Calculus Examples

$y = \frac{2 x}{x^{2} - 4}$

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

Simplify the denominator.

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Step 3.1

Rewrite $4$ as $2^{2}$ .

$y = \frac{2 x}{x^{2} - 2^{2}}$

Step 3.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 2$ .

$y = \frac{2 x}{(x + 2) (x - 2)}$

Step 4

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

$- y = \frac{2 x}{(x + 2) (x - 2)}$

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 $x$ .

$y = \frac{2 (- x)}{(- x + 2) (- x - 2)}$

Step 7

Simplify.

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Step 7.1

Multiply $- 1$ by $2$ .

$y = \frac{- 2 x}{(- x + 2) (- x - 2)}$

Step 7.2

Move the negative in front of the fraction.

$y = - \frac{2 x}{(- x + 2) (- x - 2)}$

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 $y$ .

$- y = \frac{2 (- x)}{(- x + 2) (- x - 2)}$

Step 10

Simplify.

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Step 10.1

Multiply $- 1$ by $2$ .

$- y = \frac{- 2 x}{(- x + 2) (- x - 2)}$

Step 10.2

Move the negative in front of the fraction.

$- y = - \frac{2 x}{(- x + 2) (- x - 2)}$

Step 11

Multiply both sides by $- 1$ .

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Step 11.1

Multiply each term by $- 1$ .

$- - y = - - \frac{2 x}{(- x + 2) (- x - 2)}$

Step 11.2

Multiply $- - y$ .

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Step 11.2.1

Multiply $- 1$ by $- 1$ .

$1 y = - - \frac{2 x}{(- x + 2) (- x - 2)}$

Step 11.2.2

Multiply $y$ by $1$ .

$y = - - \frac{2 x}{(- x + 2) (- x - 2)}$

Step 11.3

Multiply $- - \frac{2 x}{(- x + 2) (- x - 2)}$ .

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Step 11.3.1

Multiply $- 1$ by $- 1$ .

$y = 1 \frac{2 x}{(- x + 2) (- x - 2)}$

Step 11.3.2

Multiply $\frac{2 x}{(- x + 2) (- x - 2)}$ by $1$ .

$y = \frac{2 x}{(- x + 2) (- x - 2)}$

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