 Invariant Points for Reflection in a Line is nothing a point that remains unvaried after the transformation applied to it. Any point on a graph on the line of reflection is an invariant point. Learn more about Invariant Points Under Reflection in a Line from this article. By making this page as a reference you can know about the Invariant Points on the graph on the line of reflection which helps you to score good marks in the exams.

Invariant Points for Reflection in a Line – Definition

Invariant points means the points which lie on the line and when reflected in the line. So, only those points are invariant which lie on the y-axis.
The invariant points must have x-coordinate = 0.
Therefore, only (0, 4) is the invariant point. No points lying outside the line will be an invariant point.

Invariant Points in Reflection in a Line Examples with Answers

Example 1.
Which of the following points (-2, 0), (0, -5), (3, -3) are invariant points when reflected in the x-axis?
Solution:
Given points are (-2,0), (0,-5) and (-3,3)
We know that
The points lying on the line are invariant points. when reflected in the line.
So, only those points are invariant which lie on the x-axis.
Hence, the invariant points must have y-coordinate = 0.
Therefore, only (-2, 0) is the invariant point.

Example 2.
Which of the following points (-2, 3), (0, 6), (4, -3), (-3, 6) are invariant points when reflected in the line parallel to the x-axis at a distance of 4 on the positive side of the y-axis?
Solution:
Given points are (-2,3), (0,6), (4,-3) and (-3,6)
We know that,
The points lying on the line are invariant points when reflected in the line.
So, only those points are invariant which are on the line parallel to the x-axis at a distance of 4 on the positive side of the y-axis.
Hence, the invariant points must have a y-coordinate = 4.
Therefore, (0, 6) and (-3, 6) are the invariant points.

Example 3.
Points A and B have coordinates (4,6) respectively. Find the reflection of A’ of A under reflection in the x-axis and A” of A under reflection in the y axis.
Solution:
Given that the coordinates are (4,6)
A’ = image of A under reflection in the x axis = (4,-6).
A” = image of A under reflection in the y axis = (-4,6).

Example 4.
A point P(a,b) is reflected in the x-axis to P'(6,-5). Write down the values of a and b. P” is the image of P reflected in the y axis. Write down the coordinates of P”. Find the coordinates of P”‘, when P is reflected in the line parallel to y-axis such that x = 3.
Solution:
A point P(a,b) is reflected in the x axis P'(6,-5)
We know that Mx(x,y) = (x,-y)
Thus, the coordinates of P are (6,5)
Hence a = 6, and b = 5.
P” = image of P reflected in the y axis = (-6,5)
P”‘ = reflection of P in the line (x = 3) = (-9,5).

Example 5.
Which of the following points (-1, 0), (0, -8), (2, -7) are invariant points when reflected in the x-axis?
Solution:
Given points are (-1,0), (0,-8) and (2,-7)
We know that
The points lying on the line are invariant points. when reflected in the line.
So, only those points are invariant which lie on the x-axis.
Hence, the invariant points must have y-coordinate = 0.
Therefore, only (-1, 0) is the invariant point.

FAQs on Invariant Points Under Reflection in a Line

1. Is a line of invariant points an invariant line?

In linear transformation maps the origin to the origin.
So the origin is always an invariant point under a linear transformation. Every point on the invariant line maps to a point on the line itself.

2. What are invariant points?

Invariant points are points on a line or a shape which do not move when the specific transformation is applied.

3. How do you find invariant points on a graph?

Sketch each graph by using the key points, including invariant points, Determine the image points on the graph by square rooting the points. And locate invariant points on y = f(x) and y = g(x). When graphing the square root of a function, invariant points occur at y = 0 and y = 1.