How do you determine (without graphing) if the polynomial has line symmetry about the y-axis, point symmetry about the origin, or neither.
The function is: f(x)= -x^5 + 7x^3 + 2x
To determine if a polynomial has line symmetry about the y-axis or point symmetry about the origin without graphing, you need to examine the powers of each term in the polynomial.
For line symmetry about the y-axis, you need to check if the powers of the terms that contain x are all even. If they are, then the polynomial has line symmetry about the y-axis.
For point symmetry about the origin, you need to check if the powers of the terms that contain x are all odd. If they are, then the polynomial has point symmetry about the origin.
Let's analyze the given polynomial function f(x) = -x^5 + 7x^3 + 2x:
1. Checking for line symmetry about the y-axis:
- The highest power of x in the polynomial is 5 (from -x^5).
- The power of x in the second term (7x^3) is 3.
- The power of x in the third term (2x) is 1.
Since the power of x in the polynomial is odd (5), the polynomial does not have line symmetry about the y-axis.
2. Checking for point symmetry about the origin:
- The highest power of x in the polynomial is 5 (from -x^5).
- The power of x in the second term (7x^3) is 3.
- The power of x in the third term (2x) is 1.
Since the powers of x in the polynomial are both odd (5, 3, and 1), the polynomial does not have point symmetry about the origin.
Therefore, based on the powers of x in the polynomial, we can conclude that it has neither line symmetry about the y-axis nor point symmetry about the origin.