Unit 5.8 – Sketching Graphs of Functions and Their Derivatives

AP® Calculus AB & BC | The Art of Curve Sketching Using Calculus

Why This Matters: Curve sketching is the culmination of everything you've learned about derivatives! By analyzing \(f'(x)\) and \(f''(x)\), you can sketch accurate graphs showing all key features: intercepts, asymptotes, extrema, inflection points, and behavior. This topic synthesizes concepts from Unit 5: critical points, increasing/decreasing behavior, concavity, and extrema. You'll also learn to work backwards—given a graph of \(f'\), sketch \(f\), or vice versa. This skill is essential for understanding function behavior, solving optimization problems, and succeeding on AP® exam free-response questions!

🔗 The Complete Relationship: \(f\), \(f'\), and \(f''\)

The Master Connection Table

How Derivatives Determine Graph Features
Derivative Info	What It Tells Us	Graph Feature
\(f'(x) > 0\)	Function is increasing	Graph rises from left to right ↗
\(f'(x) < 0\)	Function is decreasing	Graph falls from left to right ↘
\(f'(x) = 0\)	Critical point (horizontal tangent)	Possible max, min, or inflection point
\(f'(x)\) undefined	Critical point (corner/cusp)	Sharp turn or vertical tangent
\(f''(x) > 0\)	Function is concave up	Curves upward ∪ (like a cup)
\(f''(x) < 0\)	Function is concave down	Curves downward ∩ (like a cap)
\(f''(x) = 0\)	Possible inflection point	Concavity may change
\(f'(c) = 0, f''(c) > 0\)	Local minimum at \(c\)	Valley bottom ∪
\(f'(c) = 0, f''(c) < 0\)	Local maximum at \(c\)	Hilltop ∩

🎯 Combined Behavior Analysis:

Understanding Function Behavior from Derivatives
Sign of \(f'\)	Sign of \(f''\)	Function Behavior	Visual
+ (positive)	+ (positive)	Increasing & Concave Up	⤴ Rising faster
+ (positive)	− (negative)	Increasing & Concave Down	⤵ Rising slower
− (negative)	+ (positive)	Decreasing & Concave Up	⤴ Falling slower
− (negative)	− (negative)	Decreasing & Concave Down	⤵ Falling faster

📋 Complete Curve Sketching Procedure

The Systematic 10-Step Process:

Find the domain: Identify where \(f(x)\) is defined
- Check for division by zero
- Check for negative values under even roots
- Check for logarithm domains
Find intercepts:
- y-intercept: Set \(x = 0\), find \(f(0)\)
- x-intercepts: Set \(f(x) = 0\), solve for \(x\)
Check for asymptotes:
- Vertical: Where denominator = 0
- Horizontal: \(\lim_{x \to \pm\infty} f(x)\)
- Slant: Long division if degree(num) = degree(den) + 1
Find \(f'(x)\) and simplify
Find critical points: Solve \(f'(x) = 0\) and find where \(f'(x)\) is undefined
Analyze sign of \(f'(x)\): Create sign chart to find increasing/decreasing intervals
Find \(f''(x)\) and simplify
Find potential inflection points: Solve \(f''(x) = 0\)
Analyze sign of \(f''(x)\): Create sign chart to find concavity
Sketch the graph: Combine all information
- Plot critical points, inflection points, intercepts
- Draw asymptotes
- Use increasing/decreasing and concavity to connect points

💡 Pro Tip: Create a Master Analysis Chart that shows all information at once:

Interval:      (-∞, c₁)    c₁    (c₁, c₂)    c₂    (c₂, ∞)

f'(x):         + + + +     0     − − − −     0     + + + +
f behavior:    Incr.            Decr.              Incr.
Extrema:                  MAX                MIN

f''(x):        − − −       −     + + + +     +     + + + +
Concavity:     Down              Up                Up
Inflection:         (somewhere between c₁ and c₂)

📊 Sketching \(f\) Given the Graph of \(f'\)

READING THE DERIVATIVE GRAPH

Key Principles for Sketching \(f\) from \(f'\)

Where \(f'(x) > 0\) (above x-axis) → \(f\) is increasing
Where \(f'(x) < 0\) (below x-axis) → \(f\) is decreasing
Where \(f'(x) = 0\) (crosses x-axis) → \(f\) has horizontal tangent (critical point)
Where \(f'\) has a maximum → \(f\) changes from concave down to concave up
Where \(f'\) has a minimum → \(f\) changes from concave up to concave down
Where \(f'\) is increasing → \(f\) is concave up
Where \(f'\) is decreasing → \(f\) is concave down

