4. Curve Sketching Assistant — Theory
What “curve sketching” means
Curve sketching is a structured way to draw an accurate graph of \(y=f(x)\) without plotting many random points.
You identify key features (domain issues, intercepts, asymptotes, increasing/decreasing behavior, concavity, and inflection points),
then assemble a sketch that reflects all of them.
Step 1 — Domain and discontinuities
First determine where \(f(x)\) is defined. Common domain restrictions:
- \(\ln(u)\): requires \(u>0\)
- \(\sqrt{u}\): requires \(u\ge 0\)
- Rational functions \(\dfrac{p(x)}{q(x)}\): undefined where \(q(x)=0\)
- \(\tan(u)\): undefined where \(\cos(u)=0\)
A discontinuity can be a vertical asymptote (function blows up to \(\pm\infty\)) or a removable hole
(a missing point that could be filled by simplifying).
Step 2 — Intercepts
\[
\text{x-intercepts: solve } f(x)=0.
\qquad
\text{y-intercept: } (0,f(0)) \text{ if defined.}
\]
These anchor the graph near the axes.
Step 3 — Asymptotes
Vertical asymptotes often come from denominator zeros or \(\tan\) blow-ups.
Horizontal/oblique asymptotes describe end behavior as \(x\to\pm\infty\).
\[
\text{Horizontal: } \lim_{x\to\pm\infty} f(x)=L \Rightarrow y=L.
\]
\[
\text{Oblique: } f(x)\approx mx+b \text{ for large }|x|.
\]
For rational functions, you can often determine these by comparing polynomial degrees; limits give the formal proof.
Step 4 — First derivative: increasing/decreasing and extrema
Compute \(f'(x)\). Critical points occur where \(f'(x)=0\) (and sometimes where \(f'\) is undefined but \(f\) exists).
\[
f'(x)>0 \Rightarrow f \text{ increasing},\qquad
f'(x)<0 \Rightarrow f \text{ decreasing}.
\]
First derivative test: if \(f'\) changes from \(+\) to \(-\), you have a local maximum; from \(-\) to \(+\), a local minimum.
Step 5 — Second derivative: concavity and inflection points
\[
f''(x)>0 \Rightarrow \text{concave up},\qquad
f''(x)<0 \Rightarrow \text{concave down}.
\]
Candidates for inflection points come from solving \(f''(x)=0\), but an inflection occurs only if concavity actually changes sign.
Second derivative test at a critical point \(x=c\): if \(f''(c)>0\) → local min; if \(f''(c)<0\) → local max.
If \(f''(c)=0\), the test is inconclusive (use the first derivative sign change).
How to use the assistant effectively
- Start with a moderate window (e.g., \(w=6\)). Increase \(w\) to capture asymptotes/intercepts further away.
- Turn on asymptotes and points to see what the tool detected.
- Enable concavity coloring to visually confirm concave up/down regions.
- If you suspect missing features, increase “Scan samples” to improve detection of multiple roots/turning points.
- Use the Limits chapter for a formal asymptote proof (the assistant’s infinity asymptotes are heuristic).
About numeric detection
The assistant combines symbolic derivatives with numeric root scanning.
Some situations (highly oscillatory functions, very tight features, or extremely steep growth)
may require a smaller window, more samples, or a formal analytic approach.
