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Reflection Over An Axis

In this video, you will learn how to do a reflection over an axis, such as the x-axis or y-axis.

To reflect a shape over an axis, you can either match the distance of a point to the axis on the other side of using the reflection notation.

To match the distance, you can count the number of units to the axis and plot a point on the corresponding point over the axis.

You can also negate the value depending on the line of reflection where the x-value is negated if the reflection is over the y-axis and the y-value is negated if the reflection is over the x-axis.

Either way, the answer is the same thing.

For example:
Triangle ABC with coordinate points A(1,2), B(3,5), and C(7,1). Determine the coordinate points of the image after a reflection over the x-axis.

Since the reflection applied is going to be over the x-axis, that means negating the y-value. As a result, points of the image are going to be:
A'(1,-2), B'(3,-5), and C'(7,-1)

By counting the units, we know that point A is located two units above the x-axis. Count two units below the x-axis and there is point A’. Do the same for the other points and the points are also
A'(1,-2), B'(3,-5), and C'(7,-1)

Reflection Notation:
rx-axis = (x,-y)
ry-axis = (-x,y)

Reflection Over An Axis

Video-Lesson Transcript

In this lesson, we’ll go over reflections on a coordinate system. This will involve changing the coordinates.

For example, try to reflect over the x-axis.

We have triangle ABC with coordinates

A (1, 2)
B (3, 5)
C (7, 1)

We’re going to reflect it over the x-axis. We’re going to flip it over.

So we’ll do what we normally do. Just one point at a time.

Now, A is above 2 units from the x-axis so we’ll move it below the x-axis by 2 units.

This will be the A^\prime (1, -2).

Let’s do the same for B. It’s 5 units above the x-axis so we’re going to go 5 units below the x-axis. Notice that it’s still in line with x = 3.

This is now B^\prime (3, -5).

Look at point C at 7, 1. It’s 1 point above the x-axis so we’ll go 1 point below the x-axis.

So, C^\prime (7, -1).

And just connect the points. Then we can see our reflection over the x-axis.

When we reflect over the x-axis, something happens to the coordinates.

The initial coordinates (x, y) change. The x coordinate stays the same but the y coordinate is the same number but now it’s negative.

(x, y) \rightarrow (x, -y)

In reflecting over the x-axis, we’ll write

r_{x-axis}

Now, the same thing goes for reflecting over the y-axis.

We’re going to reflect triangle ABC over the y-axis.

r_{y-axis} \triangle{ABC}

Similar to reflecting over the x-axis, we’ll just do one point at a time.

A (1, 2)
B (3, 5)
C (7, 1)

A is 1 unit from the y-axis so we’ll move 1 beyond the y-axis.

So, A^\prime (-1, 2).

Let’s look at B at (3, 5). That means it’s 3 units from the y-axis so we’ll move 3 coordinates on the other side of the y-axis.

Now, B^\prime (-3, 5).

Finally, C is at (7, 1) so we’ll go 7 points beyond the y-axis.

We’ll have C^\prime (-7, 1).

Now, we can draw a triangle that is a reflection of triangle ABC over the y-axis.

Let’s look at how these coordinates changed.

Originally we have coordinates (x, y) but x became negative while y stayed the same.

(x, y) \rightarrow (-x, y)

Let’s recap.

The rule of reflecting over the x-axis is

r_{x-axis} (x, y) \rightarrow (x, -y)

And for reflecting over the y-axis is

r_{y-axis} (x, y) \rightarrow (-x, y)

If you reflect it over the x-axis, x coordinate stays the same the other coordinate becomes negative.

And reflecting over the y-axis, y coordinate stays the same while the other coordinate becomes negative.