Whether you are just starting to explore reflections in math or need to brush up on your geometry, this simple, middle-school-friendly guide is for you.
Read on to find easy-to-follow definitions and explanations, solved examples, and resources to help you learn and master reflections.
Reflection in mathematics is a geometric transformation where a shape or object is flipped across a line, known as the line of reflection. This results in a mirror image that is the same size and shape as the original but appears flipped or mirrored.
You encounter the concept of reflection every time you look in a mirror.
Try it now:
Step in front of a mirror and raise your right hand.
What hand is your mirror image raising?
If you raise your right hand in front of a mirror, your reflection, i.e. your mirror image, raises its left hand.
Reflection is one of the 4 types of transformations in geometry .
Other types of geometrical transformations are:
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To truly understand what reflection is and to distinguish it from other geometrical transformations, we need to know its properties.
Reflection has 5 key properties:
If any of these five properties are missing, the geometrical transformation is either not a reflection at all or is a combination of reflection and other transformations.
Let’s go over some language we use to talk about reflections.
We use these terms to explain how reflection works:
To reflect a point or figure on the coordinate plane, we sometimes use the X-axis or Y-axis as the line of reflection.
Let’s see how each type of reflection works.
Reflection over the x-axis is a transformation where each point in a shape or a graph is flipped across the x-axis.
If you have a point (x, y), reflecting it over the x-axis will give you the point (x, -y).
In other words, the x-coordinate (how far left or right the point is) stays the same, but the y-coordinate (how far up or down the point is) becomes its opposite.
For example, if you have the point (2, 3), reflecting it over the x-axis would give you (2, -3), because the x-coordinate remains 2, but the y-coordinate changes from 3 to -3, flipping it across the x-axis.
In reflection over the Y-axis, each point in a shape or graph is mirrored horizontally along the Y-axis.
When you have a point (x, y), reflecting it over the Y-axis gives you the point (-x, y). To put it simply, the y-coordinate remains unchanged, but the x-coordinate changes to its negative value.
For example, let's take the point (4, -5). Reflecting it over the Y-axis results in (-4, -5).
Notice that while the y-coordinate remains -5, the x-coordinate changes from 4 to -4 as it mirrors along the Y-axis.
Here’s a visual representation of reflection over the Y-axis.
The line y = x represents all the points where the y-coordinate is equal to the x-coordinate.
It's a diagonal line that passes from the bottom left corner to the top right corner through the origin at a 45-degree angle.
Reflection over the y = x line means flipping a point or shape across this diagonal line.
During this reflection, the x-coordinate of each point becomes its y-coordinate, and the y-coordinate becomes its x-coordinate.
For example, if you have a point (x, y), its reflection over the y = x line would be (y, x).
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