Curve Sketching Lab
Read a graph from its derivatives: f′ tells you where the curve climbs and falls, f″ tells you how it bends. Find critical points, classify them, locate inflection points, and sketch with confidence.
1 Sign-chart lab
Shape the cubic f(x) = x3 + bx2 + cx + d with the sliders. The canvas plots f (blue) with critical points (gold) and the inflection point (teal); the table below is the live sign chart.
Critical: –
Inflection: –
Live sign chart
2 What to notice
- Gold dots sit exactly where the blue curve flattens — and the f′ row of the sign chart changes sign there.
- Slide c until the two gold dots merge: the discriminant hits zero and the max/min pair annihilates into a flat inflection.
- The teal dot (inflection) always sits halfway between the two critical points for a cubic.