Find the Symmetry y=x^3+3x

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

Step 1

There are three types of symmetry:

1. X-Axis Symmetry

2. Y-Axis Symmetry

3. Origin Symmetry

Step 2

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 3

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

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

Step 4

Simplify the left side.

Remove parentheses.

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

Step 5

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 6

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

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

Step 7

Simplify the right side.

Simplify each term.

Apply the product rule to $- x$ .

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

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

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

Multiply $- 1$ by $3$ .

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

Step 8

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 9

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

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

Step 10

Solve for $y$ .

Simplify the left side.

Remove parentheses.

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

Simplify the right side.

Simplify each term.

Apply the product rule to $- x$ .

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

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

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

Multiply $- 1$ by $3$ .

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

Simplify the left side.

Multiply $- (- y)$ .

Multiply $- 1$ by $- 1$ .

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

Multiply $y$ by $1$ .

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

Simplify the right side.

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

Simplify each term.

Apply the product rule to $- x$ .

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

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

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

Multiply $- 1$ by $3$ .

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

Apply the distributive property.

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

Multiply $- - x^{3}$ .

Multiply $- 1$ by $- 1$ .

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

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

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

Multiply $- 3$ by $- 1$ .

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

Step 11

Since the equation is identical to the original equation, it is symmetric to the origin.

Symmetric with respect to the origin

Step 12

Find the Symmetry y=x^3+3x

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