Find the Symmetry y = cube root of 6x

$y = \sqrt[3]{6 x}$

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 $y$ .

$- y = \sqrt[3]{6 x}$

Step 4

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 5

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

$y = \sqrt[3]{6 (- x)}$

Step 6

Simplify.

Multiply $- 1$ by $6$ .

$y = \sqrt[3]{- 6 x}$

Rewrite $- 6 x$ as ${({(- 1)}^{3})}^{3} \cdot (6 x)$ .

Rewrite $- 6$ as $- 1 (6)$ .

$y = \sqrt[3]{- 1 (6) x}$

Rewrite $- 1$ as ${(- 1)}^{3}$ .

$y = \sqrt[3]{{(- 1)}^{3} \cdot 6 x}$

Rewrite $- 1$ as ${(- 1)}^{3}$ .

$y = \sqrt[3]{{({(- 1)}^{3})}^{3} \cdot 6 x}$

Add parentheses.

$y = \sqrt[3]{{({(- 1)}^{3})}^{3} \cdot (6 x)}$

Pull terms out from under the radical.

$y = {(- 1)}^{3} \sqrt[3]{6 x}$

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

$y = - 1 \sqrt[3]{6 x}$

Rewrite $- 1 \sqrt[3]{6 x}$ as $- \sqrt[3]{6 x}$ .

$y = - \sqrt[3]{6 x}$

Step 7

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 8

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

$- y = \sqrt[3]{6 (- x)}$

Step 9

Simplify.

Multiply $- 1$ by $6$ .

$- y = \sqrt[3]{- 6 x}$

Rewrite $- 6 x$ as ${({(- 1)}^{3})}^{3} \cdot (6 x)$ .

Rewrite $- 6$ as $- 1 (6)$ .

$- y = \sqrt[3]{- 1 (6) x}$

Rewrite $- 1$ as ${(- 1)}^{3}$ .

$- y = \sqrt[3]{{(- 1)}^{3} \cdot 6 x}$

Rewrite $- 1$ as ${(- 1)}^{3}$ .

$- y = \sqrt[3]{{({(- 1)}^{3})}^{3} \cdot 6 x}$

Add parentheses.

$- y = \sqrt[3]{{({(- 1)}^{3})}^{3} \cdot (6 x)}$

Pull terms out from under the radical.

$- y = {(- 1)}^{3} \sqrt[3]{6 x}$

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

$- y = - 1 \sqrt[3]{6 x}$

Rewrite $- 1 \sqrt[3]{6 x}$ as $- \sqrt[3]{6 x}$ .

$- y = - \sqrt[3]{6 x}$

Step 10

Multiply both sides by $- 1$ .

Multiply each term by $- 1$ .

$- - y = - - \sqrt[3]{6 x}$

Multiply $- - y$ .

Multiply $- 1$ by $- 1$ .

$1 y = - - \sqrt[3]{6 x}$

Multiply $y$ by $1$ .

$y = - - \sqrt[3]{6 x}$

Multiply $- - \sqrt[3]{6 x}$ .

Multiply $- 1$ by $- 1$ .

$y = 1 \sqrt[3]{6 x}$

Multiply $\sqrt[3]{6 x}$ by $1$ .

$y = \sqrt[3]{6 x}$

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 = cube root of 6x

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