Calculus Examples

$y = \frac{x}{\sqrt{x^{2} + 1}}$

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

Multiply $\frac{x}{\sqrt{x^{2} + 1}}$ by $\frac{\sqrt{x^{2} + 1}}{\sqrt{x^{2} + 1}}$ .

$y = \frac{x}{\sqrt{x^{2} + 1}} \cdot \frac{\sqrt{x^{2} + 1}}{\sqrt{x^{2} + 1}}$

Step 4

Combine and simplify the denominator.

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Step 4.1

Multiply $\frac{x}{\sqrt{x^{2} + 1}}$ by $\frac{\sqrt{x^{2} + 1}}{\sqrt{x^{2} + 1}}$ .

$y = \frac{x \sqrt{x^{2} + 1}}{\sqrt{x^{2} + 1} \sqrt{x^{2} + 1}}$

Step 4.2

Raise $\sqrt{x^{2} + 1}$ to the power of $1$ .

$y = \frac{x \sqrt{x^{2} + 1}}{{\sqrt{x^{2} + 1}}^{1} \sqrt{x^{2} + 1}}$

Step 4.3

Raise $\sqrt{x^{2} + 1}$ to the power of $1$ .

$y = \frac{x \sqrt{x^{2} + 1}}{{\sqrt{x^{2} + 1}}^{1} {\sqrt{x^{2} + 1}}^{1}}$

Step 4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \frac{x \sqrt{x^{2} + 1}}{{\sqrt{x^{2} + 1}}^{1 + 1}}$

Step 4.5

Add $1$ and $1$ .

$y = \frac{x \sqrt{x^{2} + 1}}{{\sqrt{x^{2} + 1}}^{2}}$

Step 4.6

Rewrite ${\sqrt{x^{2} + 1}}^{2}$ as $x^{2} + 1$ .

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Step 4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{2} + 1}$ as ${(x^{2} + 1)}^{\frac{1}{2}}$ .

$y = \frac{x \sqrt{x^{2} + 1}}{{({(x^{2} + 1)}^{\frac{1}{2}})}^{2}}$

Step 4.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y = \frac{x \sqrt{x^{2} + 1}}{{(x^{2} + 1)}^{\frac{1}{2} \cdot 2}}$

Step 4.6.3

Combine $\frac{1}{2}$ and $2$ .

$y = \frac{x \sqrt{x^{2} + 1}}{{(x^{2} + 1)}^{\frac{2}{2}}}$

Step 4.6.4

Cancel the common factor of $2$ .

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Step 4.6.4.1

Cancel the common factor.

$y = \frac{x \sqrt{x^{2} + 1}}{{(x^{2} + 1)}^{\frac{2}{2}}}$

Step 4.6.4.2

Rewrite the expression.

$y = \frac{x \sqrt{x^{2} + 1}}{{(x^{2} + 1)}^{1}}$

Step 4.6.5

Simplify.

$y = \frac{x \sqrt{x^{2} + 1}}{x^{2} + 1}$

Step 5

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

$- y = \frac{x \sqrt{x^{2} + 1}}{x^{2} + 1}$

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 $x$ .

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

Step 8

Simplify.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Apply the product rule to $- x$ .

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

Step 8.1.2

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

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

Step 8.1.3

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

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

Step 8.2

Simplify the denominator.

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Step 8.2.1

Apply the product rule to $- x$ .

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

Step 8.2.2

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

$y = \frac{- x \sqrt{x^{2} + 1}}{1 x^{2} + 1}$

Step 8.2.3

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

$y = \frac{- x \sqrt{x^{2} + 1}}{x^{2} + 1}$

Step 8.3

Move the negative in front of the fraction.

$y = - \frac{x \sqrt{x^{2} + 1}}{x^{2} + 1}$

Step 9

Since the equation is not identical to the original equation, it is not symmetric to the y-axis.

Not symmetric to the y-axis

Step 10

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

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

Step 11

Simplify.

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Step 11.1

Simplify the numerator.

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Step 11.1.1

Apply the product rule to $- x$ .

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

Step 11.1.2

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

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

Step 11.1.3

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

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

Step 11.2

Simplify the denominator.

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Step 11.2.1

Apply the product rule to $- x$ .

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

Step 11.2.2

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

$- y = \frac{- x \sqrt{x^{2} + 1}}{1 x^{2} + 1}$

Step 11.2.3

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

$- y = \frac{- x \sqrt{x^{2} + 1}}{x^{2} + 1}$

Step 11.3

Move the negative in front of the fraction.

$- y = - \frac{x \sqrt{x^{2} + 1}}{x^{2} + 1}$

Step 12

Multiply both sides by $- 1$ .

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Step 12.1

Multiply each term by $- 1$ .

$- - y = - - \frac{x \sqrt{x^{2} + 1}}{x^{2} + 1}$

Step 12.2

Multiply $- - y$ .

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Step 12.2.1

Multiply $- 1$ by $- 1$ .

$1 y = - - \frac{x \sqrt{x^{2} + 1}}{x^{2} + 1}$

Step 12.2.2

Multiply $y$ by $1$ .

$y = - - \frac{x \sqrt{x^{2} + 1}}{x^{2} + 1}$

Step 12.3

Multiply $- - \frac{x \sqrt{x^{2} + 1}}{x^{2} + 1}$ .

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Step 12.3.1

Multiply $- 1$ by $- 1$ .

$y = 1 \frac{x \sqrt{x^{2} + 1}}{x^{2} + 1}$

Step 12.3.2

Multiply $\frac{x \sqrt{x^{2} + 1}}{x^{2} + 1}$ by $1$ .

$y = \frac{x \sqrt{x^{2} + 1}}{x^{2} + 1}$

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