To find the pre-image of the point given the transformation rule, follow these steps:
1. Understand the transformation rule:
The rule provided is [tex]\((x, y) \rightarrow (x-6, y+8)\)[/tex].
2. Identify the image point:
The image point is [tex]\((-4, 1)\)[/tex].
3. Set up equations to reverse the transformation:
The transformation changes the point [tex]\((x, y)\)[/tex] to [tex]\((x-6, y+8)\)[/tex].
Given [tex]\((-4, 1)\)[/tex] is the image, we need to find the original point [tex]\((x, y)\)[/tex].
This can be expressed as:
[tex]\[(x-6, y+8) = (-4, 1)\][/tex]
4. Solve for [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
We set up two equations based on the components of the transformation:
[tex]\[x - 6 = -4\][/tex]
[tex]\[y + 8 = 1\][/tex]
For the first equation:
[tex]\[x - 6 = -4 \implies x = -4 + 6 \implies x = 2\][/tex]
For the second equation:
[tex]\[y + 8 = 1 \implies y = 1 - 8 \implies y = -7\][/tex]
5. Determine the pre-image coordinates:
The pre-image point found from solving the equations is [tex]\((2, -7)\)[/tex].
6. Check the options provided:
[tex]\[
\begin{align*}
&(-10, 9) \\
&(2, -7) \\
&(-2, 7) \\
&(10, -9)
\end{align*}
\][/tex]
From our solution, the pre-image point [tex]\((2, -7)\)[/tex] is indeed one of the options provided.
Therefore, the pre-image point is [tex]\((2, -7)\)[/tex].
