Symmetry
Symmetry is how we're able to determine whether a Power Function ,\(f(x) = x^n\), is even or odd.
Even Functions
Even Power Functions contain even whole number exponents.
Even Functions utilize Line Symmetry, where they remain unchanged when reflected in the \(y\)-axis.
The graph of the function \(x^2\) represents an even function:
As you can see, there is an equal distance between the positive and negative sides of the function in relation to the \(y\)-axis.
Additionally, you can determine if a function is even algebraically as such:
We can verify this using the function \(x^3\):
\((-x)^2 = x^2\)
\(x^2 = x^2\)
Odd Functions
Odd Power Functions contain odd whole number exponents.
Odd Functions utilize Point Symmetry (or rotational symmetry), where they remain unchanged when rotated \(180^{\circ}\).
The graph of the function \(x^3\) represents an even function:
As you can see, there is an equal distance between the positive and negative sides of the function in relation to the origin.
Additionally, you can determine if a function is even algebraically as such:
We can verify this using the function \(x^3\):
\((-x)^3 = -x^3\)
\(-x^3 = -x^3\)
\(h(x) = 2x^5\)
Since this is an odd-degree polynomial (\(5\)), we will check for odd symmetry first:
Next, we can plug the values into the function:
Then, we can simplify:
Since both sides are equal, \(h(x)\) is odd.
\(g(x) = -\cfrac{1}{3}x^8\)
Since this is an odd-degree polynomial (\(5\)), we will check for odd symmetry first:
Next, we can plug the values into the function:
Then, we can simplify:
Since both sides are equal, \(g(x)\) is even.