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Reflection Rules – What are they? How can you use them?

By Trisha Ganguli

A topic in mathematics that is very popular is called transformations: they allow us to transform certain mathematical functions by stretching, enlarging, shrinking, or even reflecting to create an image of what was. One of these transformations is reflection – reflection is when you flip an image. In a more mathematical sense, you can flip not only a point, but a function or a line to get an image.

Why is this concept useful?

Reflection is one of the many transformations that can be used to change a function into a desired function. Such kind of transformations can also be used to find patterns between functions – one function could be a reflection of another, or just a translation of another, or maybe both? Decomposing each of the transformations into smaller changes like reflection help us to understand what happens to a function when we see a reflection of it.

Where does this concept fit into the curriculum?

High School

What happens when you look into a lake or a mirror? You see an image of yourself – a “reflection” of your real face but in the lake/mirror. This reflection is simply just a “flipped” version of the real object, which is your face. Similarly, in mathematics, you can also flip functions to get another function. This helps us understand about how we can manipulate a function to get another function.


How do you use this concept?

One of the first steps to understanding reflection is that any reflection of an object or a function has to be across a fixed line, which is often called the line of reflection. When you look at your reflection in the lake, the line (more so a plane) of reflection is the surface of the lake, somewhat like this picture:

Notice that in this picture, the size, shape, color is all the same as in the real world. The only change is that the image was flipped so the distance from the real object to the line of reflection is the same as from the image to the line of reflection. Previously, our line of reflection was the lake, but now we can change that to make the line of reflection a line on a x-y Cartesian plane (note the line of reflection can be any line in any direction – horizontal, vertical or even diagonal). We can reapply this concept in these three common scenarios - flipping across the x-axis, y-axis or the y = x line:

1. Flipping across the x-axis:

To flip across the x-axis, for all points of the real function/shape (x,y) we need to change it to (x,-y), like so:

2.  Flipping across the y-axis:

To flip across the y-axis, for all points of the real function/shape (x,y) we need to change it to (-x,y), like so:

3. Flipping across the y = x line:

Recall that the y = x line is a diagonal line of the Cartesian plane that looks like:

If you have a shape or line segment with co-ordinates, the process is similar to the previous two ways. For all points of the real function/shape (x,y) we need to change it to (y,x) (invert the co-ordinates).

Note that in any case of reflection, the distance between the real object and the line of reflection should be equal to the distance between the new (‘reflected’) image and the line of reflection. So, if co-ordinates are not provided, then you can simply just measure the distance, and draw a reflected image at the same distance away from the line of reflection.

Sample Math Problems

Question

Which of these diagrams shows a correct reflection of triangle ABC across the y-axis?

Options are:

A)

B)

C)

D)

Solution

Option C is the correct answer. This is because distance across every point from ABC to the line of reflection (i.e. the y-axis) is equal to the distance from every point from DEF to the line of reflection.

Question

Which of these diagrams shows a correct reflection of triangle ABC across the x-axis?

Options are:

A)

B)

C)

D)

Solution

Option A is the correct answer. This is because distance across every point from ABC to the line of reflection (i.e. the x-axis) is equal to the distance from every point from DEF to the line of reflection.

Question

In the diagram below, you can see a triangle ABC. If the line of reflection was x = 4, what would the co-ordinates be of an image of only point B?

Solution

(7, 5). Firstly, it is important to note that B is at (1, 5). Because we know that the line of reflection is x = 4 (a vertical line), that means there is no change in any point’s y coordinate, so we know that point B’s y coordinate continues to stay at 5. For the x coordinate, we know that the distance (horizontal distance) between B and the line of reflection is 4 – 1 = 3 units, so the reflected image should be the same distance away from the line of reflection, so 4 + 3 = 7 units. Therefore, the co-ordinates must be (7, 5).

Question

If the line of reflection is y = x, then find the correct image of triangle ABC from these options:

A)

B)

C)

D)

Solution

The answer is option (B). This is because for all respective points of ABC, the coordinates of image FGH are flipped. That is, for all points (x,y) in ABC, the image had coordinates (y,x), which is what happens when you flip an object over the y = x line.

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Practice Math Problems

1. Draw or label the coordinates of the image of a quadrilateral ABCD below if it was flipped across the x-axis:

2. Draw or label the coordinates of the image of a quadrilateral ABCD below if it was flipped across the y-axis:

3. Draw or label the coordinates of the image of a quadrilateral ABCD below if it was flipped across the x = -1 line of reflection:

4. Draw or label the coordinates of the image of a quadrilateral ABCD below if it was flipped across the y = 1 line of reflection:

FAQs on Reflection Rules – What are they? How can you use them?

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