Find the Symmetry y^2=x^3-8x

$y^{2} = x^{3} - 8 x$

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 $y$ .

${(- y)}^{2} = x^{3} - 8 x$

Step 4

Simplify the left side.

Simplify ${(- y)}^{2}$ .

Apply the product rule to $- y$ .

${(- 1)}^{2} y^{2} = x^{3} - 8 x$

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

$1 y^{2} = x^{3} - 8 x$

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

$y^{2} = x^{3} - 8 x$

Step 5

Since the equation is identical to the original equation, it is symmetric to the x-axis.

Symmetric with respect to the x-axis

Step 6

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

$y^{2} = {(- x)}^{3} - 8 (- x)$

Step 7

Simplify the right side.

Simplify each term.

Apply the product rule to $- x$ .

$y^{2} = {(- 1)}^{3} x^{3} - 8 (- x)$

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

$y^{2} = - x^{3} - 8 (- x)$

Multiply $- 1$ by $- 8$ .

$y^{2} = - x^{3} + 8 x$

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 $y$ .

${(- y)}^{2} = {(- x)}^{3} - 8 (- x)$

Step 10

Solve for $- y^{2}$ .

Simplify the left side.

Simplify ${(- y)}^{2}$ .

Apply the product rule to $- y$ .

${(- 1)}^{2} y^{2} = {(- x)}^{3} - 8 (- x)$

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

$1 y^{2} = {(- x)}^{3} - 8 (- x)$

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

$y^{2} = {(- x)}^{3} - 8 (- x)$

Simplify the right side.

Simplify each term.

Apply the product rule to $- x$ .

$y^{2} = {(- 1)}^{3} x^{3} - 8 (- x)$

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

$y^{2} = - x^{3} - 8 (- x)$

Multiply $- 1$ by $- 8$ .

$y^{2} = - x^{3} + 8 x$

Simplify the left side.

Simplify $- ({(- y)}^{2})$ .

Apply the product rule to $- y$ .

$- ({(- 1)}^{2} y^{2}) = - ({(- x)}^{3} - 8 (- x))$

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

Move ${(- 1)}^{2}$ .

${(- 1)}^{2} \cdot - 1 y^{2} = - ({(- x)}^{3} - 8 (- x))$

Multiply ${(- 1)}^{2}$ by $- 1$ .

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

${(- 1)}^{2} \cdot {(- 1)}^{1} y^{2} = - ({(- x)}^{3} - 8 (- x))$

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

${(- 1)}^{2 + 1} y^{2} = - ({(- x)}^{3} - 8 (- x))$

Add $2$ and $1$ .

${(- 1)}^{3} y^{2} = - ({(- x)}^{3} - 8 (- x))$

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

$- y^{2} = - ({(- x)}^{3} - 8 (- x))$

Simplify the right side.

Simplify $- ({(- x)}^{3} - 8 (- x))$ .

Simplify each term.

Apply the product rule to $- x$ .

$- y^{2} = - ({(- 1)}^{3} x^{3} - 8 (- x))$

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

$- y^{2} = - (- x^{3} - 8 (- x))$

Multiply $- 1$ by $- 8$ .

$- y^{2} = - (- x^{3} + 8 x)$

Apply the distributive property.

$- y^{2} = - - x^{3} - (8 x)$

Multiply $- - x^{3}$ .

Multiply $- 1$ by $- 1$ .

$- y^{2} = 1 x^{3} - (8 x)$

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

$- y^{2} = x^{3} - (8 x)$

Multiply $8$ by $- 1$ .

$- y^{2} = x^{3} - 8 x$

Step 11

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

Not symmetric to the origin

Step 12

Determine the symmetry.

Symmetric with respect to the x-axis

Step 13

Find the Symmetry y^2=x^3-8x

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