Graph f(x)=(x^3-16x)/(3x^2-12)

$f (x) = \frac{x^{3} - 16 x}{3 x^{2} - 12}$

Step 1

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

$x = - 2, x = 2$

Step 2

Since $\frac{x (x + 4) (x - 4)}{3 (x + 2) (x - 2)}$ $\to$ $\infty$ as $x$ $\to$ $- 2$ from the left and $\frac{x (x + 4) (x - 4)}{3 (x + 2) (x - 2)}$ $\to$ $- \infty$ as $x$ $\to$ $- 2$ from the right, then $x = - 2$ is a vertical asymptote.

$x = - 2$

Step 3

Since $\frac{x (x + 4) (x - 4)}{3 (x + 2) (x - 2)}$ $\to$ $\infty$ as $x$ $\to$ $2$ from the left and $\frac{x (x + 4) (x - 4)}{3 (x + 2) (x - 2)}$ $\to$ $- \infty$ as $x$ $\to$ $2$ from the right, then $x = 2$ is a vertical asymptote.

$x = 2$

Step 4

List all of the vertical asymptotes:

$x = - 2, 2$

Step 5

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 6

Find $n$ and $m$ .

$n = 3$

$m = 2$

Step 7

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

No Horizontal Asymptotes

Step 8

Find the oblique asymptote using polynomial division.

Simplify the expression.

Simplify the numerator.

Factor $x$ out of $x^{3} - 16 x$ .

Factor $x$ out of $x^{3}$ .

$\frac{x \cdot x^{2} - 16 x}{3 x^{2} - 12}$

Factor $x$ out of $- 16 x$ .

$\frac{x \cdot x^{2} + x \cdot - 16}{3 x^{2} - 12}$

Factor $x$ out of $x \cdot x^{2} + x \cdot - 16$ .

$\frac{x (x^{2} - 16)}{3 x^{2} - 12}$

Rewrite $16$ as $4^{2}$ .

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

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 4$ .

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

Simplify the denominator.

Factor $3$ out of $3 x^{2} - 12$ .

Factor $3$ out of $3 x^{2}$ .

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

Factor $3$ out of $- 12$ .

$\frac{x (x + 4) (x - 4)}{3 x^{2} + 3 \cdot - 4}$

Factor $3$ out of $3 x^{2} + 3 \cdot - 4$ .

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

Rewrite $4$ as $2^{2}$ .

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

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 2$ .

$\frac{x (x + 4) (x - 4)}{3 (x + 2) (x - 2)}$

Expand $x (x + 4) (x - 4)$ .

Apply the distributive property.

$\frac{(x \cdot x + x \cdot 4) (x - 4)}{3 (x + 2) (x - 2)}$

Apply the distributive property.

$\frac{x \cdot x (x - 4) + x \cdot (4 (x - 4))}{3 (x + 2) (x - 2)}$

Apply the distributive property.

$\frac{x \cdot x \cdot x + x \cdot x \cdot - 4 + x \cdot (4 (x - 4))}{3 (x + 2) (x - 2)}$

Apply the distributive property.

$\frac{x \cdot x \cdot x + x \cdot x \cdot - 4 + x \cdot (4 x) + x \cdot 4 \cdot - 4}{3 (x + 2) (x - 2)}$

Move $x$ .

$\frac{x \cdot x \cdot x + x \cdot (- 4 x) + x \cdot (4 x) + x \cdot 4 \cdot - 4}{3 (x + 2) (x - 2)}$

Reorder $x$ and $- 4$ .

$\frac{x \cdot x \cdot x - 4 \cdot (x \cdot x) + x \cdot (4 x) + x \cdot 4 \cdot - 4}{3 (x + 2) (x - 2)}$

Reorder $x$ and $4$ .

$\frac{x \cdot x \cdot x - 4 \cdot (x \cdot x) + 4 \cdot (x \cdot x) + x \cdot 4 \cdot - 4}{3 (x + 2) (x - 2)}$

Reorder $x$ and $4$ .

$\frac{x \cdot x \cdot x - 4 \cdot (x \cdot x) + 4 \cdot (x \cdot x) + 4 \cdot x \cdot - 4}{3 (x + 2) (x - 2)}$

Move $x$ .

