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How do you use the first and second derivatives to sketch a function, and read between the graphs of f, f-prime and f-double-prime?

Topic 5.8 Sketching Graphs of Functions and Their Derivatives: use derivative information to sketch a graph and relate the graphs of f, f-prime and f-double-prime.

A focused answer to AP Calculus AB Topic 5.8, combining increasing/decreasing and concavity information to sketch a function and to read across the graphs of f, f-prime and f-double-prime, with worked feature-by-feature analysis.

Generated by Claude Opus 4.89 min answer

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  1. What this topic is asking
  2. The translation table
  3. A worked feature-by-feature sketch
  4. Reading the derivative graph
  5. Why this is a connecting-representations skill
  6. Sketching the derivative from the function
  7. A systematic reading order

What this topic is asking

The College Board (Topic 5.8) asks you to synthesize the first and second derivative information into a graph, and to read between the graphs of ff, f′f', and f′′f''. A question may give you any one of the three and ask about features of the others.

The translation table

A worked feature-by-feature sketch

Reading the derivative graph

A frequent exam format hands you the graph of f′f' and asks about ff. The rules invert cleanly: where the graph of f′f' is above the xx-axis, ff is increasing; where f′f' crosses the axis with a sign change, ff has an extremum; where the graph of f′f' is rising, ff is concave up; and where f′f' has a local max or min, ff has an inflection point. The most common error is to treat a feature of the f′f' graph as if it were a feature of ff: a maximum of f′f' is not a maximum of ff but an inflection point of ff. Keeping straight which graph you are looking at is the core discipline.

Why this is a connecting-representations skill

This topic is where the mathematical practice of connecting representations is examined most directly. You must move fluently among the graphical, analytical, and verbal descriptions of the same function and its derivatives. The grader looks for justifications phrased in terms of the correct derivative: "since f′f' changes from positive to negative" for an extremum, "since f′f' has a local minimum" or "since f′′f'' changes sign" for an inflection point. Naming the wrong derivative, even with the right conclusion, costs the justification mark, so anchor every statement to the specific derivative graph the conclusion depends on.

Sketching the derivative from the function

The reverse direction, drawing f′f' from the graph of ff, is just as testable. The height of the f′f' graph at each xx is the slope of ff there: where ff is rising, f′f' is positive (above the axis); where ff is falling, f′f' is negative; where ff has a horizontal tangent (a peak, valley, or flat inflection), f′f' crosses zero. Steeper parts of ff push the f′f' graph farther from the axis, and the flattest points of ff pull f′f' toward the axis. To sketch f′′f'' from f′f', repeat the same logic one level down. Practicing both directions, reading features down from ff to its derivatives and constructing derivative graphs up from a given ff, builds the two-way fluency the exam rewards.

A systematic reading order

When a problem gives one graph and asks for several features of the related functions, a fixed reading order avoids mistakes. Identify which function the graph shows (ff, f′f', or f′′f''), then translate to the target using the level relationships: zeros of f′f' with sign change give extrema of ff; local extrema of f′f' give inflection points of ff; the sign of f′f' gives the increasing/decreasing intervals of ff; the slope of f′f' gives the concavity of ff. Going through these in the same order each time, and writing the justification in terms of the derivative actually graphed, keeps the answer organized and earns the reasoning marks consistently.

Exam-style practice questions

Practice questions written in the style of College Board exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

AP 2022 (style)1 marksSection I (multiple choice, no calculator). The graph of f′f' is positive and decreasing on an interval. On that interval, the graph of ff is (A) increasing and concave up (B) increasing and concave down (C) decreasing and concave up (D) decreasing and concave down
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The correct answer is (B), increasing and concave down.

f′>0f' > 0 means ff is increasing. f′f' decreasing means f′′<0f'' < 0, so ff is concave down. The combination is increasing and concave down.

AP 2024 (style)4 marksSection II (free response, no calculator). The graph of f′f' (the derivative of ff) crosses the xx-axis at x=1x = 1 (from positive to negative) and at x=3x = 3 (from negative to positive), and has a minimum at x=2x = 2. (a) Where does ff have relative extrema, and of what kind? (b) Where does ff have a point of inflection? Justify each answer using the graph of f′f'.
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A 4-point read-the-derivative question.

(a) (2 points) At x=1x = 1, f′f' changes ++ to −-, so ff has a relative maximum. At x=3x = 3, f′f' changes −- to ++, so ff has a relative minimum.
(b) (2 points) A point of inflection of ff occurs where f′′f'' changes sign, i.e. where f′f' changes from decreasing to increasing. Since f′f' has a minimum at x=2x = 2, f′f' changes from decreasing to increasing there, so ff has a point of inflection at x=2x = 2.

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