To find the image of the point [tex]\((-12, 4)\)[/tex] after a dilation by a scale factor of [tex]\(\frac{1}{4}\)[/tex] centered at the origin, follow these steps:
1. Identify the original coordinates of the point:
[tex]\[
(x, y) = (-12, 4)
\][/tex]
2. Identify the scale factor:
[tex]\[
k = \frac{1}{4}
\][/tex]
3. Apply the scale factor to the x-coordinate:
[tex]\[
x' = x \cdot k = -12 \cdot \frac{1}{4} = -3
\][/tex]
4. Apply the scale factor to the y-coordinate:
[tex]\[
y' = y \cdot k = 4 \cdot \frac{1}{4} = 1
\][/tex]
5. Combine the new coordinates to get the image of the point after dilation:
[tex]\[
(x', y') = (-3, 1)
\][/tex]
Therefore, the image of the point [tex]\((-12, 4)\)[/tex] after a dilation by a scale factor of [tex]\(\frac{1}{4}\)[/tex] centered at the origin is [tex]\((-3, 1)\)[/tex].
