### Instruction: The dilation, centered at $-1 + 4i,$ with scale factor $-2,$ takes $2i$ to which complex number?### Response: A dilation is a transformation that scales every point by a given factor, preserving the angles and proportions of the original figure.
To find the image of a point under a dilation, I can use the formula $z' = a(z - c) + c,$ where $z$ is the original point, $z'$ is the image point, $a$ is the scale factor, and $c$ is the center of dilation.
In this problem, $z = 2i,$ $a = -2,$ and $c = -1 + 4i.$
So I can plug these values into the formula and simplify.
$z' = -2(2i - (-1 + 4i)) + (-1 + 4i)$
$z' = -2(2i + 1 - 4i) - 1 + 4i$
$z' = -2(-2i + 1) - 1 + 4i$
$z' = 4i - 2 - 1 + 4i$