Lesson 13 Transformations with Matrices Solidify Understanding
Jump Start
Multiply the two matrices together to find the product matrix:
Learning Focus
Use matrices to perform geometric transformations on figures.
How can matrix operations be used to perform reflections and rotations on a coordinate grid?
Open Up the Math: Launch, Explore, Discuss
Various notations are used to denote vectors:
1.
Represent the vector labeled
Matrix multiplication can be used to transform vectors and images in a plane.
Suppose we want to reflect
2.
Find the
3.
Find the matrix whose entries consist only of
4.
Find the matrix whose entries consist only of
5.
Find the matrix whose entries consist only of
6.
Find the matrix whose entries consist only of
7.
Is there another way to obtain a rotation of
Pause and Reflect
We can represent polygons in the plane by listing the coordinates of its vertices as columns of a matrix. For example, the triangle below can be represented by the matrix
8.
Multiply the matrix which represents the vertices of
9.
How might you find the coordinates of the triangle that is formed after
10.
How might you find the coordinates of the triangle that is formed after
Ready for More?
In this task, we used matrices to reflect or rotate a geometric figure. Devise a strategy for translating a geometric figure using matrix operations. Illustrate your strategy by drawing a quadrilateral in the first quadrant of a coordinate grid and labeling its vertices. Predict what the coordinates of the vertices would be if the quadrilateral is translated by a specific vector. Then illustrate how you could use matrices to determine the coordinates of the translated figure.
Takeaways
I can use matrices to reflect or rotate a vector, such as,
To reflect over the
To reflect over the
To rotate
I can perform other transformations by
Lesson Summary
In this lesson, we learned how to use matrix multiplication to rotate the vertices of geometric figures around the origin on the coordinate grid, and to reflect figures across either of the axes.
1.
Find the measure of the angle marked by an
For problems 2–3, use
2.
Find
3.
Describe the relationship between