# How do you graph rational functions?

Jul 12, 2015

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.