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.

E 2.6 Exercises