Question

A teacher asks students to determine the accuracy of this statement.

A reflection over the x-axis and then over the y-axis results in the same transformation as a 180degree rotation about the origin of the original figure.

Kwame’s explanation of the accuracy of the statement is shown below.

Step 1: Choose the vertices of a pre-image: (1, 2), (1, 4), (2, 3).
Step 2: Reflect the pre-image over the x-axis: (1, –2), (1, –4), (2, –3).
Step 3: Reflect the figure from step 2 over the y-axis: (–1, 2), (–1, 4), (–2, 3).
Step 4: Rotate the pre-image 180degree: (–1, –2), (–1, –4), (–2, –3).
Step 5: Compare the reflected figure to the rotated figure. The vertices are not located in the same place, therefore the statement made by the teacher is inaccurate.

Which best explains Kwame’s solution?
Kwame’s explanation and solution are incorrect. He reflected the image over the y-axis in step 2 and the x-axis in step 3, instead of the reverse.
Kwame’s explanation and solution are incorrect. He reflected the pre-image at step 3, rather than the figure from step 2.
Kwame’s explanation and solution are incorrect. When reflecting over the x- and y-axes, he negated the incorrect coordinate in the ordered pair.
Kwame’s explanation and solution are incorrect. The coordinates of the rotated figure should be switched and negated, not just negated.

Answer 1

Step-by-step explanation:

1 and 2.........

switched and negated, not just negated......

Answer 2

Option 2 : Kwame’s explanation and solution are incorrect. He reflected the pre-image at step 3, rather than the figure from the previous step.

In general, any transformations on a coordinate point (x,y) is as follows :

• 180 degrees rotation : It rotates the cartesian plane by half a circle. Due to this, both the X-axis and Y-axis get reversed.

                      $(x,y)$ ⇒ $(-x,-y)$

• Reflection about X-axis : When a coordinate point is reflected with respect to the X-axis, its X-coordinate remains the same and the Y-coordinate is negated.

                      $(x,y)$ ⇒ $(x,-y)$

• Reflection about Y-axis : When a coordinate point is reflected with respect to the Y-axis, its Y-coordinate remains the same and the X-coordinate is negated.

                      $(x,y)$ ⇒ $(-x,y)$

STEP 1 of Procedure: The pre-vertices are chosen correctly

STEP 2 of Procedure:

The points need to be reflected with respect to X-axis. So, the y-coordinate is to be negated which is correctly done.

(x,y) ⇒ (x,-y)

STEP 3 of Procedure:

The resultant points of step 2 need to be reflected with respect to the Y-axis. So, the x-coordinate is to be negated which is incorrectly done by Kwame.

(x,-y) ⇒ (-x,-y)

STEP 4 of Procedure:

The initially selected pre-vertices need to be rotated by 180 degrees. So, both the coordinates will be negated which is correctly done by Kwame.

(x,y) ⇒ (-x,-y)

STEP 5 of Procedure:

The result of 180-degree rotation and reflection about X axis followed by the Y-axis is the same i.e. (-x,-y). But it is incorrect for Kwame, as his Step 3 is incorrect.

Hence, Kwame's solution and explanation both are incorrect as his STEP-3 of the procedure is incorrect.

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