Determine whether the equation has a graph that is symmetric with respect to the y-axis, the x-axis, the origin, or none of
these.
-$$y = - 5 x ^ { 3 } + 5 x$$
A) $x$-axis only
B) origin only
C) $x$-axis, $y$-axis, origin
D) $y$-axis only

Question

Quizplus · Accepted Answer

A graph is symmetric with respect to the origin if replacing $x$ with $-x$ and $y$ with $-y$ yields the original equation. For $y = -5x^3 + 5x$, replacing $x$ with $-x$ gives $y = -5(-x)^3 + 5(-x) = -(-5x^3 + 5x) = 5x^3 - 5x$, which is the original equation negated ($-y$). This confirms symmetry with respect to the origin.

Determine Whether the Equation Has a Graph That Is Symmetric y=−5x3+5xy = - 5 x ^ { 3 } + 5 xy=−5x3+5x

Determine Whether the Equation Has a Graph That Is Symmetric $y = - 5 x ^ { 3 } + 5 x$