Find the Symmetry y=2x^5

$y = 2 x^{5}$

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 $y$ .

$- y = 2 x^{5}$

Step 4

Simplify the left side.

Remove parentheses.

$- y = 2 x^{5}$

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 $x$ .

$y = 2 {(- x)}^{5}$

Step 7

Simplify the right side.

Simplify $2 {(- x)}^{5}$ .

Apply the product rule to $- x$ .

$y = 2 ({(- 1)}^{5} x^{5})$

Raise $- 1$ to the power of $5$ .

$y = 2 (- x^{5})$

Multiply $- 1$ by $2$ .

$y = - 2 x^{5}$

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 $y$ .

$- y = 2 {(- x)}^{5}$

Step 10

Solve for $y$ .

Simplify the left side.

Remove parentheses.

$- y = 2 {(- x)}^{5}$

Simplify the right side.

Simplify $2 {(- x)}^{5}$ .

Apply the product rule to $- x$ .

$- y = 2 ({(- 1)}^{5} x^{5})$

Raise $- 1$ to the power of $5$ .

$- y = 2 (- x^{5})$

Multiply $- 1$ by $2$ .

$- y = - 2 x^{5}$

Simplify the left side.

Multiply $- (- y)$ .

Multiply $- 1$ by $- 1$ .

$1 y = - (2 {(- x)}^{5})$

Multiply $y$ by $1$ .

$y = - (2 {(- x)}^{5})$

Simplify the right side.

Simplify $- (2 {(- x)}^{5})$ .

Apply the product rule to $- x$ .

$y = - (2 ({(- 1)}^{5} x^{5}))$

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

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

$y = {(- 1)}^{5} \cdot - 1 (2 (x^{5}))$

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

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

$y = {(- 1)}^{5} \cdot {(- 1)}^{1} (2 (x^{5}))$

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

$y = {(- 1)}^{5 + 1} (2 (x^{5}))$

Add $5$ and $1$ .

$y = {(- 1)}^{6} (2 (x^{5}))$

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

$y = 1 (2 (x^{5}))$

Multiply $2 (x^{5})$ by $1$ .

$y = 2 x^{5}$

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=2x^5

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