Lesson 6: 5.6 Determining Concavity of Functions over Their Domains

How does the rate of change of the derivative (the second derivative) describe the curvature or "bend" of a function’s graph?
What is the analytical relationship between the sign of f''(x) and the behavior of f'(x)?
How can we identify a Point of Inflection (POI) using both algebraic and graphical evidence?What does the concavity of a function tell us about the position of a tangent line relative to the curve?

Concave Up
Concave Down
Point of Inflection (POI)
Second Derivative
Critical Values of the First Derivative
Sign Analysis (or Sign Chart)
Tangent Line Overestimate/Underestimate

FUN-4.A: Determine the intervals on which a function is concave up or concave down and the x-coordinates of any points of inflection.
EK FUN-4.A.1: The graph of a function f is concave up on an interval if f' is increasing on that interval.
EK FUN-4.A.2: The graph of a function f is concave down on an interval if $f'$ is decreasing on that interval.
EK FUN-4.A.3: A point of inflection is a point on the graph of a differentiable function where the concavity changes.

In this lesson, students transition from analyzing the first derivative (increasing/decreasing) to analyzing the second derivative to determine the shape of the graph. Students learn to find the second derivative, identify potential points of inflection (where f''(x) = 0 or is undefined), and use sign charts to confirm intervals of concavity. The lesson also covers the "Tangent Line Test"—recognizing that if a graph is concave up, the tangent line lies below the curve, and if it is concave down, the tangent line lies above.

Purpose: To provide students with the analytical tools necessary to describe the "bending" behavior of functions, which is a critical component of curve sketching and interpreting motion (acceleration).
DOK Level 1 (Recall): Calculating second derivatives of polynomial, trig, and transcendental functions.
DOK Level 2 (Skill/Concept): Creating a second derivative sign chart and identifying intervals of concavity.
DOK Level 3 (Strategic Thinking): Justifying whether a point is a Point of Inflection based on a change in sign of f''(x), and interpreting concavity from the graph of f' or f''.

Support (Scaffolding): * Provide a "Derivative Hierarchy" graphic organizer showing the relationship of first and second derivative.
Use guided sign charts with pre-plotted critical values for students struggling with algebraic organization.
Extension (Enrichment): * Implicit Relations: Challenge students to find the concavity of a curve defined implicitly and substituting back into the expression.
Contextual Modeling: Ask students to explain what "concave down" means in the context of a "diminishing return" on a profit function.
Visual/Kinesthetic: * Have students use their hands to "trace" the slope of a curve to feel when the slope is increasing (hand tilting up = concave up) versus decreasing.

AP Classroom Assessments