analyze the change in coordinates of the vertex before and after the transformation.
The original coordinates of the vertex are (2,4). After the transformation, the new coordinates of the vertex are (4,-2).
To find the change in x-coordinate, subtract the original x-coordinate from the new x-coordinate: 4 - 2 = 2.
To find the change in y-coordinate, subtract the original y-coordinate from the new y-coordinate: -2 - 4 = -6.
Now let's analyze the possible transformations:
a. Reflection over the x-axis: This transformation would change the sign of the y-coordinate but keep the x-coordinate the same. However, in this case, the x-coordinate has changed, so it cannot be a reflection over the x-axis.
b. Reflection over the y-axis: This transformation would change the sign of the x-coordinate but keep the y-coordinate the same. However, in this case, the y-coordinate has changed, so it cannot be a reflection over the y-axis.
c. Rotation of 180º counter-clockwise: This transformation would result in a change of signs for both the x and y coordinates. In this case, the change in coordinates matches the change we observed: 2 and -6. So, the correct answer is a rotation of 180º counter-clockwise.
d. Rotation of 270º counter-clockwise: This transformation would result in a change in the signs of both the x and y coordinates, but also change the magnitude. In this case, the magnitude of the change in coordinates does not match what we observed, so it cannot be a rotation of 270º counter-clockwise.
Therefore, the correct answer is c. a rotation of 180º counter-clockwise.