Curve Sketching

Complete Curve Sketching Procedure

1. Domain: Find where $f(x)$ is defined
2. Intercepts: Find x-intercepts ($y = 0$) and y-intercept ($x = 0$)
3. Symmetry: Check for even/odd functions
4. Asymptotes: Find vertical, horizontal, and oblique asymptotes
5. First derivative: Find critical points, increasing/decreasing intervals
6. Second derivative: Find inflection points, concavity
7. Plot key points and connect with smooth curve

Symmetry

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

Asymptotes

Vertical Asymptotes

Occur where $f(x) \to \pm\infty$, typically where denominator = 0

Horizontal Asymptotes

$y = \lim_{x \to \infty} f(x) \quad \text{and/or} \quad y = \lim_{x \to -\infty} f(x)$

Oblique (Slant) Asymptotes

When degree of numerator = degree of denominator + 1

Use polynomial division: $y = mx + b$

Concavity

• Concave up: $f''(x) > 0$ (holds water, smile)
• Concave down: $f''(x) < 0$ (spills water, frown)

Inflection Points

Where $f''(x)$ changes sign (concavity changes)

Summary Table

$f'(x)$	$f''(x)$	Shape
$+$	$+$	increasing, concave up
$+$	$-$	increasing, concave down
$-$	$+$	decreasing, concave up
$-$	$-$	decreasing, concave down

Example: Complete Analysis

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

Step 1: Domain

All real numbers

Step 2: Intercepts

• y-intercept: $f(0) = 2$
• x-intercepts: $x^3 - 3x^2 + 2 = 0$
  • By inspection or factoring: $(x - 1)(x^2 - 2x - 2) = 0$
  • $x = 1$, $x = 1 \pm \sqrt{3}$

Step 3: Symmetry

$f(-x) = -x^3 - 3x^2 + 2 \neq f(x) \neq -f(x)$ No symmetry

Step 4: End Behavior

• As $x \to \infty$, $f(x) \to \infty$
• As $x \to -\infty$, $f(x) \to -\infty$ No horizontal asymptotes

Step 5: First Derivative

$f'(x) = 3x^2 - 6x = 3x(x - 2)$

$f'(x) = 0 \text{ at } x = 0, x = 2$

Interval	$f'(x)$	Behavior
$x < 0$	$+$	increasing
$0 < x < 2$	$-$	decreasing
$x > 2$	$+$	increasing

• Local max at $x = 0$: $f(0) = 2$
• Local min at $x = 2$: $f(2) = -2$

Step 6: Second Derivative

$f''(x) = 6x - 6 = 6(x - 1)$

$f''(x) = 0 \text{ at } x = 1$

Interval	$f''(x)$	Concavity
$x < 1$	$-$	concave down
$x > 1$	$+$	concave up

Inflection point at $x = 1$: $f(1) = 0$

Step 7: Key Points Summary

Point	$(x, y)$	Type
y-intercept	$(0, 2)$	local max
x-intercept	$(1, 0)$	inflection
Local min	$(2, -2)$
x-intercept	$(1-\sqrt{3}, 0) \approx (-0.73, 0)$
x-intercept	$(1+\sqrt{3}, 0) \approx (2.73, 0)$