📈 Curve Sketching

What is Curve Sketching?

Curve sketching is the art of drawing an accurate graph of a function using calculus. Instead of plotting points, we use derivatives to understand the function's behavior!

💡 Key Idea: Use $f'(x)$ and $f''(x)$ to determine where the function is increasing/decreasing and where it's curving up/down.

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The Complete Curve Sketching Checklist

Step-by-Step Guide

Step 1: Find the domain

Step 2: Find intercepts

• $y$-intercept: Set $x = 0$
• $x$-intercepts: Solve $f(x) = 0$

Step 3: Check for symmetry

• Even: $f(-x) = f(x)$ (symmetric about $y$-axis)
• Odd: $f(-x) = -f(x)$ (symmetric about origin)

Step 4: Find asymptotes

• Vertical: Where denominator = 0 (for rational functions)
• Horizontal: Check $\lim_{x \to \pm\infty} f(x)$
• Slant: If degree of numerator is 1 more than denominator

Step 5: Find critical points ($f'(x) = 0$ or undefined)

Step 6: Determine intervals of increase/decrease

• $f'(x) > 0$: increasing
• $f'(x) < 0$: decreasing

Step 7: Find local extrema (use First or Second Derivative Test)

Step 8: Find inflection points ($f''(x) = 0$ and concavity changes)

Step 9: Determine concavity

• $f''(x) > 0$: concave up ∪
• $f''(x) < 0$: concave down ∩

Step 10: Sketch the graph using all information!

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What Each Derivative Tells You

First Derivative: $f'(x)$

Sign of $f'(x)$:

• $f'(x) > 0$ → function increasing ↗
• $f'(x) < 0$ → function decreasing ↘
• $f'(x) = 0$ → horizontal tangent (potential max/min)

Critical points: Where $f'(x) = 0$ or $f'(x)$ undefined

Local extrema: Where $f'$ changes sign

Second Derivative: $f''(x)$

Sign of $f''(x)$:

• $f''(x) > 0$ → concave up ∪ (curving upward)
• $f''(x) < 0$ → concave down ∩ (curving downward)
• $f''(x) = 0$ → potential inflection point

Inflection points: Where concavity changes (and $f''$ changes sign)

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Example: Complete Curve Sketch

Sketch $f(x) = x^3 - 6x^2 + 9x + 1$

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Step 1: Domain

Polynomial → domain is all real numbers: $(-\infty, \infty)$

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Step 2: Intercepts

$y$-intercept: $f(0) = 1$ → point $(0, 1)$

$x$-intercepts: $x^3 - 6x^2 + 9x + 1 = 0$ (hard to solve, skip for now)

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Step 3: Symmetry

$f(-x) = -x^3 - 6x^2 - 9x + 1 \neq f(x)$ and $\neq -f(x)$

No symmetry

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Step 4: Asymptotes

Polynomial → no asymptotes

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Step 5: First derivative and critical points

$f'(x) = 3x^2 - 12x + 9 = 3(x^2 - 4x + 3) = 3(x-1)(x-3)$

Critical points: $x = 1$ and $x = 3$

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Step 6: Sign chart for $f'(x)$

        1         3
   ++++  |  ----  |  ++++
      ↗  ▼  ↘     ▲  ↗

• Increasing on $(-\infty, 1)$ and $(3, \infty)$
• Decreasing on $(1, 3)$

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Step 7: Local extrema

At $x = 1$: $f'$ changes + to − → local max $f(1) = 1 - 6 + 9 + 1 = 5$

At $x = 3$: $f'$ changes − to + → local min $f(3) = 27 - 54 + 27 + 1 = 1$

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Step 8: Second derivative and inflection points

$f''(x) = 6x - 12 = 6(x - 2)$

$f''(x) = 0$ when $x = 2$

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Step 9: Concavity

           2
   ----    |    ++++
      ∩         ∪

• Concave down on $(-\infty, 2)$
• Concave up on $(2, \infty)$
• Inflection point at $x = 2$: $f(2) = 8 - 24 + 18 + 1 = 3$

