Graph h(x)=-(44x^2)/(v^2)+x+6

$h (x) = - \frac{44 x^{2}}{v^{2}} + x + 6$

Step 1

Find where the expression $- \frac{44 x^{2}}{v^{2}} + x + 6$ is undefined.

$x = 0$

Step 2

The vertical asymptotes occur at areas of infinite discontinuity.

No Vertical Asymptotes

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 $m$ .

$n = 2$

$m = 0$

Step 5

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

No Horizontal Asymptotes

Step 6

Find the oblique asymptote using polynomial division.

Combine.

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

$- \frac{44 x^{2}}{v^{2}} + \frac{x v^{2}}{v^{2}} + 6$

Combine the numerators over the common denominator.

$\frac{- 44 x^{2} + x v^{2}}{v^{2}} + 6$

Factor $x$ out of $- 44 x^{2} + x v^{2}$ .

Factor $x$ out of $- 44 x^{2}$ .

$\frac{x (- 44 x) + x v^{2}}{v^{2}} + 6$

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

$\frac{x (- 44 x) + x (v^{2})}{v^{2}} + 6$

Factor $x$ out of $x (- 44 x) + x (v^{2})$ .

$\frac{x (- 44 x + v^{2})}{v^{2}} + 6$

To write $6$ as a fraction with a common denominator, multiply by $\frac{v^{2}}{v^{2}}$ .

$\frac{x (- 44 x + v^{2})}{v^{2}} + \frac{6 v^{2}}{v^{2}}$

Combine the numerators over the common denominator.

$\frac{x (- 44 x + v^{2}) + 6 v^{2}}{v^{2}}$

Simplify the numerator.

Apply the distributive property.

$\frac{x (- 44 x) + x v^{2} + 6 v^{2}}{v^{2}}$

Rewrite using the commutative property of multiplication.

$\frac{- 44 x \cdot x + x v^{2} + 6 v^{2}}{v^{2}}$

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

Move $x$ .

$\frac{- 44 (x \cdot x) + x v^{2} + 6 v^{2}}{v^{2}}$

Multiply $x$ by $x$ .

$\frac{- 44 x^{2} + x v^{2} + 6 v^{2}}{v^{2}}$

Simplify with factoring out.

Factor $- 1$ out of $- 44 x^{2}$ .

$\frac{- (44 x^{2}) + x v^{2} + 6 v^{2}}{v^{2}}$

Factor $- 1$ out of $x v^{2}$ .

$\frac{- (44 x^{2}) - (- x v^{2}) + 6 v^{2}}{v^{2}}$

Factor $- 1$ out of $- (44 x^{2}) - (- x v^{2})$ .

$\frac{- (44 x^{2} - x v^{2}) + 6 v^{2}}{v^{2}}$

Factor $- 1$ out of $6 v^{2}$ .

$\frac{- (44 x^{2} - x v^{2}) - (- 6 v^{2})}{v^{2}}$

Factor $- 1$ out of $- (44 x^{2} - x v^{2}) - (- 6 v^{2})$ .

$\frac{- (44 x^{2} - x v^{2} - 6 v^{2})}{v^{2}}$

Simplify the expression.

Rewrite $- (44 x^{2} - x v^{2} - 6 v^{2})$ as $- 1 (44 x^{2} - x v^{2} - 6 v^{2})$ .

$\frac{- 1 (44 x^{2} - x v^{2} - 6 v^{2})}{v^{2}}$

Move the negative in front of the fraction.

$- \frac{44 x^{2} - x v^{2} - 6 v^{2}}{v^{2}}$

Simplify.

$\frac{- 44 x^{2} + x v^{2} + 6 v^{2}}{v^{2}}$

Simplify the expression.

Factor $- 1$ out of $- 44 x^{2}$ .

$\frac{- (44 x^{2}) + x v^{2} + 6 v^{2}}{v^{2}}$

Factor $- 1$ out of $x v^{2}$ .

$\frac{- (44 x^{2}) - (- x v^{2}) + 6 v^{2}}{v^{2}}$

Factor $- 1$ out of $- (44 x^{2}) - (- x v^{2})$ .

