Question

1. without graphing, determine if each polynomial function has line symmetry about the ( y )-axis, point symmetry about the origin, or neither. graph the functions to verify your answers.

a) ( f(x) = -x^5 + 7x^3 + 2x )
b) ( f(x) = x^4 + 3x^2 + 1 )
c) ( f(x) = 4x^3 - 3x^2 + 8x + 1 )

Explanation:

Step1: Recall odd/even function rules

A function has y-axis symmetry (even) if $f(-x)=f(x)$; origin symmetry (odd) if $f(-x)=-f(x)$.
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Part a) $f(x) = -x^5 + 7x^3 + 2x$

Step1: Compute $f(-x)$

$f(-x) = -(-x)^5 + 7(-x)^3 + 2(-x)$
$f(-x) = x^5 -7x^3 -2x$

Step2: Compare to $-f(x)$

$-f(x) = -(-x^5 +7x^3 +2x) = x^5 -7x^3 -2x$
$f(-x) = -f(x)$
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Part b) $f(x) = x^4 + 3x^2 + 1$

Step1: Compute $f(-x)$

$f(-x) = (-x)^4 + 3(-x)^2 + 1$
$f(-x) = x^4 +3x^2 +1$

Step2: Compare to $f(x)$

$f(-x) = f(x)$
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Part c) $f(x) = 4x^3 - 3x^2 + 8x + 1$

Step1: Compute $f(-x)$

$f(-x) = 4(-x)^3 -3(-x)^2 +8(-x)+1$
$f(-x) = -4x^3 -3x^2 -8x +1$

Step2: Compare to $f(x)$ and $-f(x)$

$-f(x) = -4x^3 +3x^2 -8x -1$
$f(-x)
eq f(x)$ and $f(-x)
eq -f(x)$

Answer:

a) Point symmetry about the origin (odd function)
b) Line symmetry about the y-axis (even function)
c) Neither symmetry