Traslazioni

The translation is the isometry that moves every point of the plane by the same amount and in the same direction, described by a vector.

Definition — Translation

Given a vector $\vec{v} = (a;b)$, the translation by vector $\vec{v}$ is the transformation $T_{\vec{v}}:\ (x;y) \mapsto (x';y') = (x + a;\ y + b).$

Property — Effect on curves

Given a curve of equation $f(x,y) = 0$, the equation of the curve translated by vector $(a;b)$ is obtained by replacing $x$ with $x-a$ and $y$ with $y-b$ in the original equation: $f(x-a,\ y-b) = 0.$ The “minus” is counterintuitive but correct: if we translate every point of the curve to the right by $a$, a new point $(x';y')$ belongs to the translated curve if and only if its “ancestor” $(x'-a;\ y'-b)$ belongs to the original curve.

Example

The parabola $y = x^2$ translated by $(+3;-1)$ becomes $y + 1 = (x-3)^2$, that is $y = (x-3)^2 - 1 = x^2 - 6x + 8.$ The vertex moves from $(0;0)$ to $(3;-1)$, as expected.

Links

Topics: Plane transformations
Concepts: Parabola · Translation · Vector
Methods: Isometric transformations
Skills: Analytic geometry · Sketching a graph