Find the Symmetry f(x)=-2(x-4)^2(x^2-4)

$f (x) = - 2 {(x - 4)}^{2} (x^{2} - 4)$

Determine if the function is odd, even, or neither in order to find the symmetry.

1. If odd, the function is symmetric about the origin.

2. If even, the function is symmetric about the y-axis.

Find $f (- x)$ .

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Find $f (- x)$ by substituting $- x$ for all occurrence of $x$ in $f (x)$ .

$f (- x) = - 2 {((- x) - 4)}^{2} ({(- x)}^{2} - 4)$

Rewrite ${(- x - 4)}^{2}$ as $(- x - 4) (- x - 4)$ .

$f (- x) = - 2 ((- x - 4) (- x - 4)) ({(- x)}^{2} - 4)$

Expand $(- x - 4) (- x - 4)$ using the FOIL Method.

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Apply the distributive property.

$f (- x) = - 2 (- x (- x - 4) - 4 (- x - 4)) ({(- x)}^{2} - 4)$

Apply the distributive property.

$f (- x) = - 2 (- x (- x) - x \cdot - 4 - 4 (- x - 4)) ({(- x)}^{2} - 4)$

Apply the distributive property.

$f (- x) = - 2 (- x (- x) - x \cdot - 4 - 4 (- x) - 4 \cdot - 4) ({(- x)}^{2} - 4)$

Simplify and combine like terms.

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Simplify each term.

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Multiply $x$ by $x$ .

$f (- x) = - 2 (- 1 \cdot (- 1 x^{2}) - x \cdot - 4 - 4 (- x) - 4 \cdot - 4) ({(- x)}^{2} - 4)$

Multiply $- 1$ by $- 1$ .

$f (- x) = - 2 (1 x^{2} - x \cdot - 4 - 4 (- x) - 4 \cdot - 4) ({(- x)}^{2} - 4)$

Multiply $x^{2}$ by $1$ .

$f (- x) = - 2 (x^{2} - x \cdot - 4 - 4 (- x) - 4 \cdot - 4) ({(- x)}^{2} - 4)$

Multiply $- 4$ by $- 1$ .

$f (- x) = - 2 (x^{2} + 4 x - 4 (- x) - 4 \cdot - 4) ({(- x)}^{2} - 4)$

Multiply $- 1$ by $- 4$ .

$f (- x) = - 2 (x^{2} + 4 x + 4 x - 4 \cdot - 4) ({(- x)}^{2} - 4)$

Multiply $- 4$ by $- 4$ .

$f (- x) = - 2 (x^{2} + 4 x + 4 x + 16) ({(- x)}^{2} - 4)$

Add $4 x$ and $4 x$ .

$f (- x) = - 2 (x^{2} + 8 x + 16) ({(- x)}^{2} - 4)$

Apply the distributive property.

$f (- x) = (- 2 x^{2} - 2 (8 x) - 2 \cdot 16) ({(- x)}^{2} - 4)$

Simplify.

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Multiply $8$ by $- 2$ .

$f (- x) = (- 2 x^{2} - 16 x - 2 \cdot 16) ({(- x)}^{2} - 4)$

Multiply $- 2$ by $16$ .

$f (- x) = (- 2 x^{2} - 16 x - 32) ({(- x)}^{2} - 4)$

Simplify each term.

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Apply the product rule to $- x$ .

$f (- x) = (- 2 x^{2} - 16 x - 32) ({(- 1)}^{2} x^{2} - 4)$

Raise $- 1$ to the power of $2$ .

$f (- x) = (- 2 x^{2} - 16 x - 32) (1 x^{2} - 4)$

Multiply $x^{2}$ by $1$ .

$f (- x) = (- 2 x^{2} - 16 x - 32) (x^{2} - 4)$

Expand $(- 2 x^{2} - 16 x - 32) (x^{2} - 4)$ by multiplying each term in the first expression by each term in the second expression.

$f (- x) = - 2 x^{2} x^{2} - 2 x^{2} \cdot - 4 - 16 x \cdot x^{2} - 16 x \cdot - 4 - 32 x^{2} - 32 \cdot - 4$

Simplify terms.

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Simplify each term.

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Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Move $x^{2}$ .

$f (- x) = - 2 (x^{2} x^{2}) - 2 x^{2} \cdot - 4 - 16 x \cdot x^{2} - 16 x \cdot - 4 - 32 x^{2} - 32 \cdot - 4$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (- x) = - 2 x^{2 + 2} - 2 x^{2} \cdot - 4 - 16 x \cdot x^{2} - 16 x \cdot - 4 - 32 x^{2} - 32 \cdot - 4$

Add $2$ and $2$ .

