Find the Symmetry -3(x-2)^2(x^2-9)

$- 3 {(x - 2)}^{2} (x^{2} - 9)$

Write $- 3 {(x - 2)}^{2} (x^{2} - 9)$ as an equation.

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

Simplify the right side.

Simplify $- 3 {(x - 2)}^{2} (x^{2} - 9)$ .

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

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

Expand $(x - 2) (x - 2)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

Move $- 2$ to the left of $x$ .

$y = - 3 (x^{2} - 2 \cdot x - 2 x - 2 \cdot - 2) (x^{2} - 9)$

Multiply $- 2$ by $- 2$ .

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

Subtract $2 x$ from $- 2 x$ .

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

Apply the distributive property.

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

Simplify.

Multiply $- 4$ by $- 3$ .

$y = (- 3 x^{2} + 12 x - 3 \cdot 4) (x^{2} - 9)$

Multiply $- 3$ by $4$ .

$y = (- 3 x^{2} + 12 x - 12) (x^{2} - 9)$

Expand $(- 3 x^{2} + 12 x - 12) (x^{2} - 9)$ by multiplying each term in the first expression by each term in the second expression.

$y = - 3 x^{2} x^{2} - 3 x^{2} \cdot - 9 + 12 x \cdot x^{2} + 12 x \cdot - 9 - 12 x^{2} - 12 \cdot - 9$

Simplify terms.

Simplify each term.

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

Move $x^{2}$ .

$y = - 3 (x^{2} x^{2}) - 3 x^{2} \cdot - 9 + 12 x \cdot x^{2} + 12 x \cdot - 9 - 12 x^{2} - 12 \cdot - 9$

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

$y = - 3 x^{2 + 2} - 3 x^{2} \cdot - 9 + 12 x \cdot x^{2} + 12 x \cdot - 9 - 12 x^{2} - 12 \cdot - 9$

Add $2$ and $2$ .

$y = - 3 x^{4} - 3 x^{2} \cdot - 9 + 12 x \cdot x^{2} + 12 x \cdot - 9 - 12 x^{2} - 12 \cdot - 9$

Multiply $- 9$ by $- 3$ .

$y = - 3 x^{4} + 27 x^{2} + 12 x \cdot x^{2} + 12 x \cdot - 9 - 12 x^{2} - 12 \cdot - 9$

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

Move $x^{2}$ .

$y = - 3 x^{4} + 27 x^{2} + 12 (x^{2} x) + 12 x \cdot - 9 - 12 x^{2} - 12 \cdot - 9$

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

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

$y = - 3 x^{4} + 27 x^{2} + 12 (x^{2} x^{1}) + 12 x \cdot - 9 - 12 x^{2} - 12 \cdot - 9$

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

$y = - 3 x^{4} + 27 x^{2} + 12 x^{2 + 1} + 12 x \cdot - 9 - 12 x^{2} - 12 \cdot - 9$

Add $2$ and $1$ .

$y = - 3 x^{4} + 27 x^{2} + 12 x^{3} + 12 x \cdot - 9 - 12 x^{2} - 12 \cdot - 9$

Multiply $- 9$ by $12$ .

$y = - 3 x^{4} + 27 x^{2} + 12 x^{3} - 108 x - 12 x^{2} - 12 \cdot - 9$

Multiply $- 12$ by $- 9$ .

$y = - 3 x^{4} + 27 x^{2} + 12 x^{3} - 108 x - 12 x^{2} + 108$

Subtract $12 x^{2}$ from $27 x^{2}$ .

$y = - 3 x^{4} + 15 x^{2} + 12 x^{3} - 108 x + 108$

There are three types of symmetry:

1. X-Axis Symmetry

2. Y-Axis Symmetry

3. Origin Symmetry

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

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

$- y = - 3 x^{4} + 15 x^{2} + 12 x^{3} - 108 x + 108$

Simplify the left side.

Remove parentheses.

$- y = - 3 x^{4} + 15 x^{2} + 12 x^{3} - 108 x + 108$

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

Not symmetric to the x-axis

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

$y = - 3 {(- x)}^{4} + 15 {(- x)}^{2} + 12 {(- x)}^{3} - 108 (- x) + 108$

Simplify the right side.

Simplify each term.

Apply the product rule to $- x$ .

$y = - 3 ({(- 1)}^{4} x^{4}) + 15 {(- x)}^{2} + 12 {(- x)}^{3} - 108 (- x) + 108$

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

$y = - 3 (1 x^{4}) + 15 {(- x)}^{2} + 12 {(- x)}^{3} - 108 (- x) + 108$

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

$y = - 3 x^{4} + 15 {(- x)}^{2} + 12 {(- x)}^{3} - 108 (- x) + 108$

Apply the product rule to $- x$ .

$y = - 3 x^{4} + 15 ({(- 1)}^{2} x^{2}) + 12 {(- x)}^{3} - 108 (- x) + 108$

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

$y = - 3 x^{4} + 15 (1 x^{2}) + 12 {(- x)}^{3} - 108 (- x) + 108$

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

$y = - 3 x^{4} + 15 x^{2} + 12 {(- x)}^{3} - 108 (- x) + 108$

Apply the product rule to $- x$ .

$y = - 3 x^{4} + 15 x^{2} + 12 ({(- 1)}^{3} x^{3}) - 108 (- x) + 108$

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

$y = - 3 x^{4} + 15 x^{2} + 12 (- x^{3}) - 108 (- x) + 108$

Multiply $- 1$ by $12$ .

$y = - 3 x^{4} + 15 x^{2} - 12 x^{3} - 108 (- x) + 108$

Multiply $- 1$ by $- 108$ .

