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5.8 Sketching Graphs of Functions and Their Derivatives

Syllabus
2020
Topic
5.8
Level
—

Learning objectives

—FUN-4.A—Justify conclusions about the behavior of a function based on the behavior of its derivatives—Topic 5.8• FUN-4.A Justify conclusions about the behavior of a function based on the behavior of its derivatives.• FUN-4.A.9 Key features of functions and their derivatives can be identified and related to their graphical, numerical, and analytical representations.• FUN-4.A.10 Graphical, numerical, and analytical information from f′ and f″ can be used to predict and explain the behavior of f.• Enduring understanding FUN-4: A function’s derivative can be used to understand some behaviors of the function.

Translate $f'$ and $f''$ into the Shape of $f$

Whether derivative information is given as a graph, table, or formula, read the same mathematical features: sign, zeros, sign changes, and increasing or decreasing behavior. Translate each feature into a statement about ff before attempting a sketch.

Derivative information Behavior of ff
f′>0f'>0 / f′<0f'<0 increasing / decreasing
f′f' changes +→−+\to- / −→+-\to+ local maximum / local minimum
f′f' increasing, equivalently f′′>0f''>0 concave up
f′f' decreasing, equivalently f′′<0f''<0 concave down
f′′f'' changes sign at a point on ff inflection point

Mark domain breaks and important xx-values first. Next record intervals of increase/decrease and classify any sign-changing zeros of f′f'. Then add concavity and verified inflection points from f′′f'' or from the trend of f′f'. Finally connect the features without contradicting any interval statement.

Suppose f′f' is positive on (−∞,−2)(-\infty,-2), negative on (−2,1)(-2,1), and positive on (1,∞)(1,\infty). Then ff has a local maximum at x=−2x=-2 and a local minimum at x=1x=1. If f′′<0f''<0 for x<0x<0 and f′′>0f''>0 for x>0x>0, the sketch is concave down before 00, concave up after 00, and has an inflection point at x=0x=0 if that point lies on the graph.

Derivative information determines shape, not absolute vertical position: functions that differ by a constant have the same derivatives. A value such as f(a)f(a) is needed to anchor the sketch vertically. Also, a zero of f′f' or f′′f'' matters only when the required sign change occurs.