The Slope Detective

A function has gone missing. All the department recovered is the graph of its derivative — $f^{\prime }$, the witness. Your group are the detectives, and the witness saw everything except one thing.

The task

Your teacher hands each group a different witness: the graph of some $f^{\prime }$, and nothing else. Interrogate it. Where was the missing $f$ increasing? Decreasing? Where did it turn, and which way? Where was it climbing steepest? Where was it curving upward, and where downward? Sketch a candidate $f$ consistent with every observation — then look at the board beside yours. They drew a different $f$, and theirs fits the evidence too. How many suspects fit, what do they all have in common, and what single extra fact would identify the culprit exactly? Then swap witnesses with another group and run the interrogation again, faster.

Facilitation notes — for the teacher

The sign chart invents itself at the boards — groups shade “up” and “down” stretches without being told to, which is the point. The productive fight: is a zero of $f^{\prime }$ automatically a turn? Slip a witness whose graph touches zero without crossing to the fastest group and watch the plateau surprise them. The vertical-shift ambiguity is the payoff — infinitely many suspects, one shared shape, and a single value of $f$ pins the culprit: the $+C$ of next year’s calculus, discovered under interrogation. A witness like the upward parabola with zeros at $1$ and $3$, plus the tip $f(0)=0$, makes a satisfying closing case.

What mathematics tends to surface

Direction from the sign of $f^{\prime }$, turning points where that sign changes, and — from the slope of the witness itself — the missing function’s concavity, including the inflection point where its bending switches. And the one thing the witness never saw: how high the whole story sat. Every vertical shift of a working suspect works too. Curve Sketching turns today’s interrogation into a checklist.

Where it leads

The next two classes sketch functions from their equations using exactly today’s reasoning, run in reverse. Optimization then hunts the very turning points you located — and Motion on a Line already told this story once, with velocity as the witness.

The answer is not on this page

No witness and no verdict appear here. The interrogation happens at the boards, evidence in hand.

Curriculum connection

B1.1

sketch the graph of a derivative function, given the graph of a function that is continuous over an interval, and recognize points of inflection of the given function (i.e., points at which the concavity changes) Sample problem: Investigate the effect on the graph of the derivative of applying vertical and horizontal translations to the graph of a given function.

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B1.4

describe key features of a polynomial function, given information about its first and/or second derivatives (e.g., the graph of a derivative, the sign of a derivative over specific intervals, the x-intercepts of a derivative), sketch two or more possible graphs of the function that are consistent with the given information, and explain why an infinite number of graphs is possible Sample problem: The following is the graph of the function $g(x)$. [The document prints the graph of y = g(x) here.] If $g(x)$ is the derivative of $f(x)$, and $f(0)=0$, sketch the graph of $f(x)$. If you are now given the function equation $g(x)=(x-1)(x-3)$, determine the equation of $f^{\prime \prime }(x)$ and describe some features of the equation of $f(x)$. How would $f(x)$ change graphically and algebraically if $f(0)=2$?

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