# How do you graph f(x)=(4(x+1))/(x(x-4)) using holes, vertical and horizontal asymptotes, x and y intercepts?

Jul 19, 2018

#### Explanation:

Given: $f \left(x\right) = \frac{4 \left(x + 1\right)}{x \left(x - 4\right)}$

This type of equation is called a rational (fraction) function.

It has the form: $f \left(x\right) = \frac{N \left(x\right)}{D \left(x\right)} = \frac{{a}_{n} {x}^{n} + \ldots}{{b}_{m} {x}^{m} + \ldots}$,

where N(x)) is the numerator and $D \left(x\right)$ is the denominator,

$n$ = the degree of $N \left(x\right)$ and $m$ = the degree of $\left(D \left(x\right)\right)$

and ${a}_{n}$ is the leading coefficient of the $N \left(x\right)$ and

${b}_{m}$ is the leading coefficient of the $D \left(x\right)$

Step 1 factor : The given function is already factored.

Step 2, cancel any factors that are both in $\left(N \left(x\right)\right)$ and D(x)) (determines holes):

The given function has no holes $\text{ "=> " no factors that cancel}$

Step 3, find vertical asymptotes: $D \left(x\right) = 0$

vertical asymptote at x = 0 " and x = 4

Step 4, find horizontal asymptotes:
Compare the degrees:

If $n < m$ the horizontal asymptote is $y = 0$

If $n = m$ the horizontal asymptote is $y = {a}_{n} / {b}_{m}$

If $n > m$ there are no horizontal asymptotes

In the given equation: n = 1; m = 2 " "=> y = 0

horizontal asymptote is $y = 0$

Step 5, find x-intercept(s) : $N \left(x\right) = 0$

4(x + 1) = 0; " "x + 1 = 0 " "=> x"-intercept" (-1, 0)

Step 5, find y-intercept(s): $x = 0$

$f \left(0\right) = \frac{4 \left(1\right)}{0 \left(- 4\right)} = \text{undefined}$

no $y$-intercept.

graph{(4(x+1))/(x(x-4)) [-10, 10, -5, 5]}