Find the Symmetry xy- square root of 4-x^2=0

$x y - \sqrt{4 - x^{2}} = 0$

Simplify the left side.

Simplify each term.

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

$x y - \sqrt{2^{2} - x^{2}} = 0$

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

$x y - \sqrt{(2 + x) (2 - x)} = 0$

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 $y$ .

$x (- y) - \sqrt{(2 + x) (2 - x)} = 0$

Simplify the left side.

Rewrite using the commutative property of multiplication.

$- x y - \sqrt{(2 + x) (2 - x)} = 0$

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 $x$ .

$(- x) y - \sqrt{(2 - x) (2 - (- x))} = 0$

Simplify the left side.

Simplify $(- x) y - \sqrt{(2 - x) (2 - (- x))}$ .

Remove parentheses.

$(- x) y - \sqrt{(2 - x) (2 - (- x))} = 0$

Multiply $- - x$ .

Multiply $- 1$ by $- 1$ .

$- x y - \sqrt{(2 - x) (2 + 1 x)} = 0$

Multiply $x$ by $1$ .

$- x y - \sqrt{(2 - x) (2 + x)} = 0$

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 $y$ .

$(- x) (- y) - \sqrt{(2 - x) (2 - (- x))} = 0$

Solve for $- x y + \sqrt{(2 - x) (2 + x)}$ .

Simplify the left side.

Simplify $(- x) (- y) - \sqrt{(2 - x) (2 - (- x))}$ .

Remove parentheses.

$(- x) (- y) - \sqrt{(2 - x) (2 - (- x))} = 0$

Simplify each term.

Rewrite using the commutative property of multiplication.

$- 1 \cdot - 1 x y - \sqrt{(2 - x) (2 - (- x))} = 0$

Multiply $- 1$ by $- 1$ .

$1 x y - \sqrt{(2 - x) (2 - (- x))} = 0$

Multiply $x$ by $1$ .

$x y - \sqrt{(2 - x) (2 - (- x))} = 0$

Multiply $- - x$ .

Multiply $- 1$ by $- 1$ .

$x y - \sqrt{(2 - x) (2 + 1 x)} = 0$

Multiply $x$ by $1$ .

$x y - \sqrt{(2 - x) (2 + x)} = 0$

Simplify the left side.

Simplify $- ((- x) (- y) - \sqrt{(2 - x) (2 - (- x))})$ .

Remove parentheses.

$- ((- x) (- y) - \sqrt{(2 - x) (2 - (- x))}) = - (0)$

Simplify each term.

Rewrite using the commutative property of multiplication.

$- (- 1 \cdot - 1 x y - \sqrt{(2 - x) (2 - (- x))}) = - (0)$

Multiply $- 1$ by $- 1$ .

$- (1 x y - \sqrt{(2 - x) (2 - (- x))}) = - (0)$

Multiply $x$ by $1$ .

$- (x y - \sqrt{(2 - x) (2 - (- x))}) = - (0)$

Multiply $- - x$ .

Multiply $- 1$ by $- 1$ .

$- (x y - \sqrt{(2 - x) (2 + 1 x)}) = - (0)$

Multiply $x$ by $1$ .

$- (x y - \sqrt{(2 - x) (2 + x)}) = - (0)$

Apply the distributive property.

$- (x y) - - \sqrt{(2 - x) (2 + x)} = - (0)$

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

Multiply $- 1$ by $- 1$ .

$- x y + 1 \sqrt{(2 - x) (2 + x)} = - (0)$

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

$- x y + \sqrt{(2 - x) (2 + x)} = - (0)$

Simplify the right side.

Multiply $- 1$ by $0$ .

$- x y + \sqrt{(2 - x) (2 + x)} = 0$

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 xy- square root of 4-x^2=0

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