$f (- x) = - 2 x^{4} - 2 x^{2} \cdot - 4 - 16 x \cdot x^{2} - 16 x \cdot - 4 - 32 x^{2} - 32 \cdot - 4$

Multiply $- 4$ by $- 2$ .

$f (- x) = - 2 x^{4} + 8 x^{2} - 16 x \cdot x^{2} - 16 x \cdot - 4 - 32 x^{2} - 32 \cdot - 4$

Multiply $x$ by $x^{2}$ by adding the exponents.

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Move $x^{2}$ .

$f (- x) = - 2 x^{4} + 8 x^{2} - 16 (x^{2} x) - 16 x \cdot - 4 - 32 x^{2} - 32 \cdot - 4$

Multiply $x^{2}$ by $x$ .

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Raise $x$ to the power of $1$ .

$f (- x) = - 2 x^{4} + 8 x^{2} - 16 (x^{2} x) - 16 x \cdot - 4 - 32 x^{2} - 32 \cdot - 4$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (- x) = - 2 x^{4} + 8 x^{2} - 16 x^{2 + 1} - 16 x \cdot - 4 - 32 x^{2} - 32 \cdot - 4$

Add $2$ and $1$ .

$f (- x) = - 2 x^{4} + 8 x^{2} - 16 x^{3} - 16 x \cdot - 4 - 32 x^{2} - 32 \cdot - 4$

Multiply $- 4$ by $- 16$ .

$f (- x) = - 2 x^{4} + 8 x^{2} - 16 x^{3} + 64 x - 32 x^{2} - 32 \cdot - 4$

Multiply $- 32$ by $- 4$ .

$f (- x) = - 2 x^{4} + 8 x^{2} - 16 x^{3} + 64 x - 32 x^{2} + 128$

Subtract $32 x^{2}$ from $8 x^{2}$ .

$f (- x) = - 2 x^{4} - 24 x^{2} - 16 x^{3} + 64 x + 128$

A function is even if $f (- x) = f (x)$ .

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Check if $f (- x) = f (x)$ .

$- 2 x^{4} - 24 x^{2} - 16 x^{3} + 64 x + 128$ $\neq$ $- 2 {(x - 4)}^{2} (x^{2} - 4)$

Since $- 2 x^{4} - 24 x^{2} - 16 x^{3} + 64 x + 128$ $\neq$ $- 2 {(x - 4)}^{2} (x^{2} - 4)$ , the function is not even.

The function is not even

A function is odd if $f (- x) = - f (x)$ .

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Find $- f (x)$ .

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Rewrite ${(x - 4)}^{2}$ as $(x - 4) (x - 4)$ .

$- f (x) = - (- 2 ((x - 4) (x - 4)) (x^{2} - 4))$

Expand $(x - 4) (x - 4)$ using the FOIL Method.

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Apply the distributive property.

$- f (x) = - (- 2 (x (x - 4) - 4 (x - 4)) (x^{2} - 4))$

Apply the distributive property.

$- f (x) = - (- 2 (x \cdot x + x \cdot - 4 - 4 (x - 4)) (x^{2} - 4))$

Apply the distributive property.

$- f (x) = - (- 2 (x \cdot x + x \cdot - 4 - 4 x - 4 \cdot - 4) (x^{2} - 4))$

Simplify and combine like terms.

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Simplify each term.

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Multiply $x$ by $x$ .

$- f (x) = - (- 2 (x^{2} + x \cdot - 4 - 4 x - 4 \cdot - 4) (x^{2} - 4))$

Move $- 4$ to the left of $x$ .

$- f (x) = - (- 2 (x^{2} - 4 \cdot x - 4 x - 4 \cdot - 4) (x^{2} - 4))$

Multiply $- 4$ by $- 4$ .

$- f (x) = - (- 2 (x^{2} - 4 x - 4 x + 16) (x^{2} - 4))$

Subtract $4 x$ from $- 4 x$ .

$- f (x) = - (- 2 (x^{2} - 8 x + 16) (x^{2} - 4))$

Apply the distributive property.

$- f (x) = - ((- 2 x^{2} - 2 (- 8 x) - 2 \cdot 16) (x^{2} - 4))$

Simplify.

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Multiply $- 8$ by $- 2$ .

$- f (x) = - ((- 2 x^{2} + 16 x - 2 \cdot 16) (x^{2} - 4))$

Multiply $- 2$ by $16$ .

$- f (x) = - ((- 2 x^{2} + 16 x - 32) (x^{2} - 4))$

Expand $(- 2 x^{2} + 16 x - 32) (x^{2} - 4)$ by multiplying each term in the first expression by each term in the second expression.

