Question

reflect the given triangle over the y - axis.
$\

$$\begin{bmatrix} 3&6&3\\ -3&3&3\\ \\end{bmatrix}$$

$
$\

$$\begin{bmatrix} -3&?&\\ \\ -3&&\\ \\end{bmatrix}$$

$

Explanation:

Step1: Recall reflection over y - axis rule

When reflecting a point \((x,y)\) over the \(y\) - axis, the transformation rule is \((x,y)\to(-x,y)\). For a matrix representing points \(

$$\begin{bmatrix}x_1&x_2&x_3\\y_1&y_2&y_3\end{bmatrix}$$

\), after reflection over the \(y\) - axis, it becomes \(

$$\begin{bmatrix}-x_1&-x_2&-x_3\\y_1&y_2&y_3\end{bmatrix}$$

\).

Step2: Apply the rule to the given matrix

The original matrix is \(

$$\begin{bmatrix}3&6&3\\- 3&3&3\end{bmatrix}$$

\). For the \(x\) - values (first row), we change their signs. The \(y\) - values (second row) remain the same. So the reflected matrix is \(

$$\begin{bmatrix}-3&-6&-3\\-3&3&3\end{bmatrix}$$

\).

Answer:

The value in the green box (corresponding to the second \(x\) - value after reflection) is \(-6\), the value below the green box (corresponding to the second \(y\) - value, which remains unchanged) is \(3\), and the other values (third column) are \(-3\) (for \(x\)) and \(3\) (for \(y\)). The completed reflected matrix is \(

$$\begin{bmatrix}-3&-6&-3\\-3&3&3\end{bmatrix}$$

\).