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
$$
g(x) = \frac{a_n x^n}{b_m x^m}.
$$
$$
\begin{aligned}
f(x) & = \frac{N(x)}{D(x)} \\
& = \frac{a_n x^n +a_{n-1}x^{n-1} + \cdots +a_1 x + a_0}{b_m x^m + b_{m-1} x^{m-1} + \cdots + b_1 x + b_0}
\end{aligned}
$$
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 \(D(x).\)
$$
\begin{aligned}
f(x) & = \frac{N(x)}{D(x)} \\
& = \frac{a_n x^n +a_{n-1}x^{n-1} + \cdots +a_1 x + a_0}{b_m x^m + b_{m-1} x^m + \cdots + b_1 x + b_0}
\end{aligned}
$$
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.
Let \(f(x) = \displaystyle \frac{N(x)}{D(x)},\) where \(N(x)\) and \(D(x)\) are polynomials.
- Simplify \(f,\) if possible (i.e. reduce any common factors but keep track of the values zeros of those common factors)
- Find and plot the \(y\)-intercept (if any) by evaluating \(f(0).\)
- Find the zeros of the numerator (if any) by solving the equation \(N(x) = 0.\) Then plot the corresponding \(x\)-intercepts.
- Find the zeros of the denominator (if any) by solving the equation \(D(x) = 0.\) Then sketch and label the corresponding vertical asymptotes.
- Find, sketch, and label the horizontal asymptotes (if any) by comparing \(f\) to a function which has the same end-behavior.
- Plot at least one point between and one point beyond each \(x\)-intercept and vertical asymptote.
- Use smooth curves to complete the graph between and beyond the vertical asymptotes.