Precalculus Examples

$x^{2} - 3 = y^{2}$

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 $y$ .

$x^{2} - 3 = {(- y)}^{2}$

Step 4

Simplify.

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Step 4.1

Apply the product rule to $- y$ .

$x^{2} - 3 = {(- 1)}^{2} y^{2}$

Step 4.2

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

$x^{2} - 3 = 1 y^{2}$

Step 4.3

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

$x^{2} - 3 = y^{2}$

Step 5

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

Symmetric with respect to the x-axis

Step 6

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

${(- x)}^{2} - 3 = y^{2}$

Step 7

Simplify each term.

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Step 7.1

Apply the product rule to $- x$ .

${(- 1)}^{2} x^{2} - 3 = y^{2}$

Step 7.2

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

$1 x^{2} - 3 = y^{2}$

Step 7.3

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

$x^{2} - 3 = y^{2}$

Step 8

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

Symmetric with respect to the y-axis

Step 9

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

${(- x)}^{2} - 3 = {(- y)}^{2}$

Step 10

Simplify.

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Step 10.1

Simplify each term.

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Step 10.1.1

Apply the product rule to $- x$ .

${(- 1)}^{2} x^{2} - 3 = {(- y)}^{2}$

Step 10.1.2

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

$1 x^{2} - 3 = {(- y)}^{2}$

Step 10.1.3

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

$x^{2} - 3 = {(- y)}^{2}$

Step 10.2

Apply the product rule to $- y$ .

$x^{2} - 3 = {(- 1)}^{2} y^{2}$

Step 10.3

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

$x^{2} - 3 = 1 y^{2}$

Step 10.4

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

$x^{2} - 3 = y^{2}$

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

Determine the symmetry.

Symmetric with respect to the x-axis

Symmetric with respect to the y-axis

Symmetric with respect to the origin

Step 13