# How do you graph f(x)=(x-1)/x^2 using holes, vertical and horizontal asymptotes, x and y intercepts?

Jul 19, 2018

#### Explanation:

Given: $f \left(x\right) = \frac{x - 1}{x} ^ 2$

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 $x = 0$

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) : $N \left(x\right) = 0$

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

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

$f \left(0\right) = 0 - \frac{1}{0} ^ 2 = \text{undefined}$

no $y$-intercept.

graph{(x-1)/x^2 [-10, 10, -5, 5]}