Algebra Examples

\frac{x^{2} - 36}{x}

$\frac{x^{2} - 36}{x}$

Step 1

Write

\frac{x^{2} - 36}{x}

$\frac{x^{2} - 36}{x}$ as an equation.

y = \frac{x^{2} - 36}{x}

$y = \frac{x^{2} - 36}{x}$

Step 2

There are three types of symmetry:

1. X-Axis Symmetry

2. Y-Axis Symmetry

3. Origin Symmetry

Step 3

(x, y)

$(x, y)$ exists on the graph, then the graph is symmetric about the:

1. X-Axis if

(x, - y)

$(x, - y)$ exists on the graph

2. Y-Axis if

(- x, y)

$(- x, y)$ exists on the graph

3. Origin if

(- x, - y)

$(- x, - y)$ exists on the graph

Step 4

Simplify the numerator.

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Step 4.1

Rewrite

36

$36$ as

6^{2}

$6^{2}$ .

y = \frac{x^{2} - 6^{2}}{x}

$y = \frac{x^{2} - 6^{2}}{x}$

Step 4.2

Since both terms are perfect squares, factor using the difference of squares formula,

a^{2} - b^{2} = (a + b) (a - b)

$a^{2} - b^{2} = (a + b) (a - b)$ where

a = x

$a = x$ and

b = 6

$b = 6$ .

y = \frac{(x + 6) (x - 6)}{x}

$y = \frac{(x + 6) (x - 6)}{x}$

y = \frac{(x + 6) (x - 6)}{x}

$y = \frac{(x + 6) (x - 6)}{x}$

Step 5

Check if the graph is symmetric about the

x

$x$ -axis by plugging in

- y

$- y$ for

y

$y$ .

- y = \frac{(x + 6) (x - 6)}{x}

$- y = \frac{(x + 6) (x - 6)}{x}$

Step 6

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 7

Check if the graph is symmetric about the

y

$y$ -axis by plugging in

- x

$- x$ for

x

$x$ .

y = \frac{(- x + 6) (- x - 6)}{- x}

$y = \frac{(- x + 6) (- x - 6)}{- x}$

Step 8

Move the negative in front of the fraction.

y = - \frac{(- x + 6) (- x - 6)}{x}

$y = - \frac{(- x + 6) (- x - 6)}{x}$

Step 9

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 10

Check if the graph is symmetric about the origin by plugging in

- x

$- x$ for

x

$x$ and

- y

$- y$ for

y

$y$ .

- y = \frac{(- x + 6) (- x - 6)}{- x}

$- y = \frac{(- x + 6) (- x - 6)}{- x}$

Step 11

Move the negative in front of the fraction.

- y = - \frac{(- x + 6) (- x - 6)}{x}

$- y = - \frac{(- x + 6) (- x - 6)}{x}$

Step 12

Multiply both sides by

- 1

$- 1$ .

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Step 12.1

Multiply each term by

- 1

$- 1$ .

- - y = - - \frac{(- x + 6) (- x - 6)}{x}

$- - y = - - \frac{(- x + 6) (- x - 6)}{x}$

Step 12.2

Multiply

- - y

$- - y$ .

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Step 12.2.1

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

1 y = - - \frac{(- x + 6) (- x - 6)}{x}

$1 y = - - \frac{(- x + 6) (- x - 6)}{x}$

Step 12.2.2

Multiply

y

$y$ by

1

$1$ .

y = - - \frac{(- x + 6) (- x - 6)}{x}

$y = - - \frac{(- x + 6) (- x - 6)}{x}$

y = - - \frac{(- x + 6) (- x - 6)}{x}

$y = - - \frac{(- x + 6) (- x - 6)}{x}$

Step 12.3

Multiply

- - \frac{(- x + 6) (- x - 6)}{x}

$- - \frac{(- x + 6) (- x - 6)}{x}$ .

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Step 12.3.1

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

y = 1 \frac{(- x + 6) (- x - 6)}{x}

$y = 1 \frac{(- x + 6) (- x - 6)}{x}$

Step 12.3.2

Multiply

\frac{(- x + 6) (- x - 6)}{x}

$\frac{(- x + 6) (- x - 6)}{x}$ by

1

$1$ .

y = \frac{(- x + 6) (- x - 6)}{x}

$y = \frac{(- x + 6) (- x - 6)}{x}$

y = \frac{(- x + 6) (- x - 6)}{x}

$y = \frac{(- x + 6) (- x - 6)}{x}$

y = \frac{(- x + 6) (- x - 6)}{x}

$y = \frac{(- x + 6) (- x - 6)}{x}$

Step 13

Since the equation is identical to the original equation, it is symmetric to the origin.

Symmetric with respect to the origin

Step 14