To determine whether an equation is symmetric with respect to the origin, we need to check if replacing both variables with their negatives results in the same equation. In simpler terms, if you have an equation f(x, y)=0, it should hold that f(-x, -y) = 0 for the equation to be symmetric with respect to the origin.
For example, consider the equation y = x^3 – x.
- Original equation: y = x^3 – x
- Substituting -x for x: y = (-x)^3 – (-x) = -x^3 + x
As you can see, this does not yield the original equation; thus, it is not symmetric with respect to the origin.
On the other hand, the equation y = x^2 can be examined similarly:
- Original equation: y = x^2
- Substituting -x for x: y = (-x)^2 = x^2
This confirms that the equation remains unchanged, indicating symmetry about the y-axis rather than the origin. Therefore, while this method helps identify symmetrical graphs, keep in mind that different forms like y = -x^3 could also exhibit origin symmetry under specific transformations.
In conclusion, to find which equation maintains origin symmetry, systematically check the mathematical behavior under substitution. The key is consistency in the transformation outcome.