Find the Symmetry 4x^2+9y^2=36

$4 x^{2} + 9 y^{2} = 36$

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 $y$ .

$4 x^{2} + 9 {(- y)}^{2} = 36$

Simplify the left side.

Simplify each term.

Apply the product rule to $- y$ .

$4 x^{2} + 9 ({(- 1)}^{2} y^{2}) = 36$

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

$4 x^{2} + 9 (1 y^{2}) = 36$

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

$4 x^{2} + 9 y^{2} = 36$

Since the equation is identical to the original equation, it is symmetric to the x-axis.

Symmetric with respect to the x-axis

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

$4 {(- x)}^{2} + 9 y^{2} = 36$

Simplify the left side.

Simplify each term.

Apply the product rule to $- x$ .

$4 ({(- 1)}^{2} x^{2}) + 9 y^{2} = 36$

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

$4 (1 x^{2}) + 9 y^{2} = 36$

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

$4 x^{2} + 9 y^{2} = 36$

Since the equation is identical to the original equation, it is symmetric to the y-axis.

Symmetric with respect to the y-axis

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

$4 {(- x)}^{2} + 9 {(- y)}^{2} = 36$

Solve for $- 4 x^{2} - 9 y^{2}$ .

Simplify the left side.

Simplify each term.

Apply the product rule to $- x$ .

$4 ({(- 1)}^{2} x^{2}) + 9 {(- y)}^{2} = 36$

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

$4 (1 x^{2}) + 9 {(- y)}^{2} = 36$

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

$4 x^{2} + 9 {(- y)}^{2} = 36$

Apply the product rule to $- y$ .

$4 x^{2} + 9 ({(- 1)}^{2} y^{2}) = 36$

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

$4 x^{2} + 9 (1 y^{2}) = 36$

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

$4 x^{2} + 9 y^{2} = 36$

Simplify the left side.

Simplify $- (4 {(- x)}^{2} + 9 {(- y)}^{2})$ .

Simplify each term.

Apply the product rule to $- x$ .

$- (4 ({(- 1)}^{2} x^{2}) + 9 {(- y)}^{2}) = - (36)$

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

$- (4 (1 x^{2}) + 9 {(- y)}^{2}) = - (36)$

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

$- (4 x^{2} + 9 {(- y)}^{2}) = - (36)$

Apply the product rule to $- y$ .

$- (4 x^{2} + 9 ({(- 1)}^{2} y^{2})) = - (36)$

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

$- (4 x^{2} + 9 (1 y^{2})) = - (36)$

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

$- (4 x^{2} + 9 y^{2}) = - (36)$

Simplify by multiplying through.

Apply the distributive property.

$- (4 x^{2}) - (9 y^{2}) = - (36)$

Multiply.

Multiply $4$ by $- 1$ .

$- 4 x^{2} - (9 y^{2}) = - (36)$

Multiply $9$ by $- 1$ .

$- 4 x^{2} - 9 y^{2} = - (36)$

Simplify the right side.

Multiply $- 1$ by $36$ .

$- 4 x^{2} - 9 y^{2} = - 36$

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.

Symmetric with respect to the x-axis

Symmetric with respect to the y-axis

Symmetric with respect to the origin

Find the Symmetry 4x^2+9y^2=36

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