$\frac{- (44 x^{2} - x v^{2}) + 6 v^{2}}{v^{2}}$

Factor $- 1$ out of $6 v^{2}$ .

$\frac{- (44 x^{2} - x v^{2}) - (- 6 v^{2})}{v^{2}}$

Factor $- 1$ out of $- (44 x^{2} - x v^{2}) - (- 6 v^{2})$ .

$\frac{- (44 x^{2} - x v^{2} - 6 v^{2})}{v^{2}}$

Simplify the expression.

Rewrite $- (44 x^{2} - x v^{2} - 6 v^{2})$ as $- 1 (44 x^{2} - x v^{2} - 6 v^{2})$ .

$\frac{- 1 (44 x^{2} - x v^{2} - 6 v^{2})}{v^{2}}$

Move the negative in front of the fraction.

$- \frac{44 x^{2} - x v^{2} - 6 v^{2}}{v^{2}}$

Expand $- (44 x^{2} - x v^{2} - 6 v^{2})$ .

Negate $44 x^{2} - x v^{2} - 6 v^{2}$ .

$\frac{- (44 x^{2} - x v^{2} - 6 v^{2})}{v^{2}}$

Apply the distributive property.

$\frac{- (44 x^{2} - x v^{2}) - (- 6 v^{2})}{v^{2}}$

Apply the distributive property.

$\frac{- (44 x^{2}) - (- x v^{2}) - (- 6 v^{2})}{v^{2}}$

Remove parentheses.

$\frac{- 1 \cdot (44 x^{2}) - (- x v^{2}) - (- 6 v^{2})}{v^{2}}$

Remove parentheses.

$\frac{- 1 \cdot (44 x^{2}) + x v^{2} - (- 6 v^{2})}{v^{2}}$

Move parentheses.

$\frac{- 1 \cdot (44 x^{2}) + 1 x v^{2} - (- 6 v^{2})}{v^{2}}$

Remove parentheses.

$\frac{- 1 \cdot (44 x^{2}) + 1 x v^{2} + 6 v^{2}}{v^{2}}$

Multiply $- 1$ by $44$ .

$\frac{- 44 x^{2} + 1 x v^{2} + 6 v^{2}}{v^{2}}$

Multiply $- 1$ by $- 1$ .

$\frac{- 44 x^{2} + 1 x v^{2} + 6 v^{2}}{v^{2}}$

Multiply $x$ by $1$ .

$\frac{- 44 x^{2} + x v^{2} + 6 v^{2}}{v^{2}}$

Multiply $- 1$ by $- 6$ .

$\frac{- 44 x^{2} + x v^{2} + 6 v^{2}}{v^{2}}$

Move $x v^{2}$ .

$\frac{- 44 x^{2} + 6 v^{2} + x v^{2}}{v^{2}}$

Reorder $- 44 x^{2}$ and $6 v^{2}$ .

$\frac{6 v^{2} - 44 x^{2} + x v^{2}}{v^{2}}$

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

$v^{2}$

$44 x^{2}$

$v^{2} x$

$6 v^{2}$

Divide the highest order term in the dividend $- 44 x^{2}$ by the highest order term in divisor $v^{2}$ .

		-	$\frac{44 x^{2}}{v^{2}}$
	$v^{2}$	-	$44 x^{2}$	+	$v^{2} x$	+	$6 v^{2}$

Multiply the new quotient term by the divisor.

	-	$\frac{44 x^{2}}{v^{2}}$
$v^{2}$	-	$44 x^{2}$	+	$v^{2} x$	+	$6 v^{2}$
	-	$44 x^{2}$

The expression needs to be subtracted from the dividend, so change all the signs in $- 44 x^{2}$

	-	$\frac{44 x^{2}}{v^{2}}$
$v^{2}$	-	$44 x^{2}$	+	$v^{2} x$	+	$6 v^{2}$
	+	$44 x^{2}$

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

	-	$\frac{44 x^{2}}{v^{2}}$
$v^{2}$	-	$44 x^{2}$	+	$v^{2} x$	+	$6 v^{2}$
	+	$44 x^{2}$
		$0$

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

	-	$\frac{44 x^{2}}{v^{2}}$
$v^{2}$	-	$44 x^{2}$	+	$v^{2} x$	+	$6 v^{2}$
	+	$44 x^{2}$
		$0$	+	$v^{2} x$

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

	-	$\frac{44 x^{2}}{v^{2}}$	+	$x$
$v^{2}$	-	$44 x^{2}$	+	$v^{2} x$	+	$6 v^{2}$
	+	$44 x^{2}$
		$0$	+	$v^{2} x$

Multiply the new quotient term by the divisor.

