AP® Calculus BC · Unit 5 · Topic 5.8
Sketching Graphs of Functions and Their Derivatives
Sketching a derivative graph means plotting the original curve's slope at every x-value, not copying its height. Where the curve peaks or dips, the derivative crosses zero; where it rises or falls steeply, the derivative sits far from zero.
To sketch f' from a picture of f, stop looking at height and start looking at steepness. Walk along the curve of f from left to right and ask, at each x, what is the slope doing. Where f is climbing, f' is positive; where f is falling, f' is negative; where f has a peak, valley, or any flat spot, the slope is zero and f' touches the x-axis there. Steeper sections of f produce larger-magnitude values on f', while flatter sections pull f' close to zero. You're translating a shape into a new shape built entirely from slope readings, point by point.
This is exactly why peaks and valleys matter so much: a smooth curve only turns around at a point where its tangent line is horizontal, so every local max or min of f sits directly above a zero of f'. That single fact is the bridge between a graph's visual shape and the algebra of finding critical points — the zero of the derivative IS the x-coordinate of the turning point.
Once you can sketch f' from f, the next move is stacking three graphs — f, f', and f'' — on aligned axes and reading them together. A zero of f' lines up under a peak or valley of f. A zero of f'' lines up under a point where f' has its own peak or valley, which is an inflection point of f, where concavity flips. Reading these three graphs as one coordinated system, rather than three separate pictures, is the skill the rest of this unit builds on.
Key terms
- Critical point
- An x-value where f' equals zero or is undefined, marking a possible peak, valley, or flat spot.
- Inflection point
- A point where the concavity of f switches, corresponding to a zero (and sign change) of f''.
- Slope reading
- The method of building f' by estimating the tangent slope of f at each x-value directly from its graph.
- Stacked graphs
- Aligned graphs of f, f', and f'' drawn on the same x-axis so zeros and turning points line up vertically between them.
Watch it explained
- How it worksHow do I build a derivative's graph just by reading slopes off the original curve?
- Worked exampleDraw the derivative graph of an upward parabola with vertex at the point (2, -3).
- Common mistakeWhy isn't the derivative graph just the original graph shifted down?
- Ties togetherHow does sketching a derivative's graph set up reading stacked graphs of f, f-prime, and f-double-prime together?
Check yourself
End of topic.