Function Symmetry 1506Cd

1. **State the problem:** Determine whether the function $$y = 2x^3$$ has symmetry about the x-axis, y-axis, origin, or none. 2. **Recall symmetry rules:** - **x-axis symmetry:** Replace $$y$$ by $$-y$$ and check if the equation remains the same. - **y-axis symmetry:** Replace $$x$$ by $$-x$$ and check if the equation remains the same. - **origin symmetry:** Replace $$x$$ by $$-x$$ and $$y$$ by $$-y$$ and check if the equation remains the same. 3. **Test x-axis symmetry:** Replace $$y$$ by $$-y$$: $$-y = 2x^3$$ Multiply both sides by $$-1$$: $$y = -2x^3$$ This is not the original equation $$y = 2x^3$$, so no x-axis symmetry. 4. **Test y-axis symmetry:** Replace $$x$$ by $$-x$$: $$y = 2(-x)^3 = 2(-x^3) = -2x^3$$ This is not the original equation $$y = 2x^3$$, so no y-axis symmetry. 5. **Test origin symmetry:** Replace $$x$$ by $$-x$$ and $$y$$ by $$-y$$: $$-y = 2(-x)^3 = 2(-x^3) = -2x^3$$ Multiply both sides by $$-1$$: $$y = 2x^3$$ This matches the original equation, so the function has origin symmetry. **Final answer:** The function $$y = 2x^3$$ has **origin symmetry**.