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?


A.) 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.

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

C.) Kwame’s explanation and solution are incorrect. When reflecting over the x- and y-axes, he negated the incorrect coordinate in the ordered pair.

D.) Kwame’s explanation and solution are incorrect. The coordinates of the rotated figure should be switched and negated, not just negated.

Answer 1

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

• He got the step 3rd incorrect as it should have been (-1, -2), (-1, -4), (-2, -3) instead of (–1, 2), (–1, 4), (–2, 3) for we need to change the symbols of the y-coordinates of the points if we want to find the reflection over the x-axis and vice versa.
• Step 4 is correct
• Step 5 is incorrect since the vertices are located in the same place and therefore the teacher's statement is accurate.

Answer 2

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

Explanation:

Step 1

Given vertices of a pre-image:

• (1, 2), (1, 4), (2, 3)

Step 2

To reflect a point over the x-axis, simply negate the y-coordinate whilst keeping the x-coordinate the same. Therefore, the mapping rule to reflect a point over the x-axis is:

• (x, y) → (x, -y)

Therefore, the reflection of the vertices of the pre-image in step 1 over the x-axis gives:

• (1, -2), (1, -4), (2, -3)

So, step 2 is correct.

Step 3

To reflect a point over the y-axis, simply negate the x-coordinate whilst keeping the y-coordinate the same. Therefore, the mapping rule to reflect a point over the x-axis is:

• (x, y) → (-x, y)

Therefore, the reflection of the step 2 vertices (1, -2), (1, -4) and (2, -3) over the y-axis gives:

• (-1, -2), (-1, -4), (-2, -3)

So, step 3 is incorrect. Kwame reflected the pre-image over the y-axis in this step, rather than reflecting the figure from step 2.

Step 4

To rotate a point 180° about the origin, negate both the x and y coordinates. Therefore, the mapping rule is:

• (x, y) → (-x, -y)

Therefore, the rotation of the vertices of the pre-image by 180° about the origin gives:

• (-1, -2), (-1, -4), (-2, -3)

So, step 4 is correct.

Step 5

As Kwame made a mistake in step 3, his conclusion after comparison is also incorrect, as he is comparing the incorrect vertices in step 3 by the correct vertices in step 4.

If we compare the correct vertices in step 3, we can see that the vertices are located in the same place and therefore the statement made by the teacher is accurate.