Algebra Examples

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

Step 1

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.

Step 2

Simplify.

Tap for more steps...

Step 2.1

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

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

Step 2.2

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

Tap for more steps...

Step 2.2.1

Apply the distributive property.

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

Step 2.2.2

Apply the distributive property.

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

Step 2.2.3

Apply the distributive property.

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

Step 2.3

Simplify and combine like terms.

Tap for more steps...

Step 2.3.1

Simplify each term.

Tap for more steps...

Step 2.3.1.1

Multiply $x$ by $x$ .

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

Step 2.3.1.2

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

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

Step 2.3.1.3

Multiply $- 4$ by $- 4$ .

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

Step 2.3.2

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

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

Step 2.4

Apply the distributive property.

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

Step 2.5

Simplify.

Tap for more steps...

Step 2.5.1

Multiply $- 8$ by $- 4$ .

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

Step 2.5.2

Multiply $- 4$ by $16$ .

$f (x) = (- 4 x^{2} + 32 x - 64) (x^{2} - 4)$

Step 2.6

Expand $(- 4 x^{2} + 32 x - 64) (x^{2} - 4)$ by multiplying each term in the first expression by each term in the second expression.

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

Step 2.7

Simplify terms.

Tap for more steps...

Step 2.7.1

Simplify each term.

Tap for more steps...

Step 2.7.1.1

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

Tap for more steps...

Step 2.7.1.1.1

Move $x^{2}$ .

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

Step 2.7.1.1.2

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

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

Step 2.7.1.1.3

Add $2$ and $2$ .

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

Step 2.7.1.2

Multiply $- 4$ by $- 4$ .

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

Step 2.7.1.3

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

Tap for more steps...

Step 2.7.1.3.1

Move $x^{2}$ .

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

Step 2.7.1.3.2

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

Tap for more steps...

Step 2.7.1.3.2.1

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

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

Step 2.7.1.3.2.2

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

$f (x) = - 4 x^{4} + 16 x^{2} + 32 x^{2 + 1} + 32 x \cdot - 4 - 64 x^{2} - 64 \cdot - 4$

Step 2.7.1.3.3

Add $2$ and $1$ .

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

Step 2.7.1.4

Multiply $- 4$ by $32$ .

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

Step 2.7.1.5

Multiply $- 64$ by $- 4$ .

$f (x) = - 4 x^{4} + 16 x^{2} + 32 x^{3} - 128 x - 64 x^{2} + 256$

Step 2.7.2

Subtract $64 x^{2}$ from $16 x^{2}$ .

$f (x) = - 4 x^{4} - 48 x^{2} + 32 x^{3} - 128 x + 256$

Step 3

Find $f (- x)$ .

Tap for more steps...

Step 3.1

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

$f (- x) = - 4 {(- x)}^{4} - 48 {(- x)}^{2} + 32 {(- x)}^{3} - 128 (- x) + 256$

Step 3.2

Simplify each term.

Tap for more steps...

Step 3.2.1

Apply the product rule to $- x$ .

$f (- x) = - 4 ({(- 1)}^{4} x^{4}) - 48 {(- x)}^{2} + 32 {(- x)}^{3} - 128 (- x) + 256$

Step 3.2.2

Raise $- 1$ to the power of $4$ .

$f (- x) = - 4 (1 x^{4}) - 48 {(- x)}^{2} + 32 {(- x)}^{3} - 128 (- x) + 256$

Step 3.2.3

Multiply $x^{4}$ by $1$ .

$f (- x) = - 4 x^{4} - 48 {(- x)}^{2} + 32 {(- x)}^{3} - 128 (- x) + 256$

Step 3.2.4

Apply the product rule to $- x$ .

$f (- x) = - 4 x^{4} - 48 ({(- 1)}^{2} x^{2}) + 32 {(- x)}^{3} - 128 (- x) + 256$

Step 3.2.5

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

$f (- x) = - 4 x^{4} - 48 (1 x^{2}) + 32 {(- x)}^{3} - 128 (- x) + 256$

Step 3.2.6

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

$f (- x) = - 4 x^{4} - 48 x^{2} + 32 {(- x)}^{3} - 128 (- x) + 256$

Step 3.2.7

Apply the product rule to $- x$ .

$f (- x) = - 4 x^{4} - 48 x^{2} + 32 ({(- 1)}^{3} x^{3}) - 128 (- x) + 256$

Step 3.2.8

Raise $- 1$ to the power of $3$ .

$f (- x) = - 4 x^{4} - 48 x^{2} + 32 (- x^{3}) - 128 (- x) + 256$

Step 3.2.9

Multiply $- 1$ by $32$ .

$f (- x) = - 4 x^{4} - 48 x^{2} - 32 x^{3} - 128 (- x) + 256$

Step 3.2.10

Multiply $- 1$ by $- 128$ .

$f (- x) = - 4 x^{4} - 48 x^{2} - 32 x^{3} + 128 x + 256$

Step 4

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

Tap for more steps...

Step 4.1

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

Step 4.2

Since $- 4 x^{4} - 48 x^{2} - 32 x^{3} + 128 x + 256$ $\neq$ $- 4 x^{4} - 48 x^{2} + 32 x^{3} - 128 x + 256$ , the function is not even.

The function is not even

Step 5

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

Tap for more steps...

Step 5.1

Find $- f (x)$ .

Tap for more steps...

Step 5.1.1

Multiply $- 4 x^{4} - 48 x^{2} + 32 x^{3} - 128 x + 256$ by $- 1$ .

$- f (x) = - (- 4 x^{4} - 48 x^{2} + 32 x^{3} - 128 x + 256)$

Step 5.1.2

Apply the distributive property.

$- f (x) = - (- 4 x^{4}) - (- 48 x^{2}) - (32 x^{3}) - (- 128 x) - 1 \cdot 256$

Step 5.1.3

Simplify.

Tap for more steps...

Step 5.1.3.1

Multiply $- 4$ by $- 1$ .

$- f (x) = 4 x^{4} - (- 48 x^{2}) - (32 x^{3}) - (- 128 x) - 1 \cdot 256$

Step 5.1.3.2

Multiply $- 48$ by $- 1$ .

$- f (x) = 4 x^{4} + 48 x^{2} - (32 x^{3}) - (- 128 x) - 1 \cdot 256$

Step 5.1.3.3

Multiply $32$ by $- 1$ .

$- f (x) = 4 x^{4} + 48 x^{2} - 32 x^{3} - (- 128 x) - 1 \cdot 256$

Step 5.1.3.4

Multiply $- 128$ by $- 1$ .

$- f (x) = 4 x^{4} + 48 x^{2} - 32 x^{3} + 128 x - 1 \cdot 256$

Step 5.1.3.5

Multiply $- 1$ by $256$ .

$- f (x) = 4 x^{4} + 48 x^{2} - 32 x^{3} + 128 x - 256$

Step 5.2

Since $- 4 x^{4} - 48 x^{2} - 32 x^{3} + 128 x + 256$ $\neq$ $4 x^{4} + 48 x^{2} - 32 x^{3} + 128 x - 256$ , the function is not odd.

The function is not odd

Step 6

The function is neither odd nor even

Step 7

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

No origin symmetry

Step 8

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

No y-axis symmetry

Step 9

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

Function is not symmetric

Step 10