Graphing polynomial and rational functions means more than plotting random points. A good graph
is built from key features: roots, intercepts, holes, asymptotes, and end behavior. These features
tell you where the graph crosses the axes, where it is undefined, and what it does far from the
origin.
1. Polynomial functions
A polynomial function has the form
\[
f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0,
\qquad a_n\ne0.
\]
Its graph is continuous everywhere, so it has no holes and no vertical asymptotes. The roots are
the values of \(x\) that make
\[
f(x)=0.
\]
The end behavior of a polynomial is controlled by the leading term \(a_nx^n\). If the degree is
even, the two ends go in the same direction. If the degree is odd, the two ends go in opposite
directions. The sign of the leading coefficient determines whether the right end goes up or down.
2. Rational functions
A rational function has the form
\[
f(x)=\frac{P(x)}{D(x)},
\qquad D(x)\ne0.
\]
The denominator controls the domain. Every real value that makes the original denominator zero
must be excluded from the domain, even if a factor cancels later.
3. Simplifying before graphing
If \(P(x)\) and \(D(x)\) have a common factor, write
\[
P(x)=G(x)P_1(x),
\qquad
D(x)=G(x)D_1(x).
\]
Then the rational expression simplifies to
\[
\frac{P(x)}{D(x)}
=
\frac{G(x)P_1(x)}{G(x)D_1(x)}
=
\frac{P_1(x)}{D_1(x)},
\qquad G(x)\ne0.
\]
The simplified function describes the shape of the curve, but the canceled values are still
missing from the original graph.
4. Holes
A hole is a removable discontinuity. It occurs when a real zero of a canceled factor is not a zero
of the simplified denominator. If \(x=a\) is canceled and \(D_1(a)\ne0\), then the hole is
\[
\left(a,\frac{P_1(a)}{D_1(a)}\right).
\]
On the graph, this point is shown with an open circle. The curve approaches the point, but the
point itself is not included.
5. Vertical asymptotes
Vertical asymptotes come from real zeros of the simplified denominator:
\[
D_1(x)=0.
\]
At these values, the graph usually grows without bound. They are drawn as dashed vertical lines.
A denominator zero is not automatically a vertical asymptote; first check whether the factor
cancels. If it cancels completely, it may be a hole instead.
6. Roots and intercepts
Roots, or x-intercepts, come from zeros of the simplified numerator:
\[
P_1(x)=0.
\]
A root is only an actual x-intercept if it is not excluded from the original domain. The
y-intercept is found by evaluating \(f(0)\), provided the original denominator is not zero at
\(x=0\).
7. Horizontal, oblique, and polynomial asymptotes
For a rational function, long-run behavior depends on the degrees of the numerator and denominator.
If
\[
f(x)=\frac{P_1(x)}{D_1(x)},
\]
then:
8. Example
Consider
\[
f(x)=\frac{x^2-1}{x^2-4}.
\]
Factor numerator and denominator:
\[
x^2-1=(x-1)(x+1),
\qquad
x^2-4=(x-2)(x+2).
\]
There are no common factors, so there are no holes. The roots are
\[
x=-1,\qquad x=1.
\]
The vertical asymptotes are
\[
x=-2,\qquad x=2.
\]
Since numerator and denominator have the same degree and the leading coefficients are both \(1\),
the horizontal asymptote is
\[
y=1.
\]
9. Graphing checklist
- Rewrite the function as \(P(x)/D(x)\).
- Cancel common factors only for simplification.
- Keep original denominator zeros excluded from the domain.
- Find roots from the simplified numerator.
- Find holes from canceled real factors.
- Find vertical asymptotes from the simplified denominator.
- Use degree comparison or polynomial division for the end/asymptote model.
- Plot the curve and mark all special features clearly.
10. Common mistakes
- Do not call every denominator zero a vertical asymptote. Canceled factors may create holes.
- Do not forget that canceled x-values remain excluded from the original domain.
- Do not use the original expression to evaluate a hole; use the simplified expression.
- Do not find horizontal asymptotes by substituting large numbers only; use degree comparison.
- Do not ignore multiplicity: roots may cross or only touch the x-axis depending on multiplicity.
