Find the Symmetry x^2y^2+xy=4

$x^{2} y^{2} + x y = 4$

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

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

$x^{2} {(- y)}^{2} + x (- y) = 4$

Step 4

Simplify each term.

Apply the product rule to $- y$ .

$x^{2} ({(- 1)}^{2} y^{2}) + x (- y) = 4$

Rewrite using the commutative property of multiplication.

${(- 1)}^{2} x^{2} y^{2} + x (- y) = 4$

Raise $- 1$ to the power of $2$ .

$1 x^{2} y^{2} + x (- y) = 4$

Multiply $x^{2}$ by $1$ .

$x^{2} y^{2} + x (- y) = 4$

Rewrite using the commutative property of multiplication.

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

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 $x$ .

${(- x)}^{2} y^{2} - x y = 4$

Step 7

Simplify each term.

Apply the product rule to $- x$ .

${(- 1)}^{2} x^{2} y^{2} - x y = 4$

Raise $- 1$ to the power of $2$ .

$1 x^{2} y^{2} - x y = 4$

Multiply $x^{2}$ by $1$ .

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

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 $y$ .

${(- x)}^{2} {(- y)}^{2} - x (- y) = 4$

Step 10

Simplify each term.

Apply the product rule to $- x$ .

${(- 1)}^{2} x^{2} {(- y)}^{2} - x (- y) = 4$

Raise $- 1$ to the power of $2$ .

$1 x^{2} {(- y)}^{2} - x (- y) = 4$

Multiply $x^{2}$ by $1$ .

$x^{2} {(- y)}^{2} - x (- y) = 4$

Apply the product rule to $- y$ .

$x^{2} ({(- 1)}^{2} y^{2}) - x (- y) = 4$

Rewrite using the commutative property of multiplication.

${(- 1)}^{2} x^{2} y^{2} - x (- y) = 4$

Raise $- 1$ to the power of $2$ .

$1 x^{2} y^{2} - x (- y) = 4$

Multiply $x^{2}$ by $1$ .

$x^{2} y^{2} - x (- y) = 4$

Rewrite using the commutative property of multiplication.

$x^{2} y^{2} - 1 \cdot - 1 x y = 4$

Multiply $- 1$ by $- 1$ .

$x^{2} y^{2} + 1 x y = 4$

Multiply $x$ by $1$ .

$x^{2} y^{2} + x y = 4$

Step 11

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

Symmetric with respect to the origin

Step 12

Find the Symmetry x^2y^2+xy=4

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