Trigonometry Examples

$\frac{y + 1}{y^{2} - 4 y - 12}$

Step 1

Find where the expression $\frac{y + 1}{(y - 6) (y + 2)}$ is undefined.

$y = - 2, y = 6$

Step 2

Since $\frac{y + 1}{(y - 6) (y + 2)}$ $\to$ $- \infty$ as $y$ $\to$ $- 2$ from the left and $\frac{y + 1}{(y - 6) (y + 2)}$ $\to$ $\infty$ as $y$ $\to$ $- 2$ from the right, then $y = - 2$ is a vertical asymptote.

$y = - 2$

Step 3

Since $\frac{y + 1}{(y - 6) (y + 2)}$ $\to$ $- \infty$ as $y$ $\to$ $6$ from the left and $\frac{y + 1}{(y - 6) (y + 2)}$ $\to$ $\infty$ as $y$ $\to$ $6$ from the right, then $y = 6$ is a vertical asymptote.

$y = 6$

Step 4

List all of the vertical asymptotes:

$y = - 2, 6$

Step 5

$x = \frac{y + 1}{(y - 6) (y + 2)}$ is an equation of a line, which means there are no horizontal asymptotes.

No Horizontal Asymptotes

Step 6

Find the oblique asymptote using polynomial division.

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Step 6.1

Factor $y^{2} - 4 y - 12$ using the AC method.

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Step 6.1.1

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 $- 12$ and whose sum is $- 4$ .

$- 6, 2$

Step 6.1.2

Write the factored form using these integers.

$\frac{y + 1}{(y - 6) (y + 2)}$

Step 6.2

Expand $(y - 6) (y + 2)$ .

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Step 6.2.1

Apply the distributive property.

$\frac{y + 1}{y (y + 2) - 6 (y + 2)}$

Step 6.2.2

Apply the distributive property.

$\frac{y + 1}{y \cdot y + y \cdot 2 - 6 (y + 2)}$

Step 6.2.3

Apply the distributive property.

$\frac{y + 1}{y \cdot y + y \cdot 2 - 6 y - 6 \cdot 2}$

Step 6.2.4

Reorder $y$ and $2$ .

$\frac{y + 1}{y \cdot y + 2 y - 6 y - 6 \cdot 2}$

Step 6.2.5

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

$\frac{y + 1}{y \cdot y + 2 y - 6 y - 6 \cdot 2}$

Step 6.2.6

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

$\frac{y + 1}{y \cdot y + 2 y - 6 y - 6 \cdot 2}$

Step 6.2.7

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{y + 1}{y^{1 + 1} + 2 y - 6 y - 6 \cdot 2}$

Step 6.2.8

Add $1$ and $1$ .

$\frac{y + 1}{y^{2} + 2 y - 6 y - 6 \cdot 2}$

Step 6.2.9

Multiply $- 6$ by $2$ .

$\frac{y + 1}{y^{2} + 2 y - 6 y - 12}$

Step 6.2.10

Subtract $6 y$ from $2 y$ .

$\frac{y + 1}{y^{2} - 4 y - 12}$

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 $0$ .

$y^{2}$

$4 y$

$12$

$y$

$1$

Step 6.4

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

$0 + \frac{y^{2} - 3 y - 11}{y^{2} - 4 y - 12}$

Step 6.5

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

$y = 0$

Step 7

This is the set of all asymptotes.

Vertical Asymptotes: $y = - 2, 6$

No Horizontal Asymptotes

Oblique Asymptotes: $y = 0$

Step 8