Translated parabola

Statement

Write the equation of the parabola $y = 2x^2 - 3$ translated by vector $(1;4)$.

Solution

To translate a curve by vector $(a;b)=(1;4)$ we replace $x$ with $x-1$ and $y$ with $y-4$ in the equation: $y - 4 = 2(x-1)^2 - 3.$ Isolating $y$: $y = 2(x-1)^2 - 3 + 4 = 2(x-1)^2 + 1.$ Expanding: $y = 2(x^2 - 2x + 1) + 1 = \boxed{2x^2 - 4x + 3}.$ The vertex moves from $(0;-3)$ to $(1;1)$, consistently with the translation by vector $(1;4)$.

Links

Topics: Plane transformations
Concepts: Parabola · Translation
Methods: Isometric transformations
Skills: Analytic geometry · Using formulae
Exercise type: Geometric problem