Find the Symmetry (x-8)/(x^2-64)

$\frac{x - 8}{x^{2} - 64}$

Write $\frac{x - 8}{x^{2} - 64}$ as an equation.

$y = \frac{x - 8}{x^{2} - 64}$

Simplify the right side.

Simplify $\frac{x - 8}{x^{2} - 64}$ .

Simplify the denominator.

Rewrite $64$ as $8^{2}$ .

$y = \frac{x - 8}{x^{2} - 8^{2}}$

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 $b = 8$ .

$y = \frac{x - 8}{(x + 8) (x - 8)}$

Cancel the common factor of $x - 8$ .

Cancel the common factor.

$y = \frac{x - 8}{(x + 8) (x - 8)}$

Rewrite the expression.

$y = \frac{1}{x + 8}$

There are three types of symmetry:

1. X-Axis Symmetry

2. Y-Axis Symmetry

3. Origin Symmetry

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

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

$- y = \frac{1}{x + 8}$

Simplify the left side.

Remove parentheses.

$- y = \frac{1}{x + 8}$

Since the equation is not identical to the original equation, it is not symmetric to the x-axis.

Not symmetric to the x-axis

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

$y = \frac{1}{(- x) + 8}$

Simplify the right side.

Simplify $\frac{1}{(- x) + 8}$ .

Factor $- 1$ out of $- x$ .

$y = \frac{1}{- (x) + 8}$

Rewrite $8$ as $- 1 (- 8)$ .

$y = \frac{1}{- (x) - 1 (- 8)}$

Factor $- 1$ out of $- (x) - 1 (- 8)$ .

$y = \frac{1}{- (x - 8)}$

Rewrite negatives.

Rewrite $- (x - 8)$ as $- 1 (x - 8)$ .

$y = \frac{1}{- 1 (x - 8)}$

Move the negative in front of the fraction.

$y = - \frac{1}{x - 8}$

Since the equation is not identical to the original equation, it is not symmetric to the y-axis.

Not symmetric to the y-axis

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

$- y = \frac{1}{(- x) + 8}$

Solve for $y$ .

Simplify the left side.

Remove parentheses.

$- y = \frac{1}{(- x) + 8}$

Simplify the right side.

Simplify $\frac{1}{(- x) + 8}$ .

Factor $- 1$ out of $- x$ .

$- y = \frac{1}{- (x) + 8}$

Rewrite $8$ as $- 1 (- 8)$ .

$- y = \frac{1}{- (x) - 1 (- 8)}$

Factor $- 1$ out of $- (x) - 1 (- 8)$ .

$- y = \frac{1}{- (x - 8)}$

Rewrite negatives.

Rewrite $- (x - 8)$ as $- 1 (x - 8)$ .

$- y = \frac{1}{- 1 (x - 8)}$

Move the negative in front of the fraction.

$- y = - \frac{1}{x - 8}$

Simplify the left side.

Multiply $- (- y)$ .

Multiply $- 1$ by $- 1$ .

$1 y = - (\frac{1}{(- x) + 8})$

Multiply $y$ by $1$ .

$y = - (\frac{1}{(- x) + 8})$

Simplify the right side.

Simplify $- (\frac{1}{(- x) + 8})$ .

Factor $- 1$ out of $- x$ .

$y = - \frac{1}{- (x) + 8}$

Rewrite $8$ as $- 1 (- 8)$ .

$y = - \frac{1}{- (x) - 1 (- 8)}$

Factor $- 1$ out of $- (x) - 1 (- 8)$ .

$y = - \frac{1}{- (x - 8)}$

Simplify the expression.

Rewrite $- (x - 8)$ as $- 1 (x - 8)$ .

$y = - \frac{1}{- 1 (x - 8)}$

Move the negative in front of the fraction.

$y = - - \frac{1}{x - 8}$

Multiply $- 1$ by $- 1$ .

$y = 1 \frac{1}{x - 8}$

Multiply $\frac{1}{x - 8}$ by $1$ .

$y = \frac{1}{x - 8}$

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

Not symmetric to the origin

Determine the symmetry.

Not symmetric to the x-axis

Not symmetric to the y-axis

Not symmetric to the origin

Find the Symmetry (x-8)/(x^2-64)

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