Graph f(x)=(6x^2+7x-9)/(2x+1)

$f (x) = \frac{6 x^{2} + 7 x - 9}{2 x + 1}$

Step 1

Find where the expression $\frac{6 x^{2} + 7 x - 9}{2 x + 1}$ is undefined.

$x = - \frac{1}{2}$

Step 2

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 3

Find $n$ and $m$ .

$n = 2$

$m = 1$

Step 4

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

No Horizontal Asymptotes

Step 5

Find the oblique asymptote using polynomial division.

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

$2 x$

$1$

$6 x^{2}$

$7 x$

$9$

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

					$3 x$
	$2 x$	+	$1$		$6 x^{2}$	+	$7 x$	-	$9$

Multiply the new quotient term by the divisor.

				$3 x$
$2 x$	+	$1$		$6 x^{2}$	+	$7 x$	-	$9$
			+	$6 x^{2}$	+	$3 x$

The expression needs to be subtracted from the dividend, so change all the signs in $6 x^{2} + 3 x$

				$3 x$
$2 x$	+	$1$		$6 x^{2}$	+	$7 x$	-	$9$
			-	$6 x^{2}$	-	$3 x$

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

				$3 x$
$2 x$	+	$1$		$6 x^{2}$	+	$7 x$	-	$9$
			-	$6 x^{2}$	-	$3 x$
					+	$4 x$

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

				$3 x$
$2 x$	+	$1$		$6 x^{2}$	+	$7 x$	-	$9$
			-	$6 x^{2}$	-	$3 x$
					+	$4 x$	-	$9$

Divide the highest order term in the dividend $4 x$ by the highest order term in divisor $2 x$ .

				$3 x$	+	$2$
$2 x$	+	$1$		$6 x^{2}$	+	$7 x$	-	$9$
			-	$6 x^{2}$	-	$3 x$
					+	$4 x$	-	$9$

Multiply the new quotient term by the divisor.

				$3 x$	+	$2$
$2 x$	+	$1$		$6 x^{2}$	+	$7 x$	-	$9$
			-	$6 x^{2}$	-	$3 x$
					+	$4 x$	-	$9$
					+	$4 x$	+	$2$

The expression needs to be subtracted from the dividend, so change all the signs in $4 x + 2$

				$3 x$	+	$2$
$2 x$	+	$1$		$6 x^{2}$	+	$7 x$	-	$9$
			-	$6 x^{2}$	-	$3 x$
					+	$4 x$	-	$9$
					-	$4 x$	-	$2$

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

				$3 x$	+	$2$
$2 x$	+	$1$		$6 x^{2}$	+	$7 x$	-	$9$
			-	$6 x^{2}$	-	$3 x$
					+	$4 x$	-	$9$
					-	$4 x$	-	$2$
							-	$11$

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

$3 x + 2 - \frac{11}{2 x + 1}$

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

$y = 3 x + 2$

Step 6

This is the set of all asymptotes.

Vertical Asymptotes: $x = - \frac{1}{2}$

No Horizontal Asymptotes

Oblique Asymptotes: $y = 3 x + 2$

Step 7

Graph f(x)=(6x^2+7x-9)/(2x+1)

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