Find the Symmetry 6x^3-9x-x^5

$6 x^{3} - 9 x - x^{5}$

Step 1

Write $6 x^{3} - 9 x - x^{5}$ as an equation.

$y = 6 x^{3} - 9 x - x^{5}$

Step 2

There are three types of symmetry:

1. X-Axis Symmetry

2. Y-Axis Symmetry

3. Origin Symmetry

Step 3

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 4

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

$- y = 6 x^{3} - 9 x - x^{5}$

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 $x$ .

$y = 6 {(- x)}^{3} - 9 (- x) - {(- x)}^{5}$

Step 7

Simplify each term.

Apply the product rule to $- x$ .

$y = 6 ({(- 1)}^{3} x^{3}) - 9 (- x) - {(- x)}^{5}$

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

$y = 6 (- x^{3}) - 9 (- x) - {(- x)}^{5}$

Multiply $- 1$ by $6$ .

$y = - 6 x^{3} - 9 (- x) - {(- x)}^{5}$

Multiply $- 1$ by $- 9$ .

$y = - 6 x^{3} + 9 x - {(- x)}^{5}$

Apply the product rule to $- x$ .

$y = - 6 x^{3} + 9 x - ({(- 1)}^{5} x^{5})$

Multiply $- 1$ by ${(- 1)}^{5}$ by adding the exponents.

Move ${(- 1)}^{5}$ .

$y = - 6 x^{3} + 9 x + {(- 1)}^{5} \cdot - 1 x^{5}$

Multiply ${(- 1)}^{5}$ by $- 1$ .

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

$y = - 6 x^{3} + 9 x + {(- 1)}^{5} \cdot {(- 1)}^{1} x^{5}$

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

$y = - 6 x^{3} + 9 x + {(- 1)}^{5 + 1} x^{5}$

Add $5$ and $1$ .

$y = - 6 x^{3} + 9 x + {(- 1)}^{6} x^{5}$

Raise $- 1$ to the power of $6$ .

$y = - 6 x^{3} + 9 x + 1 x^{5}$

Multiply $x^{5}$ by $1$ .

$y = - 6 x^{3} + 9 x + x^{5}$

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 $y$ .

$- y = 6 {(- x)}^{3} - 9 (- x) - {(- x)}^{5}$

Step 10

Simplify each term.

Apply the product rule to $- x$ .

$- y = 6 ({(- 1)}^{3} x^{3}) - 9 (- x) - {(- x)}^{5}$

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

$- y = 6 (- x^{3}) - 9 (- x) - {(- x)}^{5}$

Multiply $- 1$ by $6$ .

$- y = - 6 x^{3} - 9 (- x) - {(- x)}^{5}$

Multiply $- 1$ by $- 9$ .

$- y = - 6 x^{3} + 9 x - {(- x)}^{5}$

Apply the product rule to $- x$ .

$- y = - 6 x^{3} + 9 x - ({(- 1)}^{5} x^{5})$

Multiply $- 1$ by ${(- 1)}^{5}$ by adding the exponents.

Move ${(- 1)}^{5}$ .

$- y = - 6 x^{3} + 9 x + {(- 1)}^{5} \cdot - 1 x^{5}$

Multiply ${(- 1)}^{5}$ by $- 1$ .

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

$- y = - 6 x^{3} + 9 x + {(- 1)}^{5} \cdot {(- 1)}^{1} x^{5}$

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

$- y = - 6 x^{3} + 9 x + {(- 1)}^{5 + 1} x^{5}$

Add $5$ and $1$ .

$- y = - 6 x^{3} + 9 x + {(- 1)}^{6} x^{5}$

Raise $- 1$ to the power of $6$ .

$- y = - 6 x^{3} + 9 x + 1 x^{5}$

Multiply $x^{5}$ by $1$ .

$- y = - 6 x^{3} + 9 x + x^{5}$

Step 11

Multiply both sides by $- 1$ .

Multiply each term by $- 1$ .

$- - y = - (- 6 x^{3}) - (9 x) - x^{5}$

Multiply $- - y$ .

Multiply $- 1$ by $- 1$ .

$1 y = - (- 6 x^{3}) - (9 x) - x^{5}$

Multiply $y$ by $1$ .

$y = - (- 6 x^{3}) - (9 x) - x^{5}$

Simplify each term.

Multiply $- 6$ by $- 1$ .

$y = 6 x^{3} - (9 x) - x^{5}$

Multiply $9$ by $- 1$ .

$y = 6 x^{3} - 9 x - x^{5}$

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^3-9x-x^5

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