Find the Symmetry -4(x-3)^2(x^2-4)

$- 4 {(x - 3)}^{2} (x^{2} - 4)$

Step 1

Write $- 4 {(x - 3)}^{2} (x^{2} - 4)$ as an equation.

$y = - 4 {(x - 3)}^{2} (x^{2} - 4)$

Step 2

There are three types of symmetry:

1. X-Axis Symmetry

2. Y-Axis Symmetry

3. Origin Symmetry

Step 3

If $(x, y)$ exists on the graph, then the graph is symmetric about the:

1. X-Axis if $(x, - y)$ exists on the graph

2. Y-Axis if $(- x, y)$ exists on the graph

3. Origin if $(- x, - y)$ exists on the graph

Step 4

Rewrite ${(x - 3)}^{2}$ as $(x - 3) (x - 3)$ .

$y = - 4 ((x - 3) (x - 3)) (x^{2} - 4)$

Step 5

Expand $(x - 3) (x - 3)$ using the FOIL Method.

Apply the distributive property.

$y = - 4 (x (x - 3) - 3 (x - 3)) (x^{2} - 4)$

Apply the distributive property.

$y = - 4 (x \cdot x + x \cdot - 3 - 3 (x - 3)) (x^{2} - 4)$

Apply the distributive property.

$y = - 4 (x \cdot x + x \cdot - 3 - 3 x - 3 \cdot - 3) (x^{2} - 4)$

Step 6

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$y = - 4 (x^{2} + x \cdot - 3 - 3 x - 3 \cdot - 3) (x^{2} - 4)$

Move $- 3$ to the left of $x$ .

$y = - 4 (x^{2} - 3 \cdot x - 3 x - 3 \cdot - 3) (x^{2} - 4)$

Multiply $- 3$ by $- 3$ .

$y = - 4 (x^{2} - 3 x - 3 x + 9) (x^{2} - 4)$

Subtract $3 x$ from $- 3 x$ .

$y = - 4 (x^{2} - 6 x + 9) (x^{2} - 4)$

Step 7

Apply the distributive property.

$y = (- 4 x^{2} - 4 (- 6 x) - 4 \cdot 9) (x^{2} - 4)$

Step 8

Simplify.

Multiply $- 6$ by $- 4$ .

$y = (- 4 x^{2} + 24 x - 4 \cdot 9) (x^{2} - 4)$

Multiply $- 4$ by $9$ .

$y = (- 4 x^{2} + 24 x - 36) (x^{2} - 4)$

Step 9

Expand $(- 4 x^{2} + 24 x - 36) (x^{2} - 4)$ by multiplying each term in the first expression by each term in the second expression.

$y = - 4 x^{2} x^{2} - 4 x^{2} \cdot - 4 + 24 x \cdot x^{2} + 24 x \cdot - 4 - 36 x^{2} - 36 \cdot - 4$

Step 10

Simplify each term.

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

Move $x^{2}$ .

$y = - 4 (x^{2} x^{2}) - 4 x^{2} \cdot - 4 + 24 x \cdot x^{2} + 24 x \cdot - 4 - 36 x^{2} - 36 \cdot - 4$

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

$y = - 4 x^{2 + 2} - 4 x^{2} \cdot - 4 + 24 x \cdot x^{2} + 24 x \cdot - 4 - 36 x^{2} - 36 \cdot - 4$

Add $2$ and $2$ .

$y = - 4 x^{4} - 4 x^{2} \cdot - 4 + 24 x \cdot x^{2} + 24 x \cdot - 4 - 36 x^{2} - 36 \cdot - 4$

Multiply $- 4$ by $- 4$ .

$y = - 4 x^{4} + 16 x^{2} + 24 x \cdot x^{2} + 24 x \cdot - 4 - 36 x^{2} - 36 \cdot - 4$

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

Move $x^{2}$ .

$y = - 4 x^{4} + 16 x^{2} + 24 (x^{2} x) + 24 x \cdot - 4 - 36 x^{2} - 36 \cdot - 4$

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

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

$y = - 4 x^{4} + 16 x^{2} + 24 (x^{2} x^{1}) + 24 x \cdot - 4 - 36 x^{2} - 36 \cdot - 4$

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

$y = - 4 x^{4} + 16 x^{2} + 24 x^{2 + 1} + 24 x \cdot - 4 - 36 x^{2} - 36 \cdot - 4$

Add $2$ and $1$ .

$y = - 4 x^{4} + 16 x^{2} + 24 x^{3} + 24 x \cdot - 4 - 36 x^{2} - 36 \cdot - 4$

Multiply $- 4$ by $24$ .

$y = - 4 x^{4} + 16 x^{2} + 24 x^{3} - 96 x - 36 x^{2} - 36 \cdot - 4$

Multiply $- 36$ by $- 4$ .

