4. A triangle has vertices (–2, –3), (3, 5), and (8, –1).
a. Write a matrix expression to represent reflecting the triangle across the x-axis. Evaluate the matrix expression and list the coordinates of the vertices of the transformed figure.
|+1 +0 | |-2 +3 +8 |
|+0 -1 | |-3 +5 -1 |
=
| -2 +3 +8 |
| +3 -5 +1 |
so
(-2,3), (3,-5), (8,1)
To reflect a point across the x-axis, we need to change the sign of its y-coordinate while keeping the x-coordinate unchanged.
The matrix representation for reflecting a point across the x-axis can be written as:
\begin{bmatrix}
1 & 0 \\
0 & -1 \\
\end{bmatrix}
To find the transformed coordinates, we can multiply this reflection matrix by each vertex of the triangle.
Let's evaluate the matrix expression for each vertex:
For the vertex (-2, -3):
\begin{bmatrix}
1 & 0 \\
0 & -1 \\
\end{bmatrix}
\begin{bmatrix}
-2 \\
-3 \\
\end{bmatrix}
=
\begin{bmatrix}
-2(1) + 0(-3) \\
0(-2) + (-1)(-3) \\
\end{bmatrix}
=
\begin{bmatrix}
-2 \\
3 \\
\end{bmatrix}
So, the transformed coordinates for the vertex (-2, -3) after reflecting across the x-axis are ( -2, 3).
Similarly, for the vertex (3, 5):
\begin{bmatrix}
1 & 0 \\
0 & -1 \\
\end{bmatrix}
\begin{bmatrix}
3 \\
5 \\
\end{bmatrix}
=
\begin{bmatrix}
3(1) + 0(5) \\
0(3) + (-1)(5) \\
\end{bmatrix}
=
\begin{bmatrix}
3 \\
-5 \\
\end{bmatrix}
So, the transformed coordinates for the vertex (3, 5) after reflecting across the x-axis are ( 3, -5).
And for the vertex (8, -1):
\begin{bmatrix}
1 & 0 \\
0 & -1 \\
\end{bmatrix}
\begin{bmatrix}
8 \\
-1 \\
\end{bmatrix}
=
\begin{bmatrix}
8(1) + 0(-1) \\
0(8) + (-1)(-1) \\
\end{bmatrix}
=
\begin{bmatrix}
8 \\
1 \\
\end{bmatrix}
So, the transformed coordinates for the vertex (8, -1) after reflecting across the x-axis are ( 8, 1).
Therefore, the vertices of the transformed figure after reflecting the triangle across the x-axis are ( -2, 3), (3, -5), and (8, 1).