5.8 Sketching Graphs of Functions and Their Derivatives

Connecting f, f', and f''

To sketch a graph, link all three functions: f' > 0 means f is increasing; f' < 0 means f is decreasing. f'' > 0 means f is concave up (f' increasing); f'' < 0 means concave down (f' decreasing). Zeros of f' locate possible extrema; zeros of f'' locate possible inflection points.

Reading one graph to sketch another

Given the graph of f', you can sketch f: where f' is positive, f rises; where f' crosses zero from + to −, f has a local max. The slope of f' at a point gives f''. Given f'', find where it changes sign to locate inflection points of f, where concavity of f switches.

Putting it all together

A complete sketch uses critical points (f' = 0 or undefined), increasing/decreasing intervals, concavity, and inflection points. Check behavior consistency: a function can't be both increasing and concave down forever without curving toward a max, and asymptotes or endpoints must match given information like domain restrictions or limits.

Key terms

f' > 0 on an interval

f is increasing on that interval.

f'' > 0 on an interval

f is concave up on that interval (f' is increasing).

Local extremum of f

Occurs where f' = 0 (or is undefined) and f' changes sign.

Inflection point of f

Occurs where f'' changes sign, indicating a change in concavity.

Sketching f from f' graph

Use sign of f' for increasing/decreasing, slope of f' for concavity, and zero crossings of f' for possible extrema.

Try an AP-style question

f increases up to x = 2, has a horizontal tangent there, and then decreases. What does f' do at x = 2?

1. f' crosses zero, going from positive to negative
2. f' has a relative maximum at x = 2
3. f' is undefined at x = 2
4. f' crosses zero, going from negative to positive