Lesson 9: 5.9 Connecting f, f' and f''

If I only have the graph of f', how can I determine where f is concave up?
What does a relative extremum on the graph of f' represent on the graph of f?
How do the x-intercepts of f'' relate to the slopes of f' and the curvature of f?
Can we reconstruct the "story" of a function's behavior given only its rate of change?

Derivative Graph

Slope of the Tangent

Sign Change

Increasing/Decreasing Magnitude

Concavity Linkage

Extrema of the Derivative

Point of Inflection

TRA-2.A: Interpret the meaning of a derivative within a context.

TRA-2.B: Apply the properties of derivatives to identify features of the graph of a function.

FUN-4.A: Justify conclusions about the behavior of a function based on the sign of its derivatives.

This lesson focuses on the translational logic between graphs. Students are rarely given the original function f(x) here; instead, they are given the graph of f'(x) or f''(x) and must describe the others.

Purpose: This is the most "AP-style" section of the unit. The AP exam frequently tests whether students can move fluidly between these levels without doing any actual symbolic differentiation. It develops high-level graphical literacy.

DOK Level 3 (Strategic Thinking): Using the graph of a derivative to describe the properties of the original function.

DOK Level 4 (Extended Thinking): Creating a consistent "set" of graphs (f, f', f'') or analyzing complex scenarios where a function’s derivative is given as an accumulation function.

For Struggling Learners (Scaffolding):The "Relationship Table": Provide a cheat sheet that aligns the features: If f' is... (Increasing/Decreasing/Max/Min)Then f is... (CU/CD/Inflection Point)
Color Overlays: Have students use highlighters on the f' graph—yellow for where the derivative is positive (meaning f goes up) and blue for where it is negative (meaning f goes down).
Physical Gestures: Use hand motions to show "slope" vs. "position" to help kinesthetic learners internalize the difference.

For Advanced Learners (Extension):Non-Differentiable Points: Provide a graph of f' that has a jump discontinuity or a hole and ask what that implies about the shape and continuity of f.
The "Triple Match": Provide three separate graphs and ask students to justify which is f, f', and f'' using specific coordinates as evidence.
Functional Analysis: Give the graph of f'' and one point on f' and one on f, then ask the student to sketch a possible original function.

AP College Board Assessments