Problem 19 - Algebra 2

Question

Write the vertex matrix and the reflection matrix for each figure. Then find the coordinates of the image after the reflection. Graph the preimage and the image on a coordinate plane. The vertices of rectangle \(A B D C\) are \(A(-3,5), B(5,5), D(5,-1),\) and \(C(-3,-1) .\) The rectangle is reflected over the \(x\) -axis.

Step-by-Step Solution

Answer

Reflection coordinates: \(A'(-3,-5)\), \(B'(5,-5)\), \(D'(5,1)\), \(C'(-3,1)\).

Step 1: Write the Vertex Matrix

Create a vertex matrix from the given coordinates of rectangle \(ABDC\). The vertex matrix will have the form:\[\begin{bmatrix}-3 & 5 & 5 & -3 \5 & 5 & -1 & -1\end{bmatrix}\]

Step 2: Write the Reflection Matrix

For reflection over the \(x\)-axis, the reflection matrix is:\[\begin{bmatrix}1 & 0 \0 & -1\end{bmatrix}\]

Step 3: Calculate the Image Matrix

Multiply the vertex matrix by the reflection matrix to obtain the image matrix:\[\begin{bmatrix}1 & 0 \0 & -1\end{bmatrix}\begin{bmatrix}-3 & 5 & 5 & -3 \5 & 5 & -1 & -1\end{bmatrix}=\begin{bmatrix}-3 & 5 & 5 & -3\-5 & -5 & 1 & 1\end{bmatrix}\]

Step 4: Identify the Coordinates of the Image

The coordinates of the image after reflection are:- \(A'(-3,-5)\)- \(B'(5,-5)\)- \(D'(5,1)\)- \(C'(-3,1)\)

Step 5: Graph the Preimage and the Image

Draw the preimage rectangle \(ABDC\) with the vertices \(A(-3,5), B(5,5), C(-3,-1), D(5,-1)\) on a coordinate plane. Then draw the image rectangle \(A'B'D'C'\) with the vertices \(A'(-3,-5), B'(5,-5), C'(-3,1), D'(5,1)\) on the same coordinate plane. The image should be a reflection across the \(x\)-axis.

Key Concepts

Vertex Matrix

A vertex matrix is a simple and useful tool in geometry. It helps us organize coordinates of a figure's vertices clearly and efficiently. To create the vertex matrix for rectangle \(ABDC\), simply arrange the x-coordinates in one row and the y-coordinates in another row. In this exercise, the vertex matrix is constructed from the vertices \(A(-3,5)\), \(B(5,5)\), \(D(5,-1)\), and \(C(-3,-1)\). It looks like this:

• First row: x-coordinates \([-3, 5, 5, -3]\)
• Second row: y-coordinates \([5, 5, -1, -1]\)

The vertex matrix combines these into:\[\begin{bmatrix}-3 & 5 & 5 & -3 \5 & 5 & -1 & -1\end{bmatrix}\] This structure is our foundation for applying transformations, such as reflections.

Reflection Matrix

In geometry, reflection is a flip over a line, in this case, the x-axis. To reflect any figure over the x-axis in an algebraic manner, we use a reflection matrix. The essential reflection matrix for over the x-axis is:

• First row stays the same \([1, 0]\)
• Second row is inverted \([0, -1]\)

This results in the reflection matrix as follows:\[\begin{bmatrix}1 & 0 \0 & -1\end{bmatrix}\]This matrix works by keeping x-coordinates intact while reversing the sign of the y-coordinates. By multiplying the vertex matrix of our rectangle with this reflection matrix, we transform the original figure into its mirrored counterpart across the x-axis.

Image Coordinates

Once we have our reflection matrix, we use matrix multiplication to find the image coordinates. This is where understanding how matrices interact comes in handy. We multiply each column of the vertex matrix by the reflection matrix:

• The first element of the new y-coordinate becomes its negative counterpart, flipping it across the x-axis.

For rectangle \(ABDC\), after performing this multiplication:\[\begin{bmatrix}1 & 0 \0 & -1\end{bmatrix}\begin{bmatrix}-3 & 5 & 5 & -3 \5 & 5 & -1 & -1\end{bmatrix}= \begin{bmatrix}-3 & 5 & 5 & -3 \-5 & -5 & 1 & 1\end{bmatrix}\]As a result, we get the new vertices coordinates:

• \(A'(-3,-5)\)
• \(B'(5,-5)\)
• \(D'(5,1)\)
• \(C'(-3,1)\)

These coordinates reflect the flipped positions of the vertices over the x-axis.

Coordinate Plane Graphing

Graphing is a visual way of understanding transformations. By plotting both the preimage and image on a coordinate plane, we can clearly see exactly how the rectangle \(ABDC\) reflects over the x-axis. Here's how you can do this:

• First, plot the original vertices \(A(-3,5)\), \(B(5,5)\), \(C(-3,-1)\), \(D(5,-1)\) to form rectangle \(ABDC\).
• Next, plot its transformed vertices, \(A'(-3,-5)\), \(B'(5,-5)\), \(C'(-3,1)\), \(D'(5,1)\) to form the reflected rectangle \(A'B'D'C'\).

The preimage and image should look like mirror images across the x-axis. This is a practical visual demonstration of the mathematical reflection we calculated using matrices. It helps to reinforce the concept of symmetry and the role of axes as mirrors in coordinate graphing.