Mastery of “sketch the graph of a derivative” and sketching derivative graphs can turn a tough FRQ into easy points. Therefore, every AP® Calculus student should feel comfortable with this skill. According to College Board learning objectives FUN-4.A.9 and FUN-4.A.10, students must describe how features of f appear on the graphs of f' and f''. This article explains the need-to-know facts, a clear 4-step strategy, two worked examples, and a handy reference chart.

Foundations: What a Derivative Graph Represents

A derivative measures slope—an instantaneous rate of change.

• Positive slope means the original function rises.
• Negative slope means it falls.
• Zero slope shows a horizontal tangent.

Consequently, sketching derivative graphs turns “steepness” pictures into new curves. On the AP® exam or in the AP® curriculum, the prompt or section title may read “sketch the derivative of a graph,” “5.8 sketching graphs of derivatives,” or “how to sketch the derivative of a graph.” Regardless of wording, the task is the same: translate slope information into a graph of f'.

Key Relationships Among f, f', and f''

A. Increasing vs. Decreasing

• f increases ⇔ f' > 0.
• f decreases ⇔ f' < 0.

B. Local Maxima and Minima

Local peaks and valleys occur where f' = 0 and f' changes sign.

C. Concavity and Inflection

• f'' > 0 (or f' rising) ⇔ f is concave up.
• f'' < 0 (or f' falling) ⇔ f is concave down.
• Inflection points happen where f'' = 0 and concavity changes.

D. Quick Summary

• Slopes of f become heights of f'.
• Peaks of f drop to x-intercepts on f'.
• Steep segments on f turn into high or low points on f'.

The 4-Step Strategy for Sketching Derivative Graphs

1. Mark Critical x-Values
   • Identify where the slope of f looks zero or undefined.
2. Classify Sign Intervals
   • Decide where slopes are positive or negative; label each interval.
3. Estimate Relative Steepness
   • Steeper segments give larger |f'| values, setting vertical scale.
4. Check Concavity Cues
   • Ensure f' rises when f is concave up and falls when concave down; then draw a smooth curve.

Mini-Checklist (post-it style)

• Tangent flat → f' = 0
• Slope undefined → open circle or vertical asymptote on f'
• Rising slopes → f' > 0
• Falling slopes → f' < 0

Worked Example #1 – Smooth Curve

Prompt

The graph of f is a smooth “double-hump”: rising from left, peaking at x=-2, dipping to a valley at x=1, then rising again.

Step-by-Step Solution

1. Mark critical x. Slopes appear zero at x=-2 and x=1.
2. Determine sign.
   • For x<-2 the graph rises, so f'>0.
   • Between -2 and 1 it falls, so f'<0.
   • After 1 it rises again, so f'>0.
3. Estimate steepness. The hill near -2 is steeper than the valley near 1. Therefore, the positive values of f' are higher in magnitude than the negative peak.
4. Draw f'.
   • Plot zeros at x=-2,1.
   • Sketch a larger positive hump left of -2, dip to a smaller negative value between, then rise to positive afterward.
5. Concavity check. Since f is concave down on the first hump, f' should be decreasing there. The sketch confirms this, ensuring accuracy for FUN-4.A.10.

Result: a smooth “U-shaped” curve crossing the x-axis twice.

Worked Example #2 – Piecewise & Corner Points

Prompt

Consider a graph made of straight segments:

• From x=-3 to x=0, slope = 2.
• At x=0 a sharp corner occurs.
• From x=0 to x=2, slope = -1.
• From x=2 to x=4, slope = 0 (flat line).

Solution Steps

1. Critical points arise at x=0 (corner) and at the endpoints where slope changes.
2. Sign chart: slope 2 (positive), then -1 (negative), then 0.
3. Steepness: constant, so f' will be horizontal segments at heights 2 and -1.
4. At x=0 the slope jumps, hence f' has an open circle at height 2 just left of 0, another open circle at -1 just right, and no value exactly at 0.
5. From x=2 to 4, slope 0 gives the x-axis segment.

Therefore, the derivative graph is three horizontal pieces: y=2, y=-1, and y=0, with a jump discontinuity at x=0.

AP® Tip: Multiple-choice items often hide such jump behavior among distractors.

Common Pitfalls & Quick Fixes

• Misreading a barely tilted segment as horizontal—zoom mentally!
• Ignoring vertical tangents or cusps; remember, f' is undefined there.
• Confusing height with slope; always look at angle, not position.
• Forgetting to match concavity: if f bends upward, f' should rise.

Quick Reference Chart

Conclusion

Sketching derivative graphs transforms visual slopes into precise calculus insights. The 4-step strategy, supported by the examples above, aligns with FUN-4.A.9 and FUN-4.A.10. Consequently, practicing these steps builds the confidence needed to tackle any AP® question on “sketching derivative graphs.” Keep the checklist handy, avoid the common pitfalls, and success will follow.

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