Graphing a function is not only about drawing a curve. A useful graph also reveals important
properties: intercepts, asymptotes, holes, local maximum points, local minimum points, and
domain restrictions. These features help explain how the function behaves and where special
values occur.
1. Intercepts
An \(x\)-intercept is a point where the graph crosses or touches the \(x\)-axis. At such a point,
the output is zero, so \(x\)-intercepts are found by solving:
\[
f(x)=0.
\]
If \(x=a\) is a solution, then the intercept is \((a,0)\). A function can have no
\(x\)-intercepts, one \(x\)-intercept, or many \(x\)-intercepts.
The \(y\)-intercept is found by setting \(x=0\):
\[
y=f(0).
\]
If \(f(0)\) is defined, then the \(y\)-intercept is \((0,f(0))\). If the function is undefined
at \(x=0\), then there is no \(y\)-intercept.
2. Vertical asymptotes
A vertical asymptote is a vertical line that the graph approaches while the function grows
without bound. For rational functions, vertical asymptotes often occur where the denominator is
zero and the expression does not simplify to a finite value.
\[
\text{vertical asymptote:}\qquad x=a.
\]
For example,
\[
f(x)=\frac{x^2-1}{x-2}
\]
has a denominator equal to zero at \(x=2\). Since the numerator is not zero at \(x=2\), the graph
has a vertical asymptote there.
\[
x-2=0\Rightarrow x=2.
\]
3. Holes and removable discontinuities
A hole occurs when a function is undefined at a point, but the nearby values approach a finite
limit. This often happens when the same factor appears in both the numerator and denominator.
\[
f(x)=\frac{x^2-1}{x-1}
=\frac{(x-1)(x+1)}{x-1}.
\]
Algebraically, the expression simplifies to \(x+1\) when \(x\ne1\), but the original function is
still undefined at \(x=1\). The graph has a hole at the point where the simplified function would
have been:
\[
y=1+1=2.
\]
So the hole is at \((1,2)\).
4. Horizontal asymptotes
A horizontal asymptote describes long-term behavior. If a function gets closer and closer to a
constant value as \(x\) becomes very large or very negative, then the function may have a
horizontal asymptote:
\[
y=L.
\]
For example,
\[
f(x)=3+\frac{1}{x-2}
\]
approaches \(3\) as \(x\) moves far away from \(2\). Therefore, \(y=3\) is a horizontal
asymptote.
5. Local extrema
A local maximum is a point where the graph is higher than nearby points. A local minimum is a
point where the graph is lower than nearby points. Numerically, extrema can be detected by
sampling nearby values. In calculus, extrema are often related to derivative sign changes.
\[
f'(x)=0
\]
gives candidates for local maximum and minimum points, although not every critical point is an
extremum.
6. Multiple functions on one graph
Graphing multiple functions on the same coordinate plane helps compare behavior. For example,
graphing a quadratic and a line together can reveal intersection points, growth differences, and
relative position. Each function should use a distinct color, and important features should be
labeled clearly.
7. Summary table
8. Practical cautions
A graphing calculator that uses numerical sampling depends on the selected window and scan
resolution. If an intercept or asymptote lies outside the selected interval, it will not be
shown. If a feature is very close to another feature, increasing the scan resolution can improve
detection. For exact algebraic work, a symbolic simplification step may still be needed.
9. Final checklist
- Enter the function or functions.
- Choose a graphing window that includes the important region.
- Find \(x\)-intercepts by solving \(f(x)=0\).
- Find the \(y\)-intercept by computing \(f(0)\).
- Check denominator restrictions for asymptotes and holes.
- Look for local maximum and minimum points.
- Use the graph to confirm the numerical results visually.