$\frac{x \cdot x \cdot x - 4 \cdot (x \cdot x) + 4 \cdot (x \cdot x) + 4 \cdot (- 4 x)}{3 (x + 2) (x - 2)}$

Multiply $- 4$ by $x$ .

$\frac{x \cdot x \cdot x - 4 x \cdot x + 4 \cdot (x \cdot x) + 4 \cdot (- 4 x)}{3 (x + 2) (x - 2)}$

Multiply $4$ by $x$ .

$\frac{x \cdot x \cdot x - 4 x \cdot x + 4 x \cdot x + 4 \cdot (- 4 x)}{3 (x + 2) (x - 2)}$

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

$\frac{x \cdot x \cdot x - 4 x \cdot x + 4 x \cdot x + 4 \cdot (- 4 x)}{3 (x + 2) (x - 2)}$

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

$\frac{x \cdot x \cdot x - 4 x \cdot x + 4 x \cdot x + 4 \cdot (- 4 x)}{3 (x + 2) (x - 2)}$

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

$\frac{x^{1 + 1} x - 4 x \cdot x + 4 x \cdot x + 4 \cdot (- 4 x)}{3 (x + 2) (x - 2)}$

Add $1$ and $1$ .

$\frac{x^{2} x - 4 x \cdot x + 4 x \cdot x + 4 \cdot (- 4 x)}{3 (x + 2) (x - 2)}$

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

$\frac{x^{2} x - 4 x \cdot x + 4 x \cdot x + 4 \cdot (- 4 x)}{3 (x + 2) (x - 2)}$

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

$\frac{x^{2 + 1} - 4 x \cdot x + 4 x \cdot x + 4 \cdot (- 4 x)}{3 (x + 2) (x - 2)}$

Add $2$ and $1$ .

$\frac{x^{3} - 4 x \cdot x + 4 x \cdot x + 4 \cdot (- 4 x)}{3 (x + 2) (x - 2)}$

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

$\frac{x^{3} - 4 (x \cdot x) + 4 x \cdot x + 4 \cdot (- 4 x)}{3 (x + 2) (x - 2)}$

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

$\frac{x^{3} - 4 (x \cdot x) + 4 x \cdot x + 4 \cdot (- 4 x)}{3 (x + 2) (x - 2)}$

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

$\frac{x^{3} - 4 x^{1 + 1} + 4 x \cdot x + 4 \cdot (- 4 x)}{3 (x + 2) (x - 2)}$

Add $1$ and $1$ .

$\frac{x^{3} - 4 x^{2} + 4 x \cdot x + 4 \cdot (- 4 x)}{3 (x + 2) (x - 2)}$

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

$\frac{x^{3} - 4 x^{2} + 4 (x \cdot x) + 4 \cdot (- 4 x)}{3 (x + 2) (x - 2)}$

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

$\frac{x^{3} - 4 x^{2} + 4 (x \cdot x) + 4 \cdot (- 4 x)}{3 (x + 2) (x - 2)}$

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

$\frac{x^{3} - 4 x^{2} + 4 x^{1 + 1} + 4 \cdot (- 4 x)}{3 (x + 2) (x - 2)}$

Add $1$ and $1$ .

$\frac{x^{3} - 4 x^{2} + 4 x^{2} + 4 \cdot (- 4 x)}{3 (x + 2) (x - 2)}$

Multiply $4$ by $- 4$ .

$\frac{x^{3} - 4 x^{2} + 4 x^{2} - 16 x}{3 (x + 2) (x - 2)}$

Add $- 4 x^{2}$ and $4 x^{2}$ .

$\frac{x^{3} + 0 - 16 x}{3 (x + 2) (x - 2)}$

Subtract $16 x$ from $0$ .

$\frac{x^{3} - 16 x}{3 (x + 2) (x - 2)}$

Expand $3 (x + 2) (x - 2)$ .

Apply the distributive property.

$\frac{x^{3} - 16 x}{(3 x + 3 \cdot 2) (x - 2)}$

Apply the distributive property.