$- f (x) = - (- 2 x^{2} x^{2} - 2 x^{2} \cdot - 4 + 16 x \cdot x^{2} + 16 x \cdot - 4 - 32 x^{2} - 32 \cdot - 4)$

Simplify terms.

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Simplify each term.

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Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Move $x^{2}$ .

$- f (x) = - (- 2 (x^{2} x^{2}) - 2 x^{2} \cdot - 4 + 16 x \cdot x^{2} + 16 x \cdot - 4 - 32 x^{2} - 32 \cdot - 4)$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- f (x) = - (- 2 x^{2 + 2} - 2 x^{2} \cdot - 4 + 16 x \cdot x^{2} + 16 x \cdot - 4 - 32 x^{2} - 32 \cdot - 4)$

Add $2$ and $2$ .

$- f (x) = - (- 2 x^{4} - 2 x^{2} \cdot - 4 + 16 x \cdot x^{2} + 16 x \cdot - 4 - 32 x^{2} - 32 \cdot - 4)$

Multiply $- 4$ by $- 2$ .

$- f (x) = - (- 2 x^{4} + 8 x^{2} + 16 x \cdot x^{2} + 16 x \cdot - 4 - 32 x^{2} - 32 \cdot - 4)$

Multiply $x$ by $x^{2}$ by adding the exponents.

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Move $x^{2}$ .

$- f (x) = - (- 2 x^{4} + 8 x^{2} + 16 (x^{2} x) + 16 x \cdot - 4 - 32 x^{2} - 32 \cdot - 4)$

Multiply $x^{2}$ by $x$ .

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Raise $x$ to the power of $1$ .

$- f (x) = - (- 2 x^{4} + 8 x^{2} + 16 (x^{2} x) + 16 x \cdot - 4 - 32 x^{2} - 32 \cdot - 4)$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- f (x) = - (- 2 x^{4} + 8 x^{2} + 16 x^{2 + 1} + 16 x \cdot - 4 - 32 x^{2} - 32 \cdot - 4)$

Add $2$ and $1$ .

$- f (x) = - (- 2 x^{4} + 8 x^{2} + 16 x^{3} + 16 x \cdot - 4 - 32 x^{2} - 32 \cdot - 4)$

Multiply $- 4$ by $16$ .

$- f (x) = - (- 2 x^{4} + 8 x^{2} + 16 x^{3} - 64 x - 32 x^{2} - 32 \cdot - 4)$

Multiply $- 32$ by $- 4$ .

$- f (x) = - (- 2 x^{4} + 8 x^{2} + 16 x^{3} - 64 x - 32 x^{2} + 128)$

Simplify terms.

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Subtract $32 x^{2}$ from $8 x^{2}$ .

$- f (x) = - (- 2 x^{4} - 24 x^{2} + 16 x^{3} - 64 x + 128)$

Apply the distributive property.

$- f (x) = - (- 2 x^{4}) - (- 24 x^{2}) - (16 x^{3}) - (- 64 x) - 1 \cdot 128$

Simplify.

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Multiply $- 2$ by $- 1$ .

$- f (x) = 2 x^{4} - (- 24 x^{2}) - (16 x^{3}) - (- 64 x) - 1 \cdot 128$

Multiply $- 24$ by $- 1$ .

$- f (x) = 2 x^{4} + 24 x^{2} - (16 x^{3}) - (- 64 x) - 1 \cdot 128$

Multiply $16$ by $- 1$ .

$- f (x) = 2 x^{4} + 24 x^{2} - 16 x^{3} - (- 64 x) - 1 \cdot 128$

Multiply $- 64$ by $- 1$ .

$- f (x) = 2 x^{4} + 24 x^{2} - 16 x^{3} + 64 x - 1 \cdot 128$

Multiply $- 1$ by $128$ .

$- f (x) = 2 x^{4} + 24 x^{2} - 16 x^{3} + 64 x - 128$

Since $- 2 x^{4} - 24 x^{2} - 16 x^{3} + 64 x + 128$ $\neq$ $2 x^{4} + 24 x^{2} - 16 x^{3} + 64 x - 128$ , the function is not odd.

The function is not odd

The function is neither odd nor even

Since the function is not odd, it is not symmetric about the origin.

No origin symmetry

Since the function is not even, it is not symmetric about the y-axis.

No y-axis symmetry

Since the function is neither odd nor even, there is no origin / y-axis symmetry.

Function is not symmetric

Find the Symmetry f(x)=-2(x-4)^2(x^2-4)

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