Find the Symmetry 5x^2-4y^2+2x-1=0

$5 x^{2} - 4 y^{2} + 2 x - 1 = 0$

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 $y$ .

$5 x^{2} - 4 {(- y)}^{2} + 2 x - 1 = 0$

Simplify the left side.

Simplify each term.

Apply the product rule to $- y$ .

$5 x^{2} - 4 ({(- 1)}^{2} y^{2}) + 2 x - 1 = 0$

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

$5 x^{2} - 4 (1 y^{2}) + 2 x - 1 = 0$

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

$5 x^{2} - 4 y^{2} + 2 x - 1 = 0$

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 $x$ .

$5 {(- x)}^{2} - 4 y^{2} + 2 (- x) - 1 = 0$

Simplify the left side.

Simplify each term.

Apply the product rule to $- x$ .

$5 ({(- 1)}^{2} x^{2}) - 4 y^{2} + 2 (- x) - 1 = 0$

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

$5 (1 x^{2}) - 4 y^{2} + 2 (- x) - 1 = 0$

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

$5 x^{2} - 4 y^{2} + 2 (- x) - 1 = 0$

Multiply $- 1$ by $2$ .

$5 x^{2} - 4 y^{2} - 2 x - 1 = 0$

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 $y$ .

$5 {(- x)}^{2} - 4 {(- y)}^{2} + 2 (- x) - 1 = 0$

Solve for $- 5 x^{2} + 4 y^{2} + 2 x + 1$ .

Simplify the left side.

Simplify each term.

Apply the product rule to $- x$ .

$5 ({(- 1)}^{2} x^{2}) - 4 {(- y)}^{2} + 2 (- x) - 1 = 0$

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

$5 (1 x^{2}) - 4 {(- y)}^{2} + 2 (- x) - 1 = 0$

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

$5 x^{2} - 4 {(- y)}^{2} + 2 (- x) - 1 = 0$

Apply the product rule to $- y$ .

$5 x^{2} - 4 ({(- 1)}^{2} y^{2}) + 2 (- x) - 1 = 0$

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

$5 x^{2} - 4 (1 y^{2}) + 2 (- x) - 1 = 0$

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

$5 x^{2} - 4 y^{2} + 2 (- x) - 1 = 0$

Multiply $- 1$ by $2$ .

$5 x^{2} - 4 y^{2} - 2 x - 1 = 0$

Simplify the left side.

Simplify $- (5 {(- x)}^{2} - 4 {(- y)}^{2} + 2 (- x) - 1)$ .

Simplify each term.

Apply the product rule to $- x$ .

$- (5 ({(- 1)}^{2} x^{2}) - 4 {(- y)}^{2} + 2 (- x) - 1) = - (0)$

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

$- (5 (1 x^{2}) - 4 {(- y)}^{2} + 2 (- x) - 1) = - (0)$

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

$- (5 x^{2} - 4 {(- y)}^{2} + 2 (- x) - 1) = - (0)$

Apply the product rule to $- y$ .

$- (5 x^{2} - 4 ({(- 1)}^{2} y^{2}) + 2 (- x) - 1) = - (0)$

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

$- (5 x^{2} - 4 (1 y^{2}) + 2 (- x) - 1) = - (0)$

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

$- (5 x^{2} - 4 y^{2} + 2 (- x) - 1) = - (0)$

Multiply $- 1$ by $2$ .

$- (5 x^{2} - 4 y^{2} - 2 x - 1) = - (0)$

Apply the distributive property.

$- (5 x^{2}) - (- 4 y^{2}) - (- 2 x) - - 1 = - (0)$

Simplify.

Multiply $5$ by $- 1$ .

$- 5 x^{2} - (- 4 y^{2}) - (- 2 x) - - 1 = - (0)$

Multiply $- 4$ by $- 1$ .

$- 5 x^{2} + 4 y^{2} - (- 2 x) - - 1 = - (0)$

Multiply $- 2$ by $- 1$ .

$- 5 x^{2} + 4 y^{2} + 2 x - - 1 = - (0)$

Multiply $- 1$ by $- 1$ .

$- 5 x^{2} + 4 y^{2} + 2 x + 1 = - (0)$

Simplify the right side.

Multiply $- 1$ by $0$ .

$- 5 x^{2} + 4 y^{2} + 2 x + 1 = 0$

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.

Symmetric with respect to the x-axis

Find the Symmetry 5x^2-4y^2+2x-1=0

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