Please check my answers. Also, can I see some graphs of y-axis symmetry, orgin of symmetry, or neither?
Determine whether the graph of the polynomial has y-axis symmetry, origin symmetry, or neither.
f(x) = 4x^2 - x^3
-My answer: orgin of symmetry
f(x) = 5 - x^4
-My answer: orgin of symmetry
f(x) = x^3 - 2x
-my answer: neither
To determine whether a graph has y-axis symmetry, origin symmetry, or neither, we need to consider the properties of even and odd functions.
1. Y-axis symmetry: A function f(x) has y-axis symmetry if, for every value of x, f(x) = f(-x).
2. Origin symmetry: A function f(x) has origin symmetry if, for every value of x, f(x) = -f(-x).
Let's check your answers one by one:
1. f(x) = 4x^2 - x^3
To determine y-axis symmetry, we substitute -x for x in the function and check if it remains the same:
f(-x) = 4(-x)^2 - (-x)^3 = 4x^2 + x^3
Since f(x) and f(-x) are not equal, this function does not have y-axis symmetry.
To determine origin symmetry, we substitute -x for x in the function and check if it changes sign:
-f(-x) = -(4(-x)^2 - (-x)^3) = -4x^2 + x^3
Since f(-x) and -f(-x) are not equal, this function does not have origin symmetry.
Therefore, your answer for f(x) = 4x^2 - x^3 is neither.
2. f(x) = 5 - x^4
To determine y-axis symmetry, we substitute -x for x in the function:
f(-x) = 5 - (-x)^4 = 5 - x^4
Since f(x) and f(-x) are equal, this function has y-axis symmetry.
To determine origin symmetry, we substitute -x for x and check if it changes sign:
-f(-x) = -(5 - (-x)^4) = -5 + x^4
Since f(-x) and -f(-x) are not equal, this function does not have origin symmetry.
Therefore, your answer for f(x) = 5 - x^4 is y-axis symmetry.
3. f(x) = x^3 - 2x
To determine y-axis symmetry, we substitute -x for x in the function:
f(-x) = (-x)^3 - 2(-x) = -x^3 + 2x
Since f(x) and f(-x) are not equal, this function does not have y-axis symmetry.
To determine origin symmetry, we substitute -x for x and check if it changes sign:
-f(-x) = -((-x)^3 - 2(-x)) = x^3 - 2x
Since f(-x) and -f(-x) are not equal, this function does not have origin symmetry.
Therefore, your answer for f(x) = x^3 - 2x is neither.
In summary:
- f(x) = 4x^2 - x^3 has neither symmetry.
- f(x) = 5 - x^4 has y-axis symmetry.
- f(x) = x^3 - 2x has neither symmetry.
Please note that graphing software or tools can provide visual representations of these functions and their symmetries.