Question

Algebraically determine if the relation y = x3 − 4x is symmetric with respect to the x-axis, y-axis, the origin or has no symmetry.

Answer 1

$\bf y=x^3-4x \\\\\\ \textit{check for y-axis symmetry}\\\\ \stackrel{x=-x}{y=(-x)^3-4(-x)}\implies y=(-x)(-x)(-x)+4x\implies \boxed{y=-x^3+4x} \\\\\\ \textit{checking for x-axis symmetry}\\\\ \stackrel{y=-y}{(-y)=x^3-4x}\implies -y=x^3-4x\implies \boxed{y=-x^3+4x} \\\\\\ \textit{checking for origin symmetry}\\\\ \stackrel{x=-x\qquad y=-y}{(-y)=(-x)^3-4(-x)}\implies -y=(-x)(-x)(-x)+4x \\\\\\ -y=-x^3+4x\implies y=\cfrac{-x^3+4x}{-1}\implies \boxed{y=x^3-4x}\quad \checkmark$

notice, if you replace x = -x and y = -y accordingly for each test for symmetry, if the resulting function, is equal to the original function, then it has that type of symmetry.