Find the Symmetry 6x=y^3

$6 x = y^{3}$

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 $y$ .

$6 x = {(- y)}^{3}$

Step 4

Simplify.

Apply the product rule to $- y$ .

$6 x = {(- 1)}^{3} y^{3}$

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

$6 x = - y^{3}$

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 $x$ .

$6 (- x) = y^{3}$

Step 7

Multiply $- 1$ by $6$ .

$- 6 x = y^{3}$

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 $y$ .

$6 (- x) = {(- y)}^{3}$

Step 10

Simplify.

Multiply $- 1$ by $6$ .

$- 6 x = {(- y)}^{3}$

Apply the product rule to $- y$ .

$- 6 x = {(- 1)}^{3} y^{3}$

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

$- 6 x = - y^{3}$

Step 11

Multiply both sides by $- 1$ .

Multiply each term by $- 1$ .

$- (- 6 x) = - - y^{3}$

Multiply $- 6$ by $- 1$ .

$6 x = - - y^{3}$

Multiply $- - y^{3}$ .

Multiply $- 1$ by $- 1$ .

$6 x = 1 y^{3}$

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

$6 x = y^{3}$

Step 12

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

Symmetric with respect to the origin

Step 13

Find the Symmetry 6x=y^3

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