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Sketching Graphs of Functions and Their Derivatives AB & BC

Every feature of $f$ has a matching feature on the graph of $f'$ and another on the graph of $f''$: increasing corresponds to $f'$ above the axis, a relative extremum to $f'$ crossing the axis, concave up to $f'$ rising and $f''$ above the axis, and an inflection point to $f'$ having an extremum and $f''$ crossing. A feature moves one column to the right by turning from a shape into a position and from a position into a crossing.

Assembling a sketch means running the direction chart, then the concavity chart, then the anchor points. For $f(x) = x^{4} - 4x^{3}$ the derivative $4x^{2}(x - 3)$ gives a single relative minimum at $x = 3$, and $12x(x - 2)$ gives inflection points at $x = 0$ and $x = 2$, so the origin is a horizontal tangent that is not an extremum and is an inflection point. To tell several curves apart, line up the zeros of one with the turning points of another. What never identifies a curve is which is drawn higher, steeper, or more complicated, since shifting $f$ up by a constant leaves $f'$ untouched.

ONE FEATURE, THREE DESCRIPTIONS. THE MIDDLE COLUMN IS WHERE ERRORS LIVE. FEATURE OF f ON THE GRAPH OF f′ ON THE GRAPH OF f″ f INCREASING f′ ABOVE THE AXIS NOTHING FOLLOWS f DECREASING f′ BELOW THE AXIS NOTHING FOLLOWS f HAS A RELATIVE MAX f′ CROSSES + TO − f″ < 0 THERE f HAS A RELATIVE MIN f′ CROSSES − TO + f″ > 0 THERE f CONCAVE UP f′ INCREASING f″ ABOVE THE AXIS f CONCAVE DOWN f′ DECREASING f″ BELOW THE AXIS f HAS AN INFLECTION f′ HAS AN EXTREMUM f″ CROSSES THE AXIS NOTICE THE PATTERN: A FEATURE OF f MOVES ONE COLUMN RIGHT BY BECOMING A STATEMENT ABOUT POSITION, AND A POSITION BECOMES A CROSSING.
Rows 1 and 5 use the same curve for different questions. On a graph of $f'$, above the axis is direction and rising is concavity.
f(x) = x⁴ − 4x³ WITH ITS TWO DERIVATIVES, ON A SHARED x-SCALE. f f′ f″ MIN AT 3 ZERO AT 0 AND 3 ZERO AT 0 AND 2 x = 0: f′ TOUCHES ZERO WITHOUT CROSSING, AND f″ DOES CROSS.
At $x = 0$ the middle curve touches the axis without crossing, so there is no extremum, while the bottom curve crosses, so there is an inflection point.

The work

3 ways in · any order
Lesson
Sketching Graphs of Functions and Their Derivatives ›

Lays out the translation table between features of a function and features of its first and second derivatives, assembles a full sketch of a quartic from its two sign charts including a horizontal tangent that is an inflection point rather than an extremum, gives the line-up-the-zeros method for telling several curves apart, and names the four sentences that produce wrong answers from correct pictures.

Skill check · 10 scenarios
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Diagnostic
10-item topic check ›

Ten items spanning the two failure modes of this topic: reading a feature off the wrong curve of a function, its derivative, and its second derivative, and deciding concavity from the position of a derivative graph rather than from its rise and fall.

Not yet available · 10 items
Coming soon
Targeted Practice
Drill a single misconception ›

Pick one of the failure modes you missed and drill it on its own. The round is adaptive: two correct in a row clears the misconception and moves you to the next.

Take the diagnostic to identify your misconceptions
Diagnostic first