What Is the Formula for Reflection of a Point About X Axis with Examples
Have you ever thought about how you would represent the reflection of a point about the x-axis? This is done using the concept of reflection of a point in the x-axis. If you do not know about it, do not get worried, as this article covers all the concepts of reflection in the x-axis and the rules to find the reflection using attractive images so that the children can easily grasp the topics. Let us begin with our learning.
What is the Reflection of a Point in the x-axis?
Reflection of a point in the x-axis states that when a point is reflected across the x-axis, the x-coordinate remains constant, while the y-coordinate is assumed to be the additive inverse of the given ordinate. For example, a point (x, y) is reflected across the x-axis as (x, -y).
Reflection of a Point About the x-axis
A Point on the x-axis has Coordinates
A point on the x-axis has coordinates in the form of ordered pairs having the form (h, 0), where h is the point on the x-axis. When y = 0, the value of the x-coordinate or abscissa can be anything, irrespective of the value of the ordinate.
Rules to Find the Reflection in the x-axis
There is no hard rule to find the reflection in the x-axis; you just need to do is, to follow these two simple steps, which are given below:
Keep the coordinates of the x-axis fixed
Reverse the sign of the y-coordinate
The obtained value of the x-coordinate and y-coordinate is the reflection of a point about the x-axis.
Solved Examples
Q 1. Find the reflection of a point about the x-axis of the following:
(-1, 6)
(2, 1)
Ans: For part 1, we need to follow the given steps:
Read the coordinates (-1, 6) and find out in which quadrant it lies, i.e. the 2nd quadrant
Mark the value of x = -1 and y = 6 in the appropriate quadrants
Highlight the point and write its coordinates
To find its reflection, keep the abscissa same, i.e. x-coordinate is -1, and take the additive inverse of the ordinate, i.e. -6
Now, choose the quadrant for the new coordinates, i.e. (-1, -6)
Mark the values of x = -1 and y = -6 in the respective quadrants
Highlight the point and write its coordinates
This is how you find the reflection of a point in the x-axis.
Reflection of a Point About the x-axis
For part 2, we need to follow the given steps:
Read the coordinates (2, 1) and find out in which quadrant it lies, i.e. the 1st quadrant
Mark the values of x = 2 and y = 1 in the respective quadrants
Highlight the point and write its coordinates
To find its reflection, keep the abscissa same, i.e. 2 and take the additive inverse of ordinate, i.e. -1
Now, choose the quadrant for the new coordinates, i.e. (2, -1)
Mark the values of x = 2 and y = -1 in the respective quadrant
Highlight the point and write its coordinates
This is how you find the reflection of a point (2,1) on the x-axis.
Reflection in the x-axis
Practice Problems
Q 1. Find the reflection about the x-axis of the point (2, 6).
Ans. (2, -6)
Q 2. Find the reflection of a point on the x-axis of the following:
(-7, -3)
(3, 1)
Ans. (-7, 3)
(3, -1)
Q 3. Compute the reflection of a point on the x-axis of the following:
(4, -5)
(-8, 2)
Ans. (4, 5)
(-8, -2)
Summary
Summing up here with the concept of reflection in the x-axis. This article describes all the topics, including rules to find the reflection, the coordinate of a point on the x-axis, etc. Here we have discussed in depth how to solve the problems based on the reflection of a point in the x-axis. Some practice problems are also assigned to the students along with their answers so they can practice more and gain proficiency in the concept. Hoping you enjoyed reading the article.
FAQs on Reflection of a Point About the X Axis in Coordinate Geometry
1. What is the reflection of a point about the x-axis?
The reflection of a point about the x-axis is a new point where the x-coordinate stays the same and the y-coordinate changes its sign. If a point is (x, y), its reflection across the x-axis is (x, −y).
- The x-value remains unchanged.
- The y-value becomes its negative.
- The reflected point is the same distance from the x-axis but on the opposite side.
2. What is the formula for reflection across the x-axis?
The formula for reflection across the x-axis is (x, y) → (x, −y). This rule applies to any point in the Cartesian plane.
- Original point: (x, y)
- Reflected point: (x, −y)
- Only the sign of the y-coordinate changes.
3. How do you reflect a point over the x-axis step by step?
To reflect a point over the x-axis, keep the x-coordinate the same and change the sign of the y-coordinate. Follow these steps:
- Step 1: Write the original point (x, y).
- Step 2: Keep the x-value unchanged.
- Step 3: Replace y with −y.
- Step 4: Write the new point (x, −y).
4. What happens to the coordinates when a point is reflected about the x-axis?
When a point is reflected about the x-axis, the x-coordinate remains the same and the y-coordinate changes sign. This means:
- (x, y) becomes (x, −y).
- The point moves vertically across the x-axis.
- The distance from the x-axis stays equal.
5. Can you give an example of reflection about the x-axis?
Yes, an example of reflection about the x-axis is reflecting the point (−3, 7), which gives (−3, −7). Using the formula (x, y) → (x, −y):
- Original point: (−3, 7)
- Reflected point: (−3, −7)
6. What is the difference between reflection across the x-axis and y-axis?
The difference is that reflection across the x-axis changes the y-coordinate, while reflection across the y-axis changes the x-coordinate.
- Across x-axis: (x, y) → (x, −y)
- Across y-axis: (x, y) → (−x, y)
7. Does the distance from the x-axis change after reflection?
No, the distance from the x-axis remains the same after reflection. The reflected point is equally far from the x-axis but on the opposite side.
- Distance from x-axis = |y|
- After reflection, distance = |−y| = |y|
8. How do you reflect a graph across the x-axis?
To reflect a graph across the x-axis, replace every y-value with its negative. For a function y = f(x), the reflected graph is y = −f(x).
- Each point (x, y) becomes (x, −y).
- The graph flips vertically.
- The shape and size remain unchanged.
9. Is reflection across the x-axis a rigid transformation?
Yes, reflection across the x-axis is a rigid transformation because it preserves distance and shape. This means:
- Lengths remain the same.
- Angles remain the same.
- Only orientation changes.
10. What are common mistakes when reflecting a point about the x-axis?
A common mistake when reflecting about the x-axis is changing the wrong coordinate. Remember the correct rule is (x, y) → (x, −y).
- Do not change the x-coordinate.
- Always change the sign of the y-coordinate.
- Check if the original y-value is already negative.
