The Sketching Toolkit
Curve sketching translates an abstract formula into an accurate graph by extracting features systematically. The first derivative builds the up-and-down skeleton, while the second adds bending.
| Feature | What to Look For | Formula / Rule |
|---|---|---|
| Intercepts | Axes crossings | where ; at |
| Asymptotes | Boundary lines | Vertical where ; horizontal as |
| Symmetry | Axis mirroring | Even: ; Odd: |
Working through this checklist ensures no critical behavior is missed before drawing lines.
Derivatives and Shape
Derivatives reveal how a curve rises, falls, and bends between its structural boundaries.
| Derivative | Sign | Graphical Meaning |
|---|---|---|
| First () | Positive | Function is rising (increasing) |
| First () | Negative | Function is falling (decreasing) |
| Second () | Positive | Concave up (cups upward) |
| Second () | Negative | Concave down (cups downward) |
Inflection points occur where the second derivative changes sign, marking where concavity shifts.
Pitfall: Relying on alone gets turning points right but misses concavity and inflection points, ruining the true shape. Both derivatives are required.