Step-by-Step: Sketching \(f\) from \(f'\)

Mark critical points: Where \(f'(x) = 0\) (f' crosses x-axis)
Determine extrema:
- If \(f'\) changes from + to − → Local max
- If \(f'\) changes from − to + → Local min
Find inflection points: Where \(f'\) has max or min (where \(f'' = 0\))
Sketch regions:
- Increasing where \(f' > 0\)
- Decreasing where \(f' < 0\)
- Concave up where \(f'\) is increasing
- Concave down where \(f'\) is decreasing
Connect smoothly respecting all the features

📈 Sketching \(f'\) Given the Graph of \(f\)

READING THE FUNCTION GRAPH

Key Principles for Sketching \(f'\) from \(f\)

Where \(f\) is increasing → \(f'(x) > 0\) (above x-axis)
Where \(f\) is decreasing → \(f'(x) < 0\) (below x-axis)
Where \(f\) has horizontal tangent → \(f'(x) = 0\) (crosses x-axis)
Where \(f\) has local max → \(f'\) crosses from + to − (zero)
Where \(f\) has local min → \(f'\) crosses from − to + (zero)
Where \(f\) is concave up → \(f'\) is increasing
Where \(f\) is concave down → \(f'\) is decreasing
Where \(f\) has inflection point → \(f'\) has extremum
Steep parts of \(f\) → Large magnitude of \(f'\)
Flat parts of \(f\) → \(f'\) near zero

Step-by-Step: Sketching \(f'\) from \(f\)

Identify critical points on \(f\): Local max/min → \(f' = 0\)
Mark zeros of \(f'\): At each critical point of \(f\)
Determine sign of \(f'\):
- Positive where \(f\) increases
- Negative where \(f\) decreases
Identify inflection points on \(f\): → \(f'\) has extrema there
Determine behavior of \(f'\):
- Increasing where \(f\) is concave up
- Decreasing where \(f\) is concave down
Consider magnitude: Steeper slopes on \(f\) → larger \(|f'|\)
Sketch \(f'\) connecting all features smoothly

📖 Comprehensive Worked Examples

Example 1: Complete Curve Sketch

Problem: Sketch the graph of \(f(x) = x^3 - 6x^2 + 9x + 1\) showing all key features.

Solution:

Step 1: Domain

Polynomial → Domain: \((-\infty, \infty)\) ✓

Step 2: Intercepts

y-intercept: \(f(0) = 1\) → Point: \((0, 1)\)
x-intercepts: \(x^3 - 6x^2 + 9x + 1 = 0\) (difficult to solve exactly)

Step 3: Asymptotes

None (polynomial)

Step 4-6: First Derivative Analysis

\[ f'(x) = 3x^2 - 12x + 9 = 3(x^2 - 4x + 3) = 3(x - 1)(x - 3) \]

Critical points: \(x = 1, 3\)

Interval	Test Point	\(f'(x)\)	Behavior
\((-\infty, 1)\)	\(x = 0\)	\(3(0-1)(0-3) = 9 > 0\)	Increasing ↗
\((1, 3)\)	\(x = 2\)	\(3(1)(-1) = -3 < 0\)	Decreasing ↘
\((3, \infty)\)	\(x = 4\)	\(3(3)(1) = 9 > 0\)	Increasing ↗

Extrema:

Local max at \(x = 1\): \(f(1) = 1 - 6 + 9 + 1 = 5\)
Local min at \(x = 3\): \(f(3) = 27 - 54 + 27 + 1 = 1\)

Step 7-9: Second Derivative Analysis

\[ f''(x) = 6x - 12 = 6(x - 2) \]

Potential inflection point: \(x = 2\)

Interval	Test Point	\(f''(x)\)	Concavity
\((-\infty, 2)\)	\(x = 0\)	\(-12 < 0\)	Concave Down ∩
\((2, \infty)\)	\(x = 3\)	\(6 > 0\)	Concave Up ∪

Inflection point: \(x = 2\), \(f(2) = 8 - 24 + 18 + 1 = 3\) → \((2, 3)\)

Step 10: Summary for Sketch

Key Points:
• y-intercept: (0, 1)
• Local max: (1, 5)
• Inflection: (2, 3)
• Local min: (3, 1)

Intervals:
• Increasing: (-∞, 1) ∪ (3, ∞)
• Decreasing: (1, 3)
• Concave down: (-∞, 2)
• Concave up: (2, ∞)

Graph shape:
(-∞, 1): Rising, concave down
(1, 2): Falling, concave down (past max)
(2, 3): Falling, concave up (past inflection)
(3, ∞): Rising, concave up (past min)

Sketch Instructions: Start at (0,1), rise with concave down curve to max at (1,5), fall through inflection at (2,3) changing concavity, reach min at (3,1), then rise with concave up curve.

