One common task that we learn when studying geometry is how to dilate a graph. However, we can do the same task with matrices. Matrices are highly versatile tools that can represent a broad range of mathematical structures. For instance, matrices are used heavily in animation and computer graphics.
Dilation is one of the transformations that we can perform on a two-dimensional shape. Dilation involves changing the size of the shape. In order to perform dilation on a shape, we need a center point and a scale factor. The scale factor is essentially the ratio by which a figure is reduced or enlarged. Let''s say we have a triangle , where is at the point , at , and at . We want to dilate this figure by a scale factor of two to get
The points of each vertex can be inserted into a matrix like so.
As we can see, each column in this matrix represents an individual coordinate in the triangle. Since we want to dilate this figure by a scale factor of two, we need to use scalar multiplication. All this involves is multiplying each matrix entry by the scale factor.
Therefore, the vertices of are
Let''s try another one, but this time we''ll scale it down. If we have a triangle where , , and , let's dilate it by a scale factor of .
We''ll put these coordinates in the form of a matrix, then we''ll multiply each entry by the scale factor.
Therefore, the new coordinates are .
Now that we know how to dilate an object using a matrix, let''s try some practice problems. It may be tempting to look at the answers ahead of time, but it''s a good idea to give the problem a try before doing so.
The dilated coordinates will be .
The dilated coordinates will be .
The dilated coordinates of will be .
The dilated coordinates will be .
The dilated coordinates will be .
The dilated coordinates will be .
Transformation of Graphs using Matrices
If you''re struggling with a mathematical topic like matrices, consider working with a math tutor. An Educational Director at Varsity Tutors will choose a tutor who aligns with your academic needs. Not only that, but they will create a custom learning plan that you and your tutor can follow throughout the lessons. It''s okay if you want to fine-tune this plan as you go along. A tutor can also give you practice problems to solidify your understanding of the material. If there''s a particular type of problem that you''re having difficulty with, the tutor can guide you through example problems until you''ve got the gist of it. Also, if you have a student who is struggling in math, a tutor will give them the assistance they need. Although math can be a frustrating subject for many students, there is help available. Contact Varsity Tutors today to be paired with a quality math tutor.