When applying a transformation to a curve, the key principle is to substitute the inverse transformation into the curve's equation.
For translation by (h,k): to find the equation of the curve in the new system, replace x with X+h and y with Y+k. For rotation by theta: replace x with Xcos(theta)-Ysin(theta) and y with Xsin(theta)+Ycos(theta).
To find what the curve "looks like" in the new coordinate system: substitute X=xcos(theta)+ysin(theta) and Y=-xsin(theta)+ycos(theta) (the forward transformation) to express the curve equation in terms of X,Y.
Important distinction: "find the equation of the curve after transformation" vs "find the transformed equation." Both mean expressing the same geometric curve using the new coordinates.
Composite transformations: when both rotation and translation are needed, the standard approach is (1) rotate first to eliminate the xy-term, then (2) translate to remove linear terms. The order matters -- performing them in reverse order generally does not achieve the same simplification.
For reflecting a curve across a line: replace (x,y) in the curve equation with the reflection formulas. Since reflection is an involution (its own inverse), the forward and inverse transformations are identical.
Invariance check: after any transformation, verify that the invariants (a+b, , Delta) match the original equation. If they differ, a computational error has occurred.