Determine whether the function \(f(x) = x^4 - 4x^2 + 1\) is symmetric about the y-axis, the origin, or neither.
Answer
Step 2: Check symmetry about the origin by replacing x with -x in the function. If the function changes sign, then the function is symmetric about the origin. So, \(f(-x) = (-x)^4 - 4(-x)^2 + 1 = x^4 - 4x^2 + 1 = f(x)\). Therefore, the function is not symmetric about the origin because it does not change sign when x is replaced with -x.
Step 1 ：Step 1: Check symmetry about the y-axis by replacing x with -x in the function. If the function remains the same, then the function is symmetric about the y-axis. So, \(f(-x) = (-x)^4 - 4(-x)^2 + 1 = x^4 - 4x^2 + 1 = f(x)\). Therefore, the function is symmetric about the y-axis.
Step 2 ：Step 2: Check symmetry about the origin by replacing x with -x in the function. If the function changes sign, then the function is symmetric about the origin. So, \(f(-x) = (-x)^4 - 4(-x)^2 + 1 = x^4 - 4x^2 + 1 = f(x)\). Therefore, the function is not symmetric about the origin because it does not change sign when x is replaced with -x.
