Find the Symmetry y=x^9

$y = x^{9}$

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 $y$ .

$- y = x^{9}$

Step 4

Simplify the left side.

Remove parentheses.

$- y = x^{9}$

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 $x$ .

$y = {(- x)}^{9}$

Step 7

Simplify the right side.

Simplify ${(- x)}^{9}$ .

Apply the product rule to $- x$ .

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

Raise $- 1$ to the power of $9$ .

$y = - x^{9}$

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 $y$ .

$- y = {(- x)}^{9}$

Step 10

Solve for $y$ .

Simplify the left side.

Remove parentheses.

$- y = {(- x)}^{9}$

Simplify the right side.

Simplify ${(- x)}^{9}$ .

Apply the product rule to $- x$ .

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

Raise $- 1$ to the power of $9$ .

$- y = - x^{9}$

Simplify the left side.

Multiply $- (- y)$ .

Multiply $- 1$ by $- 1$ .

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

Multiply $y$ by $1$ .

$y = - ({(- x)}^{9})$

Simplify the right side.

Simplify $- ({(- x)}^{9})$ .

Apply the product rule to $- x$ .

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

Multiply $- 1$ by ${(- 1)}^{9}$ by adding the exponents.

Move ${(- 1)}^{9}$ .

$y = {(- 1)}^{9} \cdot - 1 x^{9}$

Multiply ${(- 1)}^{9}$ by $- 1$ .

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

$y = {(- 1)}^{9} \cdot {(- 1)}^{1} x^{9}$

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

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

Add $9$ and $1$ .

$y = {(- 1)}^{10} x^{9}$

Raise $- 1$ to the power of $10$ .

$y = 1 x^{9}$

Multiply $x^{9}$ by $1$ .

$y = x^{9}$

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^9

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