Reflection over the Vertical Axis

Understanding Reflection over the Vertical Axis

Reflection over the $y$-axis is a type of geometric transformation that moves every point on an object to a new position. Imagine the $y$-axis as a mirror. Every point will have an image on the opposite side of the $y$-axis at the same distance from the $y$-axis.

Rule for Reflection over the Vertical Axis

If a point $P(x, y)$ is reflected over the $y$-axis, its image's coordinates, $P'(x', y')$, will follow the rule:

Thus, the image of point $P(x, y)$ is $P'(-x, y)$.

Note that the value of the $y$-coordinate does not change, while the value of the $x$-coordinate becomes its opposite (negative if positive, positive if negative).

Reflecting a Point

Suppose we have point $A(3, 4)$. If point $A$ is reflected over the $y$-axis, its image, $A'$, can be determined as follows:

The original $x$-coordinate is $3$, so $x' = -3$.

The original $y$-coordinate is $4$, so $y' = 4$.

Thus, the image of point $A$ is $A'(-3, 4)$.

Let's visualize this:

Reflecting a Triangle

Now, let's reflect a triangle $PQR$ with vertices $P(1, 2)$, $Q(4, 4)$, and $R(2, 0)$ over the $y$-axis.

To reflect a triangle, we need to reflect each of its vertices.

1. Point $P(1, 2)$: Its image is $P'(-1, 2)$.
2. Point $Q(4, 4)$: Its image is $Q'(-4, 4)$.
3. Point $R(2, 0)$: Its image is $R'(-2, 0)$.

By connecting the image points $P'Q'R'$, we obtain the reflected triangle.

Reflecting a Line Equation

Suppose we have a line with the equation $y = x + 2$. To find the equation of its image after reflection over the $y$-axis, we substitute $x$ with $-x$ (because $x' = -x$) and $y$ with $y$ (because $y' = y$) into the original equation.

Original equation:

Substitute $x \rightarrow -x$:

The equation of the image is:

Let's visualize these two lines:

Exercises

1. Determine the coordinates of the image of point $K(-5, 8)$ if it is reflected over the $y$-axis!
2. A triangle $ABC$ has vertices $A(2, 1)$, $B(5, 3)$, and $C(3, 6)$. Determine the coordinates of the image triangle $A'B'C'$ after reflection over the $y$-axis!
3. Determine the equation of the image of the line $3x - 2y + 6 = 0$ if it is reflected over the $y$-axis!
4. A line has the equation $y = -3x - 4$. Determine the equation of its image after reflection over the $y$-axis.

Key Answers

1. The image of point $K(-5, 8)$ is $K'(5, 8)$.

   Explanation: $x' = -(-5) = 5$, $y' = 8$.

2. The coordinates of the image triangle $A'B'C'$ are:

   • $A'(-2, 1)$
   • $B'(-5, 3)$
   • $C'(-3, 6)$

3. The equation of the image of the line $3x - 2y + 6 = 0$ is $-3x - 2y + 6 = 0$.

   Explanation: Substitute $x$ with $-x$ into the original equation:

   $$3(-x) - 2y + 6 = 0$$

   $$-3x - 2y + 6 = 0$$

4. The equation of the image of the line $y = -3x - 4$ is $y = 3x - 4$.

   Explanation: Substitute $x$ with $-x$:

   $$y = -3(-x) - 4$$

   $$y = 3x - 4$$