Lesson 7 Transformations with Matrices Solidify Understanding

Jump Start

Multiply the two matrices together to find the product matrix:

$[\begin{matrix} 3 & 2 \\ 2 & 1 \end{matrix}] \cdot [\begin{matrix} 2 & 3 & 2 \\ 4 & 2 & 1 \end{matrix}]$

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: $v$ ; a variable written with a harpoon over it, $\vec{v}$ ; or listing the horizontal and vertical components of the vector, $⟨ v_{x}, v_{y} ⟩$ . In this task we will represent vectors by listing their horizontal and vertical components in a matrix with a single column, $[\begin{matrix} v_{x} \\ v_{y} \end{matrix}]$ .

1.

Represent the vector labeled $\vec{v}$ in the diagram as a matrix with one column.

Matrix multiplication can be used to transform vectors and images in a plane.

Suppose we want to reflect $\vec{w}$ over the $y$ -axis. We can represent $\vec{w}$ with the matrix $[\begin{matrix} 2 \\ 3 \end{matrix}]$ , and the reflected vector with the matrix $[\begin{matrix} - 2 \\ 3 \end{matrix}]$ .

2.

Find the $2 \times 2$ matrix whose entries consist only of $0$ , $1$ , and $- 1$ , so that when it is multiplied by the matrix representing the original vector, the result represents the reflected vector. That is, find $a$ , $b$ , $c$ , and $d$ such that $[\begin{matrix} a & b \\ c & d \end{matrix}] \cdot [\begin{matrix} 2 \\ 3 \end{matrix}] = [\begin{matrix} - 2 \\ 3 \end{matrix}]$ .

3.

Find the matrix whose entries consist only of $0$ , $1$ , and $- 1$ that will reflect $\vec{w}$ over the $x$ -axis.

4.

Find the matrix whose entries consist only of $0$ , $1$ , and $- 1$ that will rotate $\vec{w}$ $90 °$ counterclockwise about the origin.

5.

Find the matrix whose entries consist only of $0$ , $1$ , and $- 1$ that will rotate $\vec{w}$ $180 °$ counterclockwise about the origin.

6.

Find the matrix whose entries consist only of $0$ , $1$ , and $- 1$ that will rotate $\vec{w}$ $270 °$ counterclockwise about the origin.

7.

Is there another way to obtain a rotation of $270 °$ counterclockwise about the origin other than using the matrix found in problem 6? If so, how?

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 $[\begin{matrix} 2 & 5 & 6 \\ 3 & 7 & 4 \end{matrix}]$ .

8.

Multiply the matrix which represents the vertices of $A B C$ , by the matrix found in problem 2. Interpret the product matrix as representing the coordinates of the vertices of another triangle in the plane. Plot these points, and sketch the triangle. How is this new triangle related to the original triangle?

9.

How might you find the coordinates of the triangle that is formed after $A B C$ is rotated $90 °$ counterclockwise about the origin using matrix multiplication? Find the coordinates of the rotated triangle.

10.

How might you find the coordinates of the triangle that is formed after $A B C$ is reflected over the $x$ -axis using matrix multiplication? Find the coordinates of the reflected triangle.

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 $x$ -axis, I use:

To reflect over the $y$ -axis, I use:

To rotate $90 °$ , I use:

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.

Retrieval

1.

Find the solution to the system of equations.

${\begin{matrix} - 2 x + 5 y = 10 \\ 8 x + 2 y = - 18 \end{matrix}$

For problems 2–3, use $f (x) = 3 x - 4$ .

2.

Find $g (x)$ if $g (x) = f (x) + 11$ .

3.

Describe the relationship between $f (x)$ and $g (x)$ .