Understanding Reflection across the X-Axis
Reflection across the X-axis is a type of geometric transformation that moves every point of an object to a new position symmetrical to the X-axis. Imagine the X-axis as a flat mirror.
If a point has coordinates , then its reflection, which we'll call , will have the same -coordinate, but its -coordinate will be the negative of the original value.
Mathematically, if the initial point is , then after reflection across the X-axis, its image is .
Visualizing Points and Their Reflections
Let's observe some points and their reflections after being mirrored across the X-axis.
Notice how the -coordinate changes sign, while the -coordinate remains the same.
Based on the interactive visualization above, we can observe the relationship between the original points (pre-image) and their reflections (image) as follows:
- Point becomes
- Point becomes
- Point becomes
- Point becomes
The visible pattern is that the value remains constant, and the value changes sign (becomes its opposite).
Property of Reflection across the X-Axis
Based on the observations above, we can formulate the property of reflection across the X-axis:
This means the image of point reflected across the X-axis is . The X-axis in this case acts as the line .
Application Examples
Reflecting a Triangle
Determine the image of triangle with vertices , , and reflected across the X-axis.
To determine the image of triangle , we apply the reflection property to each of its vertices:
Consequently, the image of triangle is triangle with vertices , , and .
Reflecting a Line
If a line has the equation and is reflected across the X-axis, determine the equation of its reflected line.
Alternative Solution:
Let an arbitrary point lie on the line . Then, the following holds:
The point reflected across the X-axis produces the image .
To obtain the equation of the reflected line, we substitute the coordinates of the image into new variables. Let and .
From this, we get and .
Substitute and into the original equation :
Since and are arbitrary variables representing the coordinates on the reflected line, we can rewrite them as and .
Thus, the equation of the reflected line is:
This shows how the equation of a line changes after being reflected across the X-axis.