Square root reflected across the y-axis

Statement

The curve $y = \sqrt{x}$ undergoes a symmetry with respect to the $y$-axis: find the equation of the transformed curve.

Solution

The symmetry with respect to the $y$-axis acts as $(x;y)\mapsto(-x;y)$. To obtain the equation of the transformed curve we replace $x$ with $-x$ in the original equation: $y = \sqrt{-x}.$ The curve $\boxed{y = \sqrt{-x}}$ is defined for $x \le 0$: it is the “branch” of the square root flipped into the half-plane of negative abscissae.

Links

Topics: Plane transformations
Concepts: Axial symmetry
Methods: Isometric transformations
Skills: Analytic geometry · Sketching a graph
Exercise type: Geometric problem