Trigonometry Examples

$g (x) - 2 x^{2} - 3$

Step 1

Simplify.

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Step 1.1

Move all terms not containing $y$ to the right side of the equation.

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Step 1.1.1

Add $2 x^{2}$ to both sides of the equation.

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

Step 1.1.2

Add $3$ to both sides of the equation.

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

Step 1.2

Divide each term in $y x = 2 x^{2} + 3$ by $x$ and simplify.

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Step 1.2.1

Divide each term in $y x = 2 x^{2} + 3$ by $x$ .

$\frac{y x}{x} = \frac{2 x^{2}}{x} + \frac{3}{x}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $x$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{y x}{x} = \frac{2 x^{2}}{x} + \frac{3}{x}$

Step 1.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{2 x^{2}}{x} + \frac{3}{x}$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Cancel the common factor of $x^{2}$ and $x$ .

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Step 1.2.3.1.1

Factor $x$ out of $2 x^{2}$ .

$y = \frac{x (2 x)}{x} + \frac{3}{x}$

Step 1.2.3.1.2

Cancel the common factors.

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Step 1.2.3.1.2.1

Raise $x$ to the power of $1$ .

$y = \frac{x (2 x)}{x^{1}} + \frac{3}{x}$

Step 1.2.3.1.2.2

Factor $x$ out of $x^{1}$ .

$y = \frac{x (2 x)}{x \cdot 1} + \frac{3}{x}$

Step 1.2.3.1.2.3

Cancel the common factor.

$y = \frac{x (2 x)}{x \cdot 1} + \frac{3}{x}$

Step 1.2.3.1.2.4

Rewrite the expression.

$y = \frac{2 x}{1} + \frac{3}{x}$

Step 1.2.3.1.2.5

Divide $2 x$ by $1$ .

$y = 2 x + \frac{3}{x}$

Step 2

Find where the expression $2 x + \frac{3}{x}$ is undefined.

$x = 0$

Step 3

Consider the rational function $R (x) = \frac{a x^{n}}{b x^{m}}$ where $n$ is the degree of the numerator and $m$ is the degree of the denominator.

1. If $n < m$ , then the x-axis, $y = 0$ , is the horizontal asymptote.

2. If $n = m$ , then the horizontal asymptote is the line $y = \frac{a}{b}$ .

3. If $n > m$ , then there is no horizontal asymptote (there is an oblique asymptote).

Step 4

Find $n$ and $m$ .

$n = 2$

$m = 1$

Step 5

Since $n > m$ , there is no horizontal asymptote.

No Horizontal Asymptotes

Step 6

Find the oblique asymptote using polynomial division.

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Step 6.1

Combine.

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Step 6.1.1

To write $2 x$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{2 x \cdot x}{x} + \frac{3}{x}$

Step 6.1.2

Combine the numerators over the common denominator.

$\frac{2 x \cdot x + 3}{x}$

Step 6.1.3

Multiply $x$ by $x$ by adding the exponents.

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Step 6.1.3.1

Move $x$ .

$\frac{2 (x \cdot x) + 3}{x}$

Step 6.1.3.2

Multiply $x$ by $x$ .

$\frac{2 x^{2} + 3}{x}$

Step 6.1.4

Simplify.

$\frac{2 x^{2} + 3}{x}$

Step 6.2

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x$

$0$

$2 x^{2}$

$0 x$

$3$

Step 6.3

Divide the highest order term in the dividend $2 x^{2}$ by the highest order term in divisor $x$ .

					$2 x$
	$x$	+	$0$		$2 x^{2}$	+	$0 x$	+	$3$

Step 6.4

Multiply the new quotient term by the divisor.

				$2 x$
$x$	+	$0$		$2 x^{2}$	+	$0 x$	+	$3$
			+	$2 x^{2}$	+	$0$

Step 6.5

The expression needs to be subtracted from the dividend, so change all the signs in $2 x^{2} + 0$

				$2 x$
$x$	+	$0$		$2 x^{2}$	+	$0 x$	+	$3$
			-	$2 x^{2}$	-	$0$

Step 6.6

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$2 x$
$x$	+	$0$		$2 x^{2}$	+	$0 x$	+	$3$
			-	$2 x^{2}$	-	$0$
						$0$

Step 6.7

Pull the next term from the original dividend down into the current dividend.

				$2 x$
$x$	+	$0$		$2 x^{2}$	+	$0 x$	+	$3$
			-	$2 x^{2}$	-	$0$
						$0$	+	$3$

Step 6.8

The final answer is the quotient plus the remainder over the divisor.

$2 x + \frac{3}{x}$

Step 6.9

The oblique asymptote is the polynomial portion of the long division result.

$y = 2 x$

Step 7

This is the set of all asymptotes.

Vertical Asymptotes: $x = 0$

No Horizontal Asymptotes

Oblique Asymptotes: $y = 2 x$

Step 8