Sketching functions from derivative graphs

Task goals

• Use the sign of f'(x) to identify where f increases and decreases.
• Transfer zeros of f'(x) to stationary x-values of f, then classify a local maximum, local minimum, or stationary non-turn from the sign change.
• Infer concavity from whether f'(x) is increasing or decreasing.
• Use a given point to fix vertical position.
• Sketch a possible f consistent with zeros, signs, shape and anchor.
• Explain why one derivative can correspond to vertically shifted functions.
• Identify contradictions between a proposed f and its derivative.
• Reconstruct and classify a quartic function qualitatively from a cubic derivative.
• Optionally verify a polynomial sketch by integrating f'(x) and determining the constant from an anchor.