Certainly! Let's determine the preimage of the point [tex]\((-4, 1)\)[/tex] under the given transformation rule [tex]\((x, y) \rightarrow (x - 6, y + 8)\)[/tex].
We need to reverse this transformation to find the original point, or preimage [tex]\((x, y)\)[/tex].
Given:
[tex]\[ (x, y) \rightarrow (x - 6, y + 8) \][/tex]
[tex]\[ \text{Image point} = (-4, 1) \][/tex]
Let’s denote the preimage by [tex]\((x_0, y_0)\)[/tex]. According to the transformation:
1. The x-coordinate of the image point is given by:
[tex]\[ -4 = x_0 - 6 \][/tex]
2. The y-coordinate of the image point is given by:
[tex]\[ 1 = y_0 + 8 \][/tex]
To find [tex]\(x_0\)[/tex]:
[tex]\[ -4 = x_0 - 6 \][/tex]
Adding 6 to both sides:
[tex]\[ -4 + 6 = x_0 \][/tex]
[tex]\[ x_0 = 2 \][/tex]
To find [tex]\(y_0\)[/tex]:
[tex]\[ 1 = y_0 + 8 \][/tex]
Subtracting 8 from both sides:
[tex]\[ 1 - 8 = y_0 \][/tex]
[tex]\[ y_0 = -7 \][/tex]
Thus, the preimage point is [tex]\((2, -7)\)[/tex].
So, the answer is:
[tex]\[ (2, -7) \][/tex]
