Algebra Examples

$x = y^{2} + 2$

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 $y$ .

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

Simplify the right side.

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Simplify each term.

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Apply the product rule to $- y$ .

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

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

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

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

$x = y^{2} + 2$

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 $x$ .

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

Simplify the left side.

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Remove parentheses.

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

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 $y$ .

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

Solve for $x$ .

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Simplify the left side.

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Remove parentheses.

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

Simplify the right side.

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Simplify each term.

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Apply the product rule to $- y$ .

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

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

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

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

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

Simplify the left side.

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Multiply $- (- x)$ .

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Multiply $- 1$ by $- 1$ .

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

Multiply $x$ by $1$ .

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

Simplify the right side.

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

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Simplify each term.

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Apply the product rule to $- y$ .

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

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

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

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

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

Simplify by multiplying through.

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Apply the distributive property.

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

Multiply $- 1$ by $2$ .

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

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