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Reflection across the -axis is a type of geometric transformation that moves every point of an object to a new position symmetrical to the -axis. Imagine the -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 -axis, its image is .
Let's observe some points and their reflections after being mirrored across the -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:
The visible pattern is that the value remains constant, and the value changes sign (becomes its opposite).
Based on the observations above, we can formulate the property of reflection across the -axis:
This means the image of point reflected across the -axis is . The -axis in this case acts as the line .
Determine the image of triangle with vertices , , and reflected across the -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 .
If a line has the equation and is reflected across the -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 -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 -axis.
