# 2.6 | Rational Functions

Let $$f$$ be the rational function given by
\begin{aligned} f(x) & = \frac{N(x)}{D(x)} \\ & = \frac{a_n x^n +a_{n-1}x^{n-1} + \cdots +a_1 x + a_0}{b_m x^m + b_{m-1} x^{m-1} + \cdots + b_1 x + b_0} \end{aligned}
then $$f(x)$$ has the same end-behavior as
$$g(x) = \frac{a_n x^n}{b_m x^m}.$$
Let $$f$$ be the rational function given by
\begin{aligned} f(x) & = \frac{N(x)}{D(x)} \\ & = \frac{a_n x^n +a_{n-1}x^{n-1} + \cdots +a_1 x + a_0}{b_m x^m + b_{m-1} x^m + \cdots + b_1 x + b_0} \end{aligned}
where $$N(x)$$ and $$D(x)$$ have no common factors. Then the graph of $$f$$ has a vertical asymptote at the zeros of $$D(x).$$
Guidelines for Analyzing Graphs of Rational Functions
Let $$f(x) = \displaystyle \frac{N(x)}{D(x)},$$ where $$N(x)$$ and $$D(x)$$ are polynomials.

1. Simplify $$f,$$ if possible (i.e. reduce any common factors but keep track of the values zeros of those common factors)
2. Find and plot the $$y$$-intercept (if any) by evaluating $$f(0).$$
3. Find the zeros of the numerator (if any) by solving the equation $$N(x) = 0.$$ Then plot the corresponding $$x$$-intercepts.
4. Find the zeros of the denominator (if any) by solving the equation $$D(x) = 0.$$ Then sketch and label the corresponding vertical asymptotes.
5. Find, sketch, and label the horizontal asymptotes (if any) by comparing $$f$$ to a function which has the same end-behavior.
6. Plot at least one point between and one point beyond each $$x$$-intercept and vertical asymptote.
7. Use smooth curves to complete the graph between and beyond the vertical asymptotes.