Find the Symmetry 3x^2+x^3

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Algebra

$3 x^{2} + x^{3}$

Write the polynomial as an equation.

$y = 3 x^{2} + x^{3}$

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 $y$ .

$- y = 3 x^{2} + x^{3}$

Remove parentheses.

$- y = 3 x^{2} + x^{3}$

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 $x$ .

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

Simplify each term.

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Apply the product rule to $- x$ .

$y = 3 ({(- 1)}^{2} x^{2}) + {(- x)}^{3}$

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

$y = 3 (1 x^{2}) + {(- x)}^{3}$

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

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

Apply the product rule to $- x$ .

$y = 3 x^{2} + {(- 1)}^{3} x^{3}$

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

$y = 3 x^{2} - x^{3}$

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 $y$ .

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

Solve for $y$ .

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Simplify each term.

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Apply the product rule to $- x$ .

$y = 3 ({(- 1)}^{2} x^{2}) + {(- x)}^{3}$

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

$y = 3 (1 x^{2}) + {(- x)}^{3}$

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

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

Apply the product rule to $- x$ .

$y = 3 x^{2} + {(- 1)}^{3} x^{3}$

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

$y = 3 x^{2} - x^{3}$

Remove parentheses.

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

Simplify each term.

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Apply the product rule to $- x$ .

$- y = 3 ({(- 1)}^{2} x^{2}) + {(- x)}^{3}$

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

$- y = 3 (1 x^{2}) + {(- x)}^{3}$

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

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

Apply the product rule to $- x$ .

$- y = 3 x^{2} + {(- 1)}^{3} x^{3}$

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

$- y = 3 x^{2} - x^{3}$

Multiply $- (- y)$ .

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Multiply $- 1$ by $- 1$ .

$1 y = - (3 {(- x)}^{2} + {(- x)}^{3})$

Multiply $y$ by $1$ .

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

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

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Simplify each term.

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Apply the product rule to $- x$ .

$y = - (3 ({(- 1)}^{2} x^{2}) + {(- x)}^{3})$

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

$y = - (3 (1 x^{2}) + {(- x)}^{3})$

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

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

Apply the product rule to $- x$ .

$y = - (3 x^{2} + {(- 1)}^{3} x^{3})$

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

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

Simplify by multiplying through.

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Apply the distributive property.

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

Multiply $3$ by $- 1$ .

$y = - 3 x^{2} - - x^{3}$

Multiply $- - x^{3}$ .

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Multiply $- 1$ by $- 1$ .

$y = - 3 x^{2} + 1 x^{3}$

Multiply $x^{3}$ by $1$ .

$y = - 3 x^{2} + x^{3}$

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

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