Example 2: Sketching \(f\) Given Graph of \(f'\)

Problem: Given that \(f'(x)\) is a parabola opening upward with vertex at \((2, -4)\) and zeros at \(x = 0\) and \(x = 4\), sketch \(f(x)\) if \(f(0) = 3\).

Solution:

Step 1: Analyze \(f'(x)\)

\(f'(x) = 0\) at \(x = 0, 4\) → Critical points of \(f\)
\(f'(x) < 0\) for \(x \in (0, 4)\) (below x-axis)
\(f'(x) > 0\) for \(x < 0\) or \(x > 4\) (above x-axis)
Minimum of \(f'\) at \(x = 2\) → Inflection point of \(f\)

Step 2: Determine behavior of \(f\)

\(x < 0\): \(f' > 0\) → \(f\) increasing
\(x = 0\): \(f' = 0\) → \(f\) has horizontal tangent
\(0 < x < 4\): \(f' < 0\) → \(f\) decreasing
\(x = 4\): \(f' = 0\) → \(f\) has horizontal tangent
\(x > 4\): \(f' > 0\) → \(f\) increasing

Step 3: Identify extrema

At \(x = 0\): \(f'\) changes from + to − → Local maximum
At \(x = 4\): \(f'\) changes from − to + → Local minimum

Step 4: Determine concavity

\(x < 2\): \(f'\) decreasing → \(f'' < 0\) → Concave down
\(x = 2\): \(f'\) has minimum → \(f'' = 0\) → Inflection point
\(x > 2\): \(f'\) increasing → \(f'' > 0\) → Concave up

Step 5: Sketch

Starting at \((0, 3)\) (given):

Local max at \(x = 0\), value = 3
Decrease with concave down until \(x = 2\)
Inflection point at \(x = 2\)
Continue decreasing but concave up until \(x = 4\)
Local min at \(x = 4\) (value less than 3)
Increase with concave up for \(x > 4\)

Result: \(f\) has a local max at (0, 3), decreases through an inflection point at \(x = 2\), reaches a local min at \(x = 4\), then increases. Shape: ∩ for \(x < 2\), ∪ for \(x > 2\).

Example 3: Sketching \(f'\) Given Graph of \(f\)

Problem: Given \(f(x) = x^4 - 4x^3\), sketch \(f'(x)\).

Solution:

Step 1: Find and analyze critical points of \(f\)

\[ f'(x) = 4x^3 - 12x^2 = 4x^2(x - 3) \]

Critical points: \(x = 0, 3\)

At \(x = 0\): No extremum (inflection point with horizontal tangent)
At \(x = 3\): Local minimum

→ \(f'(0) = 0\) and \(f'(3) = 0\)

Step 2: Determine sign of \(f'\)

\(x < 0\): \(f\) decreasing → \(f' < 0\)
\(0 < x < 3\): \(f\) decreasing → \(f' < 0\)
\(x > 3\): \(f\) increasing → \(f' > 0\)

Step 3: Find inflection points of \(f\)

\[ f''(x) = 12x^2 - 24x = 12x(x - 2) \]

Inflection points at \(x = 0, 2\) → \(f'\) has extrema there

Step 4: Determine behavior of \(f'\)

\(x < 0\): \(f\) concave up → \(f'\) increasing
\(x = 0\): Inflection point → \(f'\) has local max
\(0 < x < 2\): \(f\) concave down → \(f'\) decreasing
\(x = 2\): Inflection point → \(f'\) has local min
\(x > 2\): \(f\) concave up → \(f'\) increasing

Step 5: Sketch \(f'\)

Key features:

Crosses x-axis at \(x = 0\) (local max of \(f'\))
Has local min at \(x = 2\) (negative value)
Crosses x-axis again at \(x = 3\)
Below x-axis for \(0 < x < 3\)
Above x-axis for \(x > 3\)

Result: \(f'(x)\) crosses x-axis at \(x = 0\) and \(x = 3\), has a local min at \(x = 2\), is negative between 0 and 3, and positive for \(x > 3\). Shape matches \(f'(x) = 4x^2(x-3)\).