	-	$\frac{44 x^{2}}{v^{2}}$	+	$x$
$v^{2}$	-	$44 x^{2}$	+	$v^{2} x$	+	$6 v^{2}$
	+	$44 x^{2}$
		$0$	+	$v^{2} x$
			+	$x v^{2}$

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

	-	$\frac{44 x^{2}}{v^{2}}$	+	$x$
$v^{2}$	-	$44 x^{2}$	+	$v^{2} x$	+	$6 v^{2}$
	+	$44 x^{2}$
		$0$	+	$v^{2} x$
			-	$x v^{2}$

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

	-	$\frac{44 x^{2}}{v^{2}}$	+	$x$
$v^{2}$	-	$44 x^{2}$	+	$v^{2} x$	+	$6 v^{2}$
	+	$44 x^{2}$
		$0$	+	$v^{2} x$
			-	$x v^{2}$
				$0$

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

	-	$\frac{44 x^{2}}{v^{2}}$	+	$x$
$v^{2}$	-	$44 x^{2}$	+	$v^{2} x$	+	$6 v^{2}$
	+	$44 x^{2}$
		$0$	+	$v^{2} x$
			-	$x v^{2}$
				$0$	+	$6 v^{2}$

Divide the highest order term in the dividend $6 v^{2}$ by the highest order term in divisor $v^{2}$ .

	-	$\frac{44 x^{2}}{v^{2}}$	+	$x$	+	$6$
$v^{2}$	-	$44 x^{2}$	+	$v^{2} x$	+	$6 v^{2}$
	+	$44 x^{2}$
		$0$	+	$v^{2} x$
			-	$x v^{2}$
				$0$	+	$6 v^{2}$

Multiply the new quotient term by the divisor.

	-	$\frac{44 x^{2}}{v^{2}}$	+	$x$	+	$6$
$v^{2}$	-	$44 x^{2}$	+	$v^{2} x$	+	$6 v^{2}$
	+	$44 x^{2}$
		$0$	+	$v^{2} x$
			-	$x v^{2}$
				$0$	+	$6 v^{2}$
					+	$6 v^{2}$

The expression needs to be subtracted from the dividend, so change all the signs in $6 v^{2}$

	-	$\frac{44 x^{2}}{v^{2}}$	+	$x$	+	$6$
$v^{2}$	-	$44 x^{2}$	+	$v^{2} x$	+	$6 v^{2}$
	+	$44 x^{2}$
		$0$	+	$v^{2} x$
			-	$x v^{2}$
				$0$	+	$6 v^{2}$
					-	$6 v^{2}$

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

	-	$\frac{44 x^{2}}{v^{2}}$	+	$x$	+	$6$
$v^{2}$	-	$44 x^{2}$	+	$v^{2} x$	+	$6 v^{2}$
	+	$44 x^{2}$
		$0$	+	$v^{2} x$
			-	$x v^{2}$
				$0$	+	$6 v^{2}$
					-	$6 v^{2}$
						$0$

Since the remander is $0$ , the final answer is the quotient.

$- \frac{44 x^{2}}{v^{2}} + x + 6$

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

$y = - \frac{44 x^{2}}{v^{2}} + x$

Step 7

This is the set of all asymptotes.

No Vertical Asymptotes

No Horizontal Asymptotes

Oblique Asymptotes: $y = - \frac{44 x^{2}}{v^{2}} + x$

Step 8

Graph h(x)=-(44x^2)/(v^2)+x+6

Related Questions

Support LCMGCF.com by disabling your adblocker.