# Graphs of Rational Functions

## Key Questions

• How to Find Horizontal Asymptotes of Rational Functions

Let $f \left(x\right) = \frac{p \left(x\right)}{q \left(x\right)}$, where p(x) is a polynomial of degree $m$ with leading coefficient $a$, and q(x) is a polynomial of degree $n$ with leading coefficient $b$. There are three cases:

Case 1: If $m > n$, then $f$ has no horizontal asymptotes.
Case 2: If $m = n$, then $y = \frac{a}{b}$ is the horizontal asymptote of $f$.
Case 3: If $m < n$, then $y = 0$ is the horizontal asymptote of $f$.

How to Find Vertical Asymptotes of Rational Functions

If there are any common factors between the numerator and the denominator, then cancel all common factors. Set the denominator equal to zero then solve for $x$.

I hope that this was helpful.

• Asymptotes are lines that a particular function can get very very close to but never intersect.

For example, the function $y = \frac{1}{x}$ is asymptotic to $y = 0$.
As $x$ goes larger and larger, y goes smaller and smaller. y tends to approach 0, but it will never reach hit that value.

See explanation...

#### Explanation:

Suppose $f \left(x\right) = g \frac{x}{h \left(x\right)} = \frac{{a}_{n} {x}^{n} + {a}_{n - 1} {x}^{n - 1} + . . + {a}_{0}}{{b}_{m} {x}^{m} + {b}_{m - 1} {x}^{m - 1} + \ldots + {b}_{0}}$

If $g \left(x\right)$ and $h \left(x\right)$ have some common factor $k \left(x\right)$ then let

${g}_{1} \left(x\right) = g \frac{x}{k \left(x\right)}$ and ${h}_{1} \left(x\right) = g \frac{x}{k \left(x\right)}$.

The graph of ${g}_{1} \frac{x}{{h}_{1} \left(x\right)}$ will be the same as the graph of $g \frac{x}{h \left(x\right)}$,

except that any $x$ where $k \left(x\right) = 0$ is an excluded value.

Assuming $g \left(x\right)$ and $h \left(x\right)$ have no common factor, then there will be vertical asymptotes wherever $h \left(x\right) = 0$. If a root is not a repeated root (or is repeated an odd number of times) then the limit on one side of the asymptote will be $\infty$ and on the other $- \infty$. If the root has even multiplicity then the limit on both sides of the asymptote will be the same: $\infty$ or $- \infty$.

If $n < m$ then $f \left(x\right) \to 0$ as $x \to \pm \infty$

If $n \ge m$ then divide $g \frac{x}{h \left(x\right)}$ to get a polynomial quotient and remainder. The polynomial quotient is the oblique asymptote as $x \to \pm \infty$.

For example, if $f \left(x\right) = \frac{{x}^{3} + 3}{{x}^{2} + 2}$, then:

$f \left(x\right) = \frac{{x}^{3} + 3}{{x}^{2} + 2} = \frac{{x}^{3} + 2 x - 2 x + 3}{{x}^{2} + 2} = x - \frac{2 x - 3}{{x}^{2} + 2}$

So the oblique asymptote of $f \left(x\right)$ is $y = x$

Intercepts with the $x$ axis are where $f \left(x\right) = 0$, which mean where $g \left(x\right) = 0$.

The intercept with the $y$ axis is where $x = 0$, so just substitute $x = 0$ into the equation for $f \left(x\right)$ to find $f \left(0\right) = {a}_{0} / {b}_{0}$

Apart from all this, just pick some $x$ values and calculate $f \left(x\right)$ to give you coordinates $\left(x , f \left(x\right)\right)$ through which the graph must pass.

• Rational functions are functions, which are created by dividing two function. Formally, they are represented as $\frac{f \left(x\right)}{g \left(x\right)}$, where $f \left(x\right)$ and $g \left(x\right)$ are both functions.

For example: $\frac{2 {x}^{2} + 3 x - 5}{5 x - 7}$ is a rational function where $f \left(x\right) = 2 {x}^{2} + 3 x - 5$ and $g \left(x\right) = 5 x - 7$.