🎨 Common Graph Patterns to Recognize

Standard Function Behaviors:

Quick Recognition Guide
Function Type	Key Features	\(f'\) Behavior	\(f''\) Behavior
Linear: \(y = mx + b\)	Straight line	\(f' = m\) (constant)	\(f'' = 0\) everywhere
Quadratic: \(y = ax^2 + bx + c\)	Parabola, 1 vertex	Linear	\(f'' = 2a\) (constant)
Cubic: \(y = ax^3 + \ldots\)	S-curve, 1 inflection	Parabola	Linear
Quartic: \(y = ax^4 + \ldots\)	W or M shape	Cubic	Parabola
Exponential: \(y = e^x\)	Always increasing, concave up	\(f' = e^x\)	\(f'' = e^x > 0\)
Logarithm: \(y = \ln(x)\)	Always increasing, concave down	\(f' = \frac{1}{x}\)	\(f'' = -\frac{1}{x^2} < 0\)
Sine/Cosine	Periodic waves	Cosine/Negative sine	Negative sine/Negative cosine

💡 Tips, Tricks & Strategies

✅ Essential Curve Sketching Tips:

Always start with domain: Know where the function exists
Find ALL critical points: Both where \(f' = 0\) and undefined
Create sign charts: Visual organization prevents errors
Mark inflection points: Where concavity changes
Plot key points first: Intercepts, extrema, inflection points
Use asymptotes as guides: Function approaches but doesn't cross
Respect concavity: Curves must bend correctly
Connect smoothly: No sharp corners unless derivative undefined
Check end behavior: What happens as \(x \to \pm\infty\)?
Verify with calculus: Does sketch match derivative information?

🔥 For Sketching \(f\) from \(f'\):

X-axis crossings of \(f'\): → Critical points of \(f\)
Sign of \(f'\): → Increasing/decreasing behavior of \(f\)
Extrema of \(f'\): → Inflection points of \(f\)
Slope of \(f'\): → Concavity of \(f\)
Large \(|f'|\): → Steep parts of \(f\)
Remember: You're sketching ONE antiderivative (infinite family possible)

🔥 For Sketching \(f'\) from \(f\):

Extrema of \(f\): → Zeros of \(f'\)
Increasing \(f\): → \(f' > 0\) (above x-axis)
Decreasing \(f\): → \(f' < 0\) (below x-axis)
Inflection points of \(f\): → Extrema of \(f'\)
Concave up \(f\): → \(f'\) increasing
Concave down \(f\): → \(f'\) decreasing
Steep slopes on \(f\): → Large values of \(|f'|\)

❌ Common Mistakes to Avoid

Mistake 1: Confusing zeros of \(f\) with zeros of \(f'\) (x-intercepts vs critical points)
Mistake 2: Drawing sharp corners where derivative is defined (must be smooth)
Mistake 3: Ignoring concavity when sketching (curves must bend correctly)
Mistake 4: Not checking if \(f'\) changes sign at critical points
Mistake 5: Forgetting that \(f\) can increase even if \(f'\) is decreasing
Mistake 6: Assuming every zero of \(f'\) is an extremum (could be inflection point)
Mistake 7: Crossing asymptotes (can approach but never cross vertical asymptotes)
Mistake 8: Not marking inflection points (where \(f''\) changes sign)
Mistake 9: Forgetting to verify end behavior
Mistake 10: Making \(f'\) cross x-axis at wrong places (only at \(f\)'s critical points)
Mistake 11: Drawing \(f'\) with wrong concavity based on \(f\)'s concavity
Mistake 12: Not respecting the relative steepness (magnitude of slopes)

📝 Practice Problems

Set A: Complete Sketches

Sketch \(f(x) = x^3 - 3x + 1\) showing all features
Sketch \(f(x) = \frac{x^2}{x-1}\) including asymptotes
Sketch \(f(x) = x^4 - 4x^2 + 3\)

Key Features to Find:

Critical pts: \(x = \pm 1\); Inflection: \(x = 0\); Local max: \((-1, 3)\); Local min: \((1, -1)\)
VA at \(x = 1\); HA: \(y = x + 1\) (slant); Local min at \(x = 0\), max at \(x = 2\)
Critical pts: \(x = 0, \pm\sqrt{2}\); Inflection: \(x = \pm\frac{\sqrt{2}}{\sqrt{3}}\)

