Graph Symmetry 2420F6

1. **State the problem:** We need to determine if the graph of the equation $$y=-7x^3+9x$$ is symmetric with respect to the $x$-axis, $y$-axis, and the origin. 2. **Recall symmetry rules:** - Symmetry about the $x$-axis: Replace $y$ by $-y$ and check if the equation remains unchanged. - Symmetry about the $y$-axis: Replace $x$ by $-x$ and check if the equation remains unchanged. - Symmetry about the origin: Replace $x$ by $-x$ and $y$ by $-y$ and check if the equation remains unchanged. 3. **Test for $x$-axis symmetry:** Replace $y$ by $-y$: $$-y = -7x^3 + 9x$$ Multiply both sides by $-1$: $$y = 7x^3 - 9x$$ This is not the original equation, so no symmetry about the $x$-axis. 4. **Test for $y$-axis symmetry:** Replace $x$ by $-x$: $$y = -7(-x)^3 + 9(-x) = -7(-x^3) - 9x = 7x^3 - 9x$$ This is not the original equation, so no symmetry about the $y$-axis. 5. **Test for origin symmetry:** Replace $x$ by $-x$ and $y$ by $-y$: $$-y = -7(-x)^3 + 9(-x) = -7(-x^3) - 9x = 7x^3 - 9x$$ Multiply both sides by $-1$: $$y = -7x^3 + 9x$$ This matches the original equation. 6. **Conclusion:** The graph is symmetric with respect to the origin only.