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Step 10: Sketch

Key points:

• $(0, 1)$ - $y$-intercept
• $(1, 5)$ - local max
• $(2, 3)$ - inflection point
• $(3, 1)$ - local min

The graph:

• Increases to $(1, 5)$, then decreases to $(3, 1)$, then increases again
• Concave down until $x = 2$, then concave up
• Passes through $(0, 1)$

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Special Features to Look For

Cusps and Corners

Where $f$ is continuous but $f'$ doesn't exist

Example: $f(x) = |x|$ has a corner at $x = 0$

Vertical Tangents

Where $f'(x) = \pm\infty$

Example: $f(x) = x^{1/3}$ has vertical tangent at $x = 0$

Discontinuities

• Jump discontinuity
• Removable discontinuity
• Infinite discontinuity (vertical asymptote)

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Analyzing Rational Functions

For $f(x) = \frac{p(x)}{q(x)}$:

Vertical Asymptotes

Occur where $q(x) = 0$ (denominator zero, numerator non-zero)

Example: $f(x) = \frac{1}{x-2}$ has vertical asymptote at $x = 2$

Horizontal Asymptotes

Compare degrees of $p(x)$ and $q(x)$:

1. If deg($p$) < deg($q$): $y = 0$ is horizontal asymptote
2. If deg($p$) = deg($q$): $y = \frac{\text{leading coef of } p}{\text{leading coef of } q}$
3. If deg($p$) > deg($q$): No horizontal asymptote (may have slant asymptote)

Example: $f(x) = \frac{2x^2 + 1}{x^2 - 4}$ has horizontal asymptote $y = 2$

Slant (Oblique) Asymptotes

When deg($p$) = deg($q$) + 1, use polynomial division

Example: $f(x) = \frac{x^2 + 1}{x}$ has slant asymptote $y = x$

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Quick Summary Table

| Feature | How to Find | What It Tells You | |---------|-------------|-------------------| | Domain | Where $f$ is defined | Valid $x$ values | | Intercepts | $f(0)$ and $f(x) = 0$ | Where graph crosses axes | | Critical pts | $f'(x) = 0$ or undef | Potential max/min | | Inc/Dec | Sign of $f'(x)$ | Direction of graph | | Local extrema | Sign change in $f'$ | Peaks and valleys | | Inflection pts | $f''(x) = 0$ + sign change | Where concavity changes | | Concavity | Sign of $f''(x)$ | Curvature direction | | Asymptotes | Limits, denominators | Boundary behavior |

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⚠️ Common Mistakes

Mistake 1: Not Checking Sign Changes

$f''(c) = 0$ doesn't guarantee an inflection point - concavity must change!

Mistake 2: Forgetting Domain

Always consider where the function is actually defined.

Mistake 3: Plotting Only Critical Points

Include inflection points, intercepts, and other key features!

Mistake 4: Wrong Asymptote Analysis

Vertical asymptotes: denominator = 0 Horizontal asymptotes: check limits at infinity

Mistake 5: Ignoring Behavior at Infinity

Always check what happens as $x \to \pm\infty$

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The Big Picture

Putting It All Together

1. $f(x)$ tells you the height of the graph
2. $f'(x)$ tells you the slope (increasing/decreasing)
3. $f''(x)$ tells you the curvature (concave up/down)

All three work together to give you a complete picture!

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Shortcut for Simple Polynomials

For polynomials, you can often skip some steps:

• No asymptotes
• Domain is always $\mathbb{R}$
• Continuous everywhere
• Focus on critical points, extrema, and inflection points

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📝 Practice Strategy

1. Follow the checklist systematically
2. Make a sign chart for both $f'$ and $f''$
3. Calculate key points (don't just mark $x$-values, find $y$ too!)
4. Sketch lightly first, then refine
5. Check your work: Does the sketch match your analysis?
6. Label everything: maxima, minima, inflection points, asymptotes