Precalculus Examples

$f (x) = \frac{20 x^{2} + 28 x - 3}{- 10 x + 1}$

Step 1

Find where the expression $\frac{20 x^{2} + 28 x - 3}{- 10 x + 1}$ is undefined.

$x = \frac{1}{10}$

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 by grouping.

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Step 6.1.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 20 \cdot - 3 = - 60$ and whose sum is $b = 28$ .

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Step 6.1.1.1.1

Factor $28$ out of $28 x$ .

$\frac{20 x^{2} + 28 (x) - 3}{- 10 x + 1}$

Step 6.1.1.1.2

Rewrite $28$ as $- 2$ plus $30$

$\frac{20 x^{2} + (- 2 + 30) x - 3}{- 10 x + 1}$

Step 6.1.1.1.3

Apply the distributive property.

$\frac{20 x^{2} - 2 x + 30 x - 3}{- 10 x + 1}$

Step 6.1.1.2

Factor out the greatest common factor from each group.

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Step 6.1.1.2.1

Group the first two terms and the last two terms.

$\frac{(20 x^{2} - 2 x) + 30 x - 3}{- 10 x + 1}$

Step 6.1.1.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{2 x (10 x - 1) + 3 (10 x - 1)}{- 10 x + 1}$

Step 6.1.1.3

Factor the polynomial by factoring out the greatest common factor, $10 x - 1$ .

$\frac{(10 x - 1) (2 x + 3)}{- 10 x + 1}$

Step 6.1.2

Cancel the common factor of $10 x - 1$ and $- 10 x + 1$ .

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Step 6.1.2.1

Factor $- 1$ out of $10 x$ .

$\frac{(- (- 10 x) - 1) (2 x + 3)}{- 10 x + 1}$

Step 6.1.2.2

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

$\frac{(- (- 10 x) - 1 (1)) (2 x + 3)}{- 10 x + 1}$

Step 6.1.2.3

Factor $- 1$ out of $- (- 10 x) - 1 (1)$ .

$\frac{- (- 10 x + 1) (2 x + 3)}{- 10 x + 1}$

Step 6.1.2.4

Rewrite $- (- 10 x + 1)$ as $- 1 (- 10 x + 1)$ .

$\frac{- 1 (- 10 x + 1) (2 x + 3)}{- 10 x + 1}$

Step 6.1.2.5

Cancel the common factor.

$\frac{- 1 (- 10 x + 1) (2 x + 3)}{- 10 x + 1}$

Step 6.1.2.6

Divide $- 1 (2 x + 3)$ by $1$ .

$- 1 (2 x + 3)$

Step 6.1.3

Rewrite $- 1 (2 x + 3)$ as $- (2 x + 3)$ .

$- (2 x + 3)$

Step 6.1.4

Apply the distributive property.

$- (2 x) - 1 \cdot 3$

Step 6.1.5

Multiply.

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Step 6.1.5.1

Multiply $2$ by $- 1$ .

$- 2 x - 1 \cdot 3$

Step 6.1.5.2

Multiply $- 1$ by $3$ .

$- 2 x - 3$

Step 6.2

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

$y = - 2 x - 3$

Step 7

This is the set of all asymptotes.

No Vertical Asymptotes

No Horizontal Asymptotes

Oblique Asymptotes: $y = - 2 x - 3$

Step 8