Find the Symmetry (x^2+4x-45)/(x^2-9)

$\frac{x^{2} + 4 x - 45}{x^{2} - 9}$

Step 1

Write $\frac{x^{2} + 4 x - 45}{x^{2} - 9}$ as an equation.

$y = \frac{x^{2} + 4 x - 45}{x^{2} - 9}$

Step 2

There are three types of symmetry:

1. X-Axis Symmetry

2. Y-Axis Symmetry

3. Origin Symmetry

Step 3

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 4

Factor $x^{2} + 4 x - 45$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 45$ and whose sum is $4$ .

$- 5, 9$

Write the factored form using these integers.

$y = \frac{(x - 5) (x + 9)}{x^{2} - 9}$

Step 5

Simplify the denominator.

Rewrite $9$ as $3^{2}$ .

$y = \frac{(x - 5) (x + 9)}{x^{2} - 3^{2}}$

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 $b = 3$ .

$y = \frac{(x - 5) (x + 9)}{(x + 3) (x - 3)}$

Step 6

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

$- y = \frac{(x - 5) (x + 9)}{(x + 3) (x - 3)}$

Step 7

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 8

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

$y = \frac{(- x - 5) (- x + 9)}{(- x + 3) (- x - 3)}$

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 $- x$ for $x$ and $- y$ for $y$ .

$- y = \frac{(- x - 5) (- x + 9)}{(- x + 3) (- x - 3)}$

Step 11

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

Not symmetric to the origin

Step 12

Determine the symmetry.

Not symmetric to the x-axis

Not symmetric to the y-axis

Not symmetric to the origin

Step 13

Find the Symmetry (x^2+4x-45)/(x^2-9)

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