How to Plot Functions — Cartesian, Parametric & Polar

The Three Coordinