Algebra Examples

$y = \sqrt{9 - x^{2}}$

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

Rewrite $9$ as $3^{2}$ .

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

Step 4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 3$ and $b = x$ .

$y = \sqrt{(3 + x) (3 - x)}$

Step 5

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

$- y = \sqrt{(3 + x) (3 - x)}$

Step 6

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 7

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

$y = \sqrt{(3 - x) (3 - - x)}$

Step 8

Multiply $- - x$ .

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Step 8.1

Multiply $- 1$ by $- 1$ .

$y = \sqrt{(3 - x) (3 + 1 x)}$

Step 8.2

Multiply $x$ by $1$ .

$y = \sqrt{(3 - x) (3 + x)}$

Step 9

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

Symmetric with respect to the y-axis

Step 10

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

$- y = \sqrt{(3 - x) (3 - - x)}$

Step 11

Multiply $- - x$ .

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Step 11.1

Multiply $- 1$ by $- 1$ .

$- y = \sqrt{(3 - x) (3 + 1 x)}$

Step 11.2

Multiply $x$ by $1$ .

$- y = \sqrt{(3 - x) (3 + x)}$

Step 12

Multiply both sides by $- 1$ .

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Step 12.1

Multiply each term by $- 1$ .

$- - y = - \sqrt{(3 - x) (3 + x)}$

Step 12.2

Multiply $- - y$ .

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Step 12.2.1

Multiply $- 1$ by $- 1$ .

$1 y = - \sqrt{(3 - x) (3 + x)}$

Step 12.2.2

Multiply $y$ by $1$ .

$y = - \sqrt{(3 - x) (3 + x)}$

Step 13

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

Symmetric with respect to the origin

Step 14

Determine the symmetry.

Symmetric with respect to the y-axis

Step 15