How do you test for symmetry in an equation? For example, y = x to the fifth power + x to the third power + x?
To test for symmetry, first you need to know what kind of symmetry you want to test for. Is it Over-The-Origin Symmetry, or Over-The-X-Axis Symmetry, or Over-The-Y-Axis Symmetry?
First, you should rewrite your equation with new variables:
b = a^5 + a^3 + a
For ORIGIN SYMMETRY, substitute (-a,-b) for your x's and y's, respectively. Then solve your new equation to make it look like the equation above (so the b is not negative). Your solution should look like:
-b = (-a)^5 + (-3a)^3 + (-a)
b = -(-a^5) + -(-3a^3) + -(-a)
b = a^5 + 3a^3 + a
Because this equation is exactly the same with the equation above (b = a^5 + 3a^3 + a), it is symmetrical across the ORIGIN.
Now test it for X-axis and Y-axis using these variable substitutes.
X-Axis (a,-b)
Y-Axis (-a,b)
Make sure you double check your work!
Oops! I'm sorry for the typo, it should be (-a)^3 instead of (-3a)^3. All the typos after that are the same. My mistake, I'm sorry.
To test for symmetry in an equation, you can examine the symmetry properties of its graph. In the case of the equation y = x to the fifth power + x to the third power + x, we can analyze its symmetry by checking for two types: even symmetry (also called y-axis symmetry) and odd symmetry (also called origin symmetry).
1. Even symmetry (y-axis symmetry): To check for even symmetry, substitute -x for x in the equation and simplify. If the equation remains unchanged, it indicates even symmetry.
- Substitute -x for x in the given equation: y = (-x)^5 + (-x)^3 + (-x)
- Simplify the equation: y = -x^5 - x^3 - x
- Compare the original equation (y = x^5 + x^3 + x) with the equation after substituting -x (y = -x^5 - x^3 - x)
- If the two equations are identical, the original equation has even symmetry.
2. Odd symmetry (origin symmetry): To evaluate for odd symmetry, substitute -x for x in the equation and negate the result. If the equation turns into its original form upon negating, it suggests odd symmetry.
- Substitute -x for x in the given equation: y = (-x)^5 + (-x)^3 + (-x)
- Simplify the equation: y = -x^5 - x^3 - x
- Negate the result: -y = x^5 + x^3 + x
- Compare the original equation (y = x^5 + x^3 + x) with the negated equation (-y = x^5 + x^3 + x)
- If the two equations are identical, the original equation has odd symmetry.
In the case of the equation y = x^5 + x^3 + x, it does not have even symmetry because the original equation and the equation after substituting -x are not identical. Additionally, it does not possess odd symmetry because the original equation and the negated equation are not identical either. Hence, the equation y = x^5 + x^3 + x has neither even nor odd symmetry.