Precalculus Examples

$\frac{x^{2} - 2 x - 3}{- 4 x + 12}$

Step 1

Find where the expression $\frac{x^{2} - 2 x - 3}{- 4 x + 12}$ is undefined.

$x = 3$

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 $m$ .

$n = 2$

$m = 1$

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

Simplify the expression.

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Step 6.1.1

Factor $x^{2} - 2 x - 3$ using the AC method.

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Step 6.1.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 3$ and whose sum is $- 2$ .

$- 3, 1$

Step 6.1.1.2

Write the factored form using these integers.

$\frac{(x - 3) (x + 1)}{- 4 x + 12}$

Step 6.1.2

Simplify terms.

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Step 6.1.2.1

Factor $4$ out of $- 4 x + 12$ .

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Step 6.1.2.1.1

Factor $4$ out of $- 4 x$ .

$\frac{(x - 3) (x + 1)}{4 (- x) + 12}$

Step 6.1.2.1.2

Factor $4$ out of $12$ .

$\frac{(x - 3) (x + 1)}{4 (- x) + 4 (3)}$

Step 6.1.2.1.3

Factor $4$ out of $4 (- x) + 4 (3)$ .

$\frac{(x - 3) (x + 1)}{4 (- x + 3)}$

Step 6.1.2.2

Cancel the common factor of $x - 3$ and $- x + 3$ .

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Step 6.1.2.2.1

Factor $- 1$ out of $x$ .

$\frac{(- 1 (- x) - 3) (x + 1)}{4 (- x + 3)}$

Step 6.1.2.2.2

Rewrite $- 3$ as $- 1 (3)$ .

$\frac{(- 1 (- x) - 1 (3)) (x + 1)}{4 (- x + 3)}$

Step 6.1.2.2.3

Factor $- 1$ out of $- 1 (- x) - 1 (3)$ .

$\frac{- 1 (- x + 3) (x + 1)}{4 (- x + 3)}$

Step 6.1.2.2.4

Cancel the common factor.

$\frac{- 1 (- x + 3) (x + 1)}{4 (- x + 3)}$

Step 6.1.2.2.5

Rewrite the expression.

$\frac{- 1 (x + 1)}{4}$

Step 6.1.2.3

Move the negative in front of the fraction.

$- \frac{x + 1}{4}$

Step 6.2

Expand $- (x + 1)$ .

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Step 6.2.1

Negate $x + 1$ .

$\frac{- (x + 1)}{4}$

Step 6.2.2

Apply the distributive property.

$\frac{- x - 1 \cdot 1}{4}$

Step 6.2.3

Multiply $- 1$ by $1$ .

$\frac{- x - 1}{4}$

Step 6.3

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

$4$

$x$

$1$

Step 6.4

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

		-	$\frac{x}{4}$
	$4$	-	$x$	-	$1$

Step 6.5

Multiply the new quotient term by the divisor.

	-	$\frac{x}{4}$
$4$	-	$x$	-	$1$
	-	$x$

Step 6.6

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

	-	$\frac{x}{4}$
$4$	-	$x$	-	$1$
	+	$x$

Step 6.7

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

	-	$\frac{x}{4}$
$4$	-	$x$	-	$1$
	+	$x$
		$0$

Step 6.8

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

	-	$\frac{x}{4}$
$4$	-	$x$	-	$1$
	+	$x$
		$0$	-	$1$

Step 6.9

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

$- \frac{x}{4} - \frac{1}{4}$

Step 6.10

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

$y = - \frac{x}{4}$

Step 7

This is the set of all asymptotes.

No Vertical Asymptotes

No Horizontal Asymptotes

Oblique Asymptotes: $y = - \frac{x}{4}$

Step 8