Problem 75 - Algebra 2

Question

Each matrix represents the vertices of a polygon. Translate each figure 3 units left and 2 units down. Express your answer as a matrix. $$ \left[\begin{array}{rrrr}{1} & {2} & {1} & {2} \\ {-1} & {-1} & {-2} & {-2}\end{array}\right] $$

Step-by-Step Solution

Answer

\left[\begin{array}{rrrr}-2 & -1 & -2 & -1 \ -3 & -3 & -4 & -4\end{array}\right]

Step 1: Identify the Translation Matrix

To translate a figure 3 units left and 2 units down, you create a translation matrix that will be added to the original coordinates. Since 3 units left is a horizontal shift, it corresponds to the x-values of the vertices. 2 units down is a vertical shift, affecting the y-values. Thus, the translation matrix is \[\left[\begin{array}{rrrr}{-3} & {-3} & {-3} & {-3} \ {-2} & {-2} & {-2} & {-2}\end{array}\right]\]

Step 2: Add the Translation Matrix to the Original Matrix

Add the translation matrix to the original coordinates matrix to get the new matrix of translated coordinates. \[\left[\begin{array}{rrrr}1 & 2 & 1 & 2 \ -1 & -1 & -2 & -2\end{array}\right]\] + \[\left[\begin{array}{rrrr}-3 & -3 & -3 & -3 \ -2 & -2 & -2 & -2\end{array}\right]\] equals \[\left[\begin{array}{cc}1-3 & 2-3 & 1-3 & 2-3 \ -1-2 & -1-2 & -2-2 & -2-2\end{array}\right]\]

Step 3: Calculate the Resulting Matrix

Perform the addition of the two matrices to find the coordinates of the translated polygon. \[\left[\begin{array}{rrrr}-2 & -1 & -2 & -1 \ -3 & -3 & -4 & -4\end{array}\right]\] This is the resulting matrix that represents the new vertices of the translated polygon.

Key Concepts

Translation Matrix

Imagine picking up a shape and moving it to a new position on a plane without rotating it or changing its size. This action is precisely what a translation matrix does in coordinate geometry. This special type of matrix is used to move figures in a two-dimensional plane by a certain amount horizontally (left/right) and vertically (up/down).

When you need to perform a translation, like moving a polygon 3 units to the left and 2 units down, you create a translation matrix that reflects these changes. Here’s how it’s done: each entry in the translation matrix corresponds to the change you want to make. For a leftward and downward move, we use negative values. Thus, for our exercise, the translation matrix is created with \textbf{all} entries being -3 for the horizontal shift and -2 for the vertical shift, as follows:
\[\begin{equation}\left[\begin{array}{cccc}-3 & -3 & -3 & -3 \-2 & -2 & -2 & -2 \end{array}\right]\end{equation}\]
This matrix, when added to the vertices of our shape, will give us the new positions of these points after the translation.

Matrix Addition

In order to obtain the new coordinates after a translation, we perform an operation known as matrix addition. Matrix addition is possible when we have two matrices of the same size; that is, they must have the same number of rows and columns. The process involves adding corresponding entries from each matrix to form a new matrix.

To illustrate, when we take our original matrix of coordinates and our translation matrix:\[\begin{equation}\left[\begin{array}{cccc}1 & 2 & 1 & 2 \-1 & -1 & -2 & -2 \end{array}\right]+\left[\begin{array}{cccc}-3 & -3 & -3 & -3 \-2 & -2 & -2 & -2 \end{array}\right]\end{equation}\]
We add each corresponding entry (e.g., top-left with top-left, top-right with top-right, and so on) to find the vertices of the translated shape. The resulting matrix will represent the new, translated shape. It's like moving every point of our original shape following the guidelines set by our translation matrix, one coordinate at a time.

Coordinate Transformation

Coordinate transformation is the process of changing a figure’s coordinates to represent the figure's movement or alteration within a space. The translation we discussed is a simple form of coordinate transformation, where points are shifted in a consistent direction by a fixed amount.

In our example, when applying the translation matrix to the original matrix, we are actually transforming the coordinates of the polygon from one location to another within our plane, adhering to the instructions of moving 3 units left and 2 units down. Coordinate transformations can be more complex, involving rotations, scaling (resizing), or reflection in addition to translation. However, for a straightforward translation, as we've seen, the transformation is simply a matter of applying a linear shift to each coordinate point of the figure.