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If a point is shifted to the right, the new x co-ordinate is given by adding the shift to the old x co-ordinate. If a point is shifted to the left, the new x co-ordinate is given by subtracting the shift from the old x co-ordinate.

Reflection of a point

When you stand in front of a mirror your reflection is located the same distance ( d ) behind the mirror as you are standing in front of the mirror.

We can apply the same idea to a point that is reflected on the x -axis, the y -axis and the line y = x .

Reflection on the x -axis

If a point is reflected on the x -axis, then the reflection must be the same distance below the x -axis as the point is above the x -axis and vice-versa, as though it were a mirror image.

Points A and B are reflected on the x -axis. The original points are shown with and the reflected points are shown with .
When a point is reflected about the x -axis, only the y co-ordinate of the point changes.

Find the co-ordinates of the reflection of the point P, if P is reflected on the x -axis. The co-ordinates of P are (5;10).

  1. We are given the point P with co-ordinates (5;10) and need to find the co-ordinates of the point if it is reflected on the x -axis.

  2. The point P is above the x -axis, therefore its reflection will be the same distance below the x -axis as the point P is above the x -axis. Therefore, y =-10.

    For a reflection on the x -axis, the x co-ordinate remains unchanged. Therefore, x =5.

  3. The co-ordinates of the reflected point are (5;-10).

Reflection on the y -axis

If a point is reflected on the y -axis, then the reflection must be the same distance to the left of the y -axis as the point is to the right of the y -axis and vice-versa.

Points A and B are reflected on the y -axis. The original points are shown with and the reflected points are shown with .
When a point is reflected on the y -axis, only the x co-ordinate of the point changes. The y co-ordinate remains unchanged.

Find the co-ordinates of the reflection of the point Q, if Q is reflected on the y -axis. The co-ordinates of Q are (15;5).

  1. We are given the point Q with co-ordinates (15;5) and need to find the co-ordinates of the point if it is reflected on the y -axis.

  2. The point Q is to the right of the y -axis, therefore its reflection will be the same distance to the left of the y -axis as the point Q is to the right of the y -axis. Therefore, x =-15.

    For a reflection on the y -axis, the y co-ordinate remains unchanged. Therefore, y =5.

  3. The co-ordinates of the reflected point are (-15;5).

Reflection on the line y = x

The final type of reflection you will learn about is the reflection of a point on the line y = x .

Casestudy : reflection of a point on the line y = x

Study the information given and complete the following table:

Point Reflection
A (2;1) (1;2)
B (- 1 1 2 ;-2) (-2;-1 1 2 )
C (-1;1)
D (2;-3)

What can you deduce about the co-ordinates of points that are reflected about the line y = x ?

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Source:  OpenStax, Siyavula textbooks: grade 10 maths [ncs]. OpenStax CNX. Aug 05, 2011 Download for free at http://cnx.org/content/col11239/1.2
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