$y = - 4 x^{4} + 16 x^{2} + 24 x^{3} - 96 x - 36 x^{2} + 144$

Step 11

Subtract $36 x^{2}$ from $16 x^{2}$ .

$y = - 4 x^{4} - 20 x^{2} + 24 x^{3} - 96 x + 144$

Step 12

Check if the graph is symmetric about the $x$ -axis by plugging in $- y$ for $y$ .

$- y = - 4 x^{4} - 20 x^{2} + 24 x^{3} - 96 x + 144$

Step 13

Since the equation is not identical to the original equation, it is not symmetric to the x-axis.

Not symmetric to the x-axis

Step 14

Check if the graph is symmetric about the $y$ -axis by plugging in $- x$ for $x$ .

$y = - 4 {(- x)}^{4} - 20 {(- x)}^{2} + 24 {(- x)}^{3} - 96 (- x) + 144$

Step 15

Simplify each term.

Apply the product rule to $- x$ .

$y = - 4 ({(- 1)}^{4} x^{4}) - 20 {(- x)}^{2} + 24 {(- x)}^{3} - 96 (- x) + 144$

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

$y = - 4 (1 x^{4}) - 20 {(- x)}^{2} + 24 {(- x)}^{3} - 96 (- x) + 144$

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

$y = - 4 x^{4} - 20 {(- x)}^{2} + 24 {(- x)}^{3} - 96 (- x) + 144$

Apply the product rule to $- x$ .

$y = - 4 x^{4} - 20 ({(- 1)}^{2} x^{2}) + 24 {(- x)}^{3} - 96 (- x) + 144$

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

$y = - 4 x^{4} - 20 (1 x^{2}) + 24 {(- x)}^{3} - 96 (- x) + 144$

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

$y = - 4 x^{4} - 20 x^{2} + 24 {(- x)}^{3} - 96 (- x) + 144$

Apply the product rule to $- x$ .

$y = - 4 x^{4} - 20 x^{2} + 24 ({(- 1)}^{3} x^{3}) - 96 (- x) + 144$

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

$y = - 4 x^{4} - 20 x^{2} + 24 (- x^{3}) - 96 (- x) + 144$

Multiply $- 1$ by $24$ .

$y = - 4 x^{4} - 20 x^{2} - 24 x^{3} - 96 (- x) + 144$

Multiply $- 1$ by $- 96$ .

$y = - 4 x^{4} - 20 x^{2} - 24 x^{3} + 96 x + 144$

Step 16

Since the equation is not identical to the original equation, it is not symmetric to the y-axis.

Not symmetric to the y-axis

Step 17

Check if the graph is symmetric about the origin by plugging in $- x$ for $x$ and $- y$ for $y$ .

$- y = - 4 {(- x)}^{4} - 20 {(- x)}^{2} + 24 {(- x)}^{3} - 96 (- x) + 144$

Step 18

Simplify each term.

Apply the product rule to $- x$ .

$- y = - 4 ({(- 1)}^{4} x^{4}) - 20 {(- x)}^{2} + 24 {(- x)}^{3} - 96 (- x) + 144$

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

$- y = - 4 (1 x^{4}) - 20 {(- x)}^{2} + 24 {(- x)}^{3} - 96 (- x) + 144$

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

$- y = - 4 x^{4} - 20 {(- x)}^{2} + 24 {(- x)}^{3} - 96 (- x) + 144$

Apply the product rule to $- x$ .

$- y = - 4 x^{4} - 20 ({(- 1)}^{2} x^{2}) + 24 {(- x)}^{3} - 96 (- x) + 144$

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

$- y = - 4 x^{4} - 20 (1 x^{2}) + 24 {(- x)}^{3} - 96 (- x) + 144$

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

$- y = - 4 x^{4} - 20 x^{2} + 24 {(- x)}^{3} - 96 (- x) + 144$

Apply the product rule to $- x$ .

$- y = - 4 x^{4} - 20 x^{2} + 24 ({(- 1)}^{3} x^{3}) - 96 (- x) + 144$

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

$- y = - 4 x^{4} - 20 x^{2} + 24 (- x^{3}) - 96 (- x) + 144$

Multiply $- 1$ by $24$ .

$- y = - 4 x^{4} - 20 x^{2} - 24 x^{3} - 96 (- x) + 144$

Multiply $- 1$ by $- 96$ .

$- y = - 4 x^{4} - 20 x^{2} - 24 x^{3} + 96 x + 144$

Step 19

Since the equation is not identical to the original equation, it is not symmetric to the origin.

Not symmetric to the origin

Step 20

Determine the symmetry.

Not symmetric to the x-axis

Not symmetric to the y-axis

Not symmetric to the origin

Step 21

Find the Symmetry -4(x-3)^2(x^2-4)

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