A rational function is a quotient of two polynomials. Its graph can have intercepts, holes,
vertical asymptotes, horizontal or oblique asymptotes, and sign changes across critical points.
A complete analysis combines algebraic simplification with graph interpretation.
1. Rational function form
A rational function has the form
\[
\begin{aligned}
f(x)&=\frac{P(x)}{Q(x)}
\end{aligned}
\]
The denominator controls where the original function is defined. Every real solution of
\(Q(x)=0\) is excluded from the domain.
\[
\begin{aligned}
Q(x)&\ne0
\end{aligned}
\]
2. Simplification, holes, and vertical asymptotes
If numerator and denominator share a common factor, write
\[
\begin{aligned}
P(x)&=G(x)P_1(x)\\
Q(x)&=G(x)Q_1(x)
\end{aligned}
\]
Then the expression simplifies to
\[
\begin{aligned}
f(x)&=\frac{G(x)P_1(x)}{G(x)Q_1(x)}\\
&=\frac{P_1(x)}{Q_1(x)}
\end{aligned}
\]
However, the zeros of \(G(x)\) remain excluded from the original domain. If the simplified
denominator is not zero at such a value, the graph has a removable discontinuity, or hole.
\[
\begin{aligned}
\text{hole at }x=a
\quad\Rightarrow\quad
\left(a,\frac{P_1(a)}{Q_1(a)}\right)
\end{aligned}
\]
Vertical asymptotes come from real zeros that remain in the simplified denominator:
\[
\begin{aligned}
Q_1(x)&=0
\end{aligned}
\]
3. Intercepts
The x-intercepts come from zeros of the simplified numerator, as long as those points are still
in the original domain.
\[
\begin{aligned}
P_1(x)&=0
\end{aligned}
\]
The y-intercept is found by evaluating the function at \(x=0\), provided the original denominator
is not zero there.
\[
\begin{aligned}
y&=f(0)
\end{aligned}
\]
4. Horizontal, oblique, and polynomial asymptotes
The long-run behavior depends on the degrees of the simplified numerator and denominator.
Let \(m=\deg P_1\) and \(n=\deg Q_1\).
| Degree comparison |
End behavior model |
Meaning |
| \(m<n\) |
\(y=0\) |
The x-axis is a horizontal asymptote. |
| \(m=n\) |
\(y=\dfrac{\text{leading coefficient of }P_1}{\text{leading coefficient of }Q_1}\) |
The graph approaches a horizontal line. |
| \(m=n+1\) |
Quotient from polynomial division |
The graph approaches an oblique asymptote. |
| \(m>n+1\) |
Polynomial quotient from division |
The graph approaches a higher-degree polynomial curve. |
5. Sign chart
A sign chart divides the real number line at every real zero, hole, and vertical asymptote.
Then one test value is chosen in each interval.
\[
\begin{aligned}
\text{test sign on interval}
&=\operatorname{sign}\left(f(x_{\text{test}})\right)
\end{aligned}
\]
The sign chart is especially useful for solving inequalities such as \(f(x)>0\),
\(f(x)\ge0\), \(f(x)<0\), and \(f(x)\le0\). Zeros may be included for weak inequalities,
but holes and vertical asymptotes are never included.
6. Example
Consider the rational function
\[
\begin{aligned}
f(x)&=\frac{x^2+3x-4}{x^2-9}
\end{aligned}
\]
Factor numerator and denominator.
\[
\begin{aligned}
x^2+3x-4&=(x+4)(x-1)\\
x^2-9&=(x-3)(x+3)
\end{aligned}
\]
There are no common factors, so there are no holes. The domain excludes the denominator zeros.
\[
\begin{aligned}
\text{Domain}
&=\mathbb{R}\setminus\{-3,3\}
\end{aligned}
\]
The x-intercepts come from the numerator:
\[
\begin{aligned}
x&=-4\\
x&=1
\end{aligned}
\]
The vertical asymptotes come from the denominator:
\[
\begin{aligned}
x&=-3\\
x&=3
\end{aligned}
\]
Since the numerator and denominator have the same degree and the leading coefficients are both
\(1\), the horizontal asymptote is
\[
\begin{aligned}
y&=1
\end{aligned}
\]
The critical x-values for the sign chart are \(-4\), \(-3\), \(1\), and \(3\). Testing one
point in each interval gives the sign pattern:
\[
\begin{aligned}
(-\infty,-4)&:\ +\\
(-4,-3)&:\ -\\
(-3,1)&:\ +\\
(1,3)&:\ -\\
(3,\infty)&:\ +
\end{aligned}
\]
7. Common mistakes
- Do not call a canceled denominator factor a vertical asymptote; it usually creates a hole.
- Do not add canceled x-values back into the domain.
- Do not include holes or vertical asymptotes in inequality solutions.
- Do not find horizontal asymptotes by substituting only large numbers; use degree comparison.
- Do not forget to test every interval in the sign chart.