$y = - 3 x^{4} + 15 x^{2} - 12 x^{3} + 108 x + 108$

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

Not symmetric to the y-axis

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

$- y = - 3 {(- x)}^{4} + 15 {(- x)}^{2} + 12 {(- x)}^{3} - 108 (- x) + 108$

Solve for $y$ .

Simplify the left side.

Remove parentheses.

$- y = - 3 {(- x)}^{4} + 15 {(- x)}^{2} + 12 {(- x)}^{3} - 108 (- x) + 108$

Simplify the right side.

Simplify each term.

Apply the product rule to $- x$ .

$- y = - 3 ({(- 1)}^{4} x^{4}) + 15 {(- x)}^{2} + 12 {(- x)}^{3} - 108 (- x) + 108$

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

$- y = - 3 (1 x^{4}) + 15 {(- x)}^{2} + 12 {(- x)}^{3} - 108 (- x) + 108$

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

$- y = - 3 x^{4} + 15 {(- x)}^{2} + 12 {(- x)}^{3} - 108 (- x) + 108$

Apply the product rule to $- x$ .

$- y = - 3 x^{4} + 15 ({(- 1)}^{2} x^{2}) + 12 {(- x)}^{3} - 108 (- x) + 108$

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

$- y = - 3 x^{4} + 15 (1 x^{2}) + 12 {(- x)}^{3} - 108 (- x) + 108$

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

$- y = - 3 x^{4} + 15 x^{2} + 12 {(- x)}^{3} - 108 (- x) + 108$

Apply the product rule to $- x$ .

$- y = - 3 x^{4} + 15 x^{2} + 12 ({(- 1)}^{3} x^{3}) - 108 (- x) + 108$

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

$- y = - 3 x^{4} + 15 x^{2} + 12 (- x^{3}) - 108 (- x) + 108$

Multiply $- 1$ by $12$ .

$- y = - 3 x^{4} + 15 x^{2} - 12 x^{3} - 108 (- x) + 108$

Multiply $- 1$ by $- 108$ .

$- y = - 3 x^{4} + 15 x^{2} - 12 x^{3} + 108 x + 108$

Simplify the left side.

Multiply $- (- y)$ .

Multiply $- 1$ by $- 1$ .

$1 y = - (- 3 {(- x)}^{4} + 15 {(- x)}^{2} + 12 {(- x)}^{3} - 108 (- x) + 108)$

Multiply $y$ by $1$ .

$y = - (- 3 {(- x)}^{4} + 15 {(- x)}^{2} + 12 {(- x)}^{3} - 108 (- x) + 108)$

Simplify the right side.

Simplify $- (- 3 {(- x)}^{4} + 15 {(- x)}^{2} + 12 {(- x)}^{3} - 108 (- x) + 108)$ .

Simplify each term.

Apply the product rule to $- x$ .

$y = - (- 3 ({(- 1)}^{4} x^{4}) + 15 {(- x)}^{2} + 12 {(- x)}^{3} - 108 (- x) + 108)$

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

$y = - (- 3 (1 x^{4}) + 15 {(- x)}^{2} + 12 {(- x)}^{3} - 108 (- x) + 108)$

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

$y = - (- 3 x^{4} + 15 {(- x)}^{2} + 12 {(- x)}^{3} - 108 (- x) + 108)$

Apply the product rule to $- x$ .

$y = - (- 3 x^{4} + 15 ({(- 1)}^{2} x^{2}) + 12 {(- x)}^{3} - 108 (- x) + 108)$

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

$y = - (- 3 x^{4} + 15 (1 x^{2}) + 12 {(- x)}^{3} - 108 (- x) + 108)$

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

$y = - (- 3 x^{4} + 15 x^{2} + 12 {(- x)}^{3} - 108 (- x) + 108)$

Apply the product rule to $- x$ .

$y = - (- 3 x^{4} + 15 x^{2} + 12 ({(- 1)}^{3} x^{3}) - 108 (- x) + 108)$

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

$y = - (- 3 x^{4} + 15 x^{2} + 12 (- x^{3}) - 108 (- x) + 108)$

Multiply $- 1$ by $12$ .

$y = - (- 3 x^{4} + 15 x^{2} - 12 x^{3} - 108 (- x) + 108)$

Multiply $- 1$ by $- 108$ .

$y = - (- 3 x^{4} + 15 x^{2} - 12 x^{3} + 108 x + 108)$

Apply the distributive property.

$y = - (- 3 x^{4}) - (15 x^{2}) - (- 12 x^{3}) - (108 x) - 1 \cdot 108$

Simplify.

Multiply $- 3$ by $- 1$ .

$y = 3 x^{4} - (15 x^{2}) - (- 12 x^{3}) - (108 x) - 1 \cdot 108$

Multiply $15$ by $- 1$ .

$y = 3 x^{4} - 15 x^{2} - (- 12 x^{3}) - (108 x) - 1 \cdot 108$

Multiply $- 12$ by $- 1$ .

$y = 3 x^{4} - 15 x^{2} + 12 x^{3} - (108 x) - 1 \cdot 108$

Multiply $108$ by $- 1$ .

$y = 3 x^{4} - 15 x^{2} + 12 x^{3} - 108 x - 1 \cdot 108$

Multiply $- 1$ by $108$ .

$y = 3 x^{4} - 15 x^{2} + 12 x^{3} - 108 x - 108$

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

Not symmetric to the origin

Determine the symmetry.

Not symmetric to the x-axis

Not symmetric to the y-axis

Not symmetric to the origin

Find the Symmetry -3(x-2)^2(x^2-9)

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