Image (Transformation)
The image is the resulting figure after a transformation has been applied to the original figure (preimage).
Definition
The image is what a shape looks like after a transformation is applied; the original shape is the preimage, and the resulting figure after moving, flipping, or resizing is the image, conventionally labeled with prime marks ($A'$, $B'$, $C'$) to show it came from the preimage. Formally, the image of a figure $F$ under transformation $T$ is the set $T(F) = \{T(P) : P \in F\}$; for rigid motions the image is congruent to the preimage, and for dilations it is similar. For a transformation $T: \mathbb{R}^2 \to \mathbb{R}^2$, the image of a linear map is its column space (range), and for general differentiable maps, the image near a regular point is approximated by the image of the linearization (the Jacobian); if the Jacobian is invertible at a point, the inverse function theorem guarantees the image is locally a smooth curve.
Example
Reflecting triangle $ABC$ across the y-axis makes $ABC$ the preimage and the new triangle $A'B'C'$ the image. Under translation $T(x,y) = (x+2, y-3)$, the image of the segment from $(1,4)$ to $(5,4)$ is the segment from $(3,1)$ to $(7,1)$, the same length ($4$ units) but a new position. The image of the unit circle under the linear map $T(x,y) = (2x, y)$ is the ellipse $\{(x,y) : (x/2)^2 + y^2 = 1\}$, with Jacobian $\begin{bmatrix}2&0\\0&1\end{bmatrix}$ and determinant $2$, so areas scale by factor $2$.
Key Insight
Think of "prime" as "after transformation": reading the prime mark tells you which original point it came from. The word "image" in transformation geometry is the same as in function notation: if $f(x) = 2x$, the image of $x=3$ is $f(3)=6$, and similarly if $T$ is a geometric transformation, the image of point $A$ is $T(A)$, geometry and algebra using identical language.