Set B: Sketching from Derivatives

Given \(f'(x) = (x+1)(x-2)\), sketch possible \(f(x)\)
If \(f'(x)\) is a sine wave, what can you say about \(f(x)\)?
Given \(f(x) = \sin(x)\) on \([0, 2\pi]\), sketch \(f'(x)\)

Key Points:

\(f\) has local max at \(x = -1\), local min at \(x = 2\); decreases between them
\(f\) is a cosine wave (shifted vertically by constant of integration)
\(f'(x) = \cos(x)\): starts at 1, crosses axis at \(\frac{\pi}{2}, \frac{3\pi}{2}\)

Set C: Conceptual

Can \(f\) be increasing while \(f'\) is decreasing? Explain with example.
If \(f'\) has 3 zeros, how many local extrema can \(f\) have? Explain.

Answers:

Yes! If \(f' > 0\) but decreasing, \(f\) increases but at slower rate. Ex: \(f(x) = x^2\) for \(x < 0\)
At most 3, but possibly fewer (some zeros might not be sign changes)

✏️ AP® Exam Success Tips

What AP® Graders Look For:

Clear labeling: Mark critical points, inflection points, intercepts
Proper behavior: Increasing/decreasing where appropriate
Correct concavity: Curves bend the right way
Asymptote awareness: Shown and approached correctly
Smooth curves: No unnecessary corners
Relative positioning: Points in correct order
Justification: Explain WHY graph has each feature
Sign analysis: Show work for \(f'\) and \(f''\)

Common FRQ Formats:

"Sketch the graph of f showing all key features"
"Given the graph of f', sketch a possible graph of f"
"Sketch the graph of f' given the graph of f"
"Justify that your sketch satisfies the given conditions"
"Label all critical points and inflection points"
"Identify intervals where f is increasing and concave down"

💯 Earning Full Credit:

Show calculus work: Find \(f'\) and \(f''\), identify critical points
Create sign charts: Demonstrate systematic analysis
Label clearly: Mark and identify all key features
Justify features: Explain why graph behaves as shown
Check consistency: Does sketch match all derivative information?

⚡ Quick Reference Card

Curve Sketching Quick Reference
Derivative Info	Graph Feature	How to Identify
\(f' > 0\)	Increasing	Graph rises ↗
\(f' < 0\)	Decreasing	Graph falls ↘
\(f' = 0\)	Critical point	Horizontal tangent
\(f'' > 0\)	Concave up	Curves upward ∪
\(f'' < 0\)	Concave down	Curves downward ∩
\(f'' = 0\), sign change	Inflection point	Concavity changes
\(f' = 0, f'' > 0\)	Local minimum	Valley ∪
\(f' = 0, f'' < 0\)	Local maximum	Peak ∩

🔗 Connections to Other Topics

Topic 5.8 Synthesizes:

Topic 5.2 (Critical Points): Where \(f' = 0\) or undefined
Topic 5.3 (Inc/Dec): Sign of \(f'\) determines behavior
Topic 5.4 (1st Deriv Test): Classifies critical points
Topic 5.5 (Candidates Test): Finding absolute extrema
Topic 5.6 (Concavity): Sign of \(f''\) determines curvature
Topic 5.7 (2nd Deriv Test): Alternative for classifying extrema
All of Unit 5: Complete analysis of function behavior
Unit 6 (Integration): Going from \(f'\) to \(f\) is antiderivative

Master Curve Sketching! This culminating topic brings together ALL derivative concepts: use \(f'(x)\) to find critical points (where \(f' = 0\)) and determine increasing/decreasing behavior (sign of \(f'\)); use \(f''(x)\) to find inflection points (where \(f''\) changes sign) and determine concavity (sign of \(f''\)). The complete procedure: find domain, intercepts, asymptotes, \(f'\), critical points, sign of \(f'\), \(f''\), potential inflection points, sign of \(f''\), then sketch combining all features. When sketching \(f\) from \(f'\): zeros of \(f'\) are critical points of \(f\), sign of \(f'\) shows inc/dec of \(f\), extrema of \(f'\) are inflection points of \(f\). When sketching \(f'\) from \(f\): extrema of \(f\) are zeros of \(f'\), inc/dec of \(f\) shows sign of \(f'\), inflection points of \(f\) are extrema of \(f'\). Always create sign charts, label key features, respect concavity, and verify consistency! 🎯✨