$\frac{x^{3} - 16 x}{3 x (x - 2) + 3 \cdot (2 (x - 2))}$

Apply the distributive property.

$\frac{x^{3} - 16 x}{3 x \cdot x + 3 x \cdot - 2 + 3 \cdot (2 (x - 2))}$

Apply the distributive property.

$\frac{x^{3} - 16 x}{3 x \cdot x + 3 x \cdot - 2 + 3 \cdot (2 x) + 3 \cdot 2 \cdot - 2}$

Move $x$ .

$\frac{x^{3} - 16 x}{3 x \cdot x + 3 \cdot (- 2 x) + 3 \cdot (2 x) + 3 \cdot 2 \cdot - 2}$

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

$\frac{x^{3} - 16 x}{3 (x \cdot x) + 3 \cdot (- 2 x) + 3 \cdot (2 x) + 3 \cdot 2 \cdot - 2}$

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

$\frac{x^{3} - 16 x}{3 (x \cdot x) + 3 \cdot (- 2 x) + 3 \cdot (2 x) + 3 \cdot 2 \cdot - 2}$

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

$\frac{x^{3} - 16 x}{3 x^{1 + 1} + 3 \cdot (- 2 x) + 3 \cdot (2 x) + 3 \cdot 2 \cdot - 2}$

Add $1$ and $1$ .

$\frac{x^{3} - 16 x}{3 x^{2} + 3 \cdot (- 2 x) + 3 \cdot (2 x) + 3 \cdot 2 \cdot - 2}$

Multiply $3$ by $- 2$ .

$\frac{x^{3} - 16 x}{3 x^{2} - 6 x + 3 \cdot (2 x) + 3 \cdot 2 \cdot - 2}$

Multiply $3$ by $2$ .

$\frac{x^{3} - 16 x}{3 x^{2} - 6 x + 6 x + 3 \cdot 2 \cdot - 2}$

Multiply $3$ by $2$ .

$\frac{x^{3} - 16 x}{3 x^{2} - 6 x + 6 x + 6 \cdot - 2}$

Multiply $6$ by $- 2$ .

$\frac{x^{3} - 16 x}{3 x^{2} - 6 x + 6 x - 12}$

Add $- 6 x$ and $6 x$ .

$\frac{x^{3} - 16 x}{3 x^{2} + 0 - 12}$

Subtract $12$ from $0$ .

$\frac{x^{3} - 16 x}{3 x^{2} - 12}$

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

$3 x^{2}$

$0 x$

$12$

$x^{3}$

$0 x^{2}$

$16 x$

$0$

Divide the highest order term in the dividend $x^{3}$ by the highest order term in divisor $3 x^{2}$ .

$\frac{x}{3}$

$3 x^{2}$

$0 x$

$12$

$x^{3}$

$0 x^{2}$

$16 x$

$0$

Multiply the new quotient term by the divisor.

$\frac{x}{3}$

$3 x^{2}$

$0 x$

$12$

$x^{3}$

$0 x^{2}$

$16 x$

$0$

$x^{3}$

$0$

$4 x$

The expression needs to be subtracted from the dividend, so change all the signs in $x^{3} + 0 - 4 x$

$\frac{x}{3}$

$3 x^{2}$

$0 x$

$12$

$x^{3}$

$0 x^{2}$

$16 x$

$0$

$x^{3}$

$0$

$4 x$

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

$\frac{x}{3}$

$3 x^{2}$

$0 x$

$12$

$x^{3}$

$0 x^{2}$

$16 x$

$0$

$x^{3}$

$0$

$4 x$

$12 x$

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

$\frac{x}{3}$

$3 x^{2}$

$0 x$

$12$

$x^{3}$

$0 x^{2}$

$16 x$

$0$

$x^{3}$

$0$

$4 x$

$12 x$

$0$

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

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

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

$y = \frac{x}{3}$

Step 9

This is the set of all asymptotes.

Vertical Asymptotes: $x = - 2, 2$

No Horizontal Asymptotes

Oblique Asymptotes: $y = \frac{x}{3}$

Step 10

Graph f(x)=(x^3-16x)/(3x^2-12)

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