Reflection over the Horizontal Axis

How Reflection Changes the Vertical Coordinate

In a reflection across the $x$-axis, the image of a point outside the axis lies on the opposite side of the $x$-axis at the same distance. The $x$-axis acts like a horizontal 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

Look for point pairs on the same vertical line. A point above the axis reflects below it, while a point below reflects above. Each point and its image remain equally far from the axis.

The plotted pairs show the relationship between each original point (pre-image) and its reflection (image):

• 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 $D'(4,2)$

Every pair keeps the same $x$ value and changes the sign of its $y$ value.

Property of Reflection across the Horizontal Axis

Reflection keeps the horizontal coordinate and reverses the vertical coordinate sign. The midpoint between an original point and its image lies on the mirror axis:

Both points are at distance $|y|$ from the axis. If the original point lies on the axis, its vertical coordinate is zero and it remains fixed:

Reflecting Triangles and Lines

Reflection acts on every point of a figure. Three vertices are enough to determine the reflected triangle. For a line, substitute the coordinate rule into its equation so it represents every point on the line at once.

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.

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 $y$.

The equation of the reflected line is:

Reflection across the $x$-axis changes $2x-3y=0$ into $2x+3y=0$ by replacing $y$ with $-y$.