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Math 103: Precalculus11 chapters | 81 lessons

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Lesson Transcript

Instructor:
*Jennifer Beddoe*

A graph can be reflected in three ways - across the axes, the origin and the line y = x. There are specific rules to perform each reflection. This lesson will describe those rules and show you how to perform these reflections.

When I was in high school, I thought it would be cool to paint my face in school colors before the homecoming game. I even went so far as to write 'Go Spartans' (our school mascot) on one cheek. I thought I was so clever. What I didn't remember was that the mirror was showing me a reverse image of what my face actually looked like. So what I really did was write 'snatrapS oG' on my face for all to see. It took a long time to live that down. A mirror does not show you a true image but a reflection of what is really going on.

Parts of mathematics also deal with **reflections**. A reflection is a transformation in which each point of a figure has an image that is equal in distance from the line of reflection but on the opposite side. A reflection is a type of transformation known as a flip. The figure will not change size or shape. Mathematical reflections are shown using lines or figures on a coordinate plane. There are three basic ways a graph can be reflected on the coordinate plane.

The first is across either the *x* or *y* axis.

The second is around the origin.

The third is across the line *x* = *y*.

There are rules that govern each of these types of reflections. In order to reflect the graph of an equation across the *y*-axis, you need to pick 3 or 4 points on the graph using their coordinates (*a*, *b*) and plot them as (*-a*, *b*). So the point (4,5) would be reflected to (-4,5).

Once you have reflected the points you have chosen, you can connect the dots. It is important to remember when you choose your points to pick the end points of the graph and any points where the graph changes direction at the very least. You can choose as many points as you want to reflect, but these are critical to making sure your graph is reflected correctly.

If you need to reflect your graph across the *x*-axis, the procedure is the same; however, the translation for each point is (*a*, *b*) translates to (*a*,*-b*).

In order to reflect a graph about the origin, the rule of translation for each point is (*a*,*b*) reflects to (*-a*,*-b*).

The reflection of the point (*a*,*b*) across the line *y* = *x* is (*b*,*a*).

By following these rules, you can reflect any line or figure across any of the three most common lines of reflection or the origin.

Let's try some examples

Reflect this figure across the *x* axis.

Here are the steps to draw the reflection of this figure across the *x* axis.

1. Determine the coordinates of the points that make up the vertices of the triangle. Plotting the reflections of the vertex points will allow you to 'connect the dots' and create the reflection of the entire image. For this figure, the coordinates of the vertices are (2,2), (5,2) and (3,4).

2. Using the rule for reflecting a figure across the *x* axis, convert each of the coordinates to their reflection. The rule for this type of reflection is (*a*,*b*) reflects to (*a*, *-b*). Therefore, each point will be reflected in the following way:

(2,2) reflects to (2,-2)

(5,2) reflects to (5,-2)

(3,4) reflects to (3,-4)

3. Plot the points and connect the dots.

As you can see, the triangle has been reflected across the *x* axis.

Let's try another example. Reflect this line across the origin.

As before, we first determine the coordinates for the end points of the line. In this case, they are (2,4) and (6,1). The rule for reflecting a figure across the origin is (*a*,*b*) reflects to (*-a*,*-b*).

The reflections of the end points of this particular line are (2,4) reflects to (-2,-4) and (6,1) reflects to (-6,-1). Then, we can plot these points and draw the line that is the reflection of our original line.

Let's try one more example.

Reflect this figure across the line *y* = *x*.

The vertex points for this square are:

(1,-1)

(1,-3)

(3,-1)

(3,-3)

The rule for reflecting across the line *y* = *x* is (*a*,*b*) reflects to (*b*,*a*). So, the reflections of each of these coordinates are:

(1,-1) reflects to (-1,1)

(1,-3) reflects to (-3,1)

(3,-1) reflects to (-1,3)

(3,-3) reflects to (-3,3)

By plotting these points and connecting them, we get the reflection of our square, which looks like this:

Reflections are mathematical transformations where each point of a figure is reflected across a line or point on the coordinate plane. There are three main places where reflections occur - across either axis, around the origin or across the line *y* = *x*.

The rules for performing each of these types of reflections are:

*y*-axis: (*a*,*b*) reflects to (*-a*,*b*)

*x*-axis: (*a*,*b*) reflects to (*a*,*-b*)

origin: (*a*,*b*) reflects to (*-a*,*-b*)

*y* = *x*: (*a*,*b*) reflects to (*b*,*a*)

By using these rules, you can perform most any reflection.

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Math 103: Precalculus11 chapters | 81 lessons

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