Reflection over the Horizontal Axis

Understanding Reflection across the Horizontal 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 $P$ has coordinates $(x,y)$, then its reflection, which we'll call $P'$, will have the same $x$-coordinate, but its $y$-coordinate will be the negative of the original $y$ value.

Mathematically, if the initial point is $P(x,y)$, then after reflection across the $x$-axis, its image is $P'(x,-y)$.

Visualizing Points and Their Reflections

Let's observe some points and their reflections after being mirrored across the $x$-axis.

Notice how the $y$-coordinate changes sign, while the $x$-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 $A(-5,2)$ becomes $A'(-5,-2)$
• Point $B(-3,1)$ becomes $B'(-3,-1)$
• Point $C(1,2)$ becomes $C'(1,-2)$
• Point $D(4,-2)$ becomes $D'(4,2)$

The visible pattern is that the $x$ value remains constant, and the $y$ value changes sign (becomes its opposite).

Property of Reflection across the Horizontal Axis

Based on the observations above, we can formulate the property of reflection across the $x$-axis:

This means the image of point $P(x,y)$ reflected across the $x$-axis is $P'(x,-y)$. The $x$-axis in this case acts as the line $y=0$.

Application Examples

Reflecting a Triangle

Determine the image of triangle $ABC$ with vertices $A(-1,4)$, $B(2,1)$, and $C(-2,-1)$ reflected across the $x$-axis.

To determine the image of triangle $ABC$, we apply the reflection property to each of its vertices:

Consequently, the image of triangle $ABC$ is triangle $A'B'C'$ with vertices $A'(-1,-4)$, $B'(2,-1)$, and $C'(-2,1)$.

Reflecting a Line

If a line has the equation $2x - 3y = 0$ and is reflected across the $x$-axis, determine the equation of its reflected line.

Alternative Solution:

Let an arbitrary point $P(a,b)$ lie on the line $2x - 3y = 0$. Then, the following holds:

The point $P(a,b)$ reflected across the $x$-axis produces the image $P'(a,-b)$.

To obtain the equation of the reflected line, we substitute the coordinates of the image into new variables. Let $x' = a$ and $y' = -b$.

From this, we get $a = x'$ and $b = -y'$.

Substitute $a = x'$ and $b = -y'$ into the original equation $2a - 3b = 0$:

Since $x'$ and $y'$ are arbitrary variables representing the coordinates on the reflected line, we can rewrite them as $x$ and $y$.

Thus, the equation of the reflected line is:

This shows how the equation of a line changes after being reflected across the $x$-axis.