Reflections are some of the most interesting transformations we can carry out on a graph. As the name implies, this creates a "mirror image" of our preimage, and it can be useful in a number of different circumstances. But can we reflect images using matrices instead of our usual algebraic methods? Let''s find out:
As we may recall, a reflection is a type of translation that involves "flipping" an image. An easy example of a reflection is picking up a paper star, flipping it over, and putting it back down on a table. Unlike a few other transformations, reflection does not alter an image''s size, shape, or orientation.
We also know that a reflection maps every point of a figure to an image across a line of symmetry.
But what if we wanted to reflect our image using matrices instead?
We use three reflection matrices to accomplish this:
-- Reflection over the x-axis.
-- Reflection over the y-axis.
-- Reflection over the line
We always place these reflection matrices on the left for our multiplication operations.
Let''s say we have a pentagon with the following coordinates:
Can we use matrices to reflect this pentagon over the y-axis?
We can start by writing out our coordinate points in matrix form. Note that the x-coordinates go in row 1, while the y-coordinates go in row 2.
Our next step is to select the correct reflection matrix. Recall that in order to reflect over the y-axis, we need to use this reflection matrix:
Now let''s put these two matrices together into a multiplication equation -- remembering to put our reflection matrix on the left:
This last resulting matrix represents the coordinates of the reflected image. Here are those coordinates:
What does this reflected pentagon look like compared to the original image? Let''s take a look:
Transformation of Graphs using Matrices - Rotations
Scalar Multiplication of Matrices
Reflections with matrices follow the same basic principles that students are familiar with. But there are a few important changes -- and these changes can easily catch students out. A solid choice is to revisit these new concepts alongside a tutor during 1-on-1 sessions. Tutors can answer questions you or your student didn''t have time to ask during class. They can also try a range of different explanations until a student finally "gets it." Speak with our Educational Directors today to learn more, and rest assured: Varsity Tutors will pair you student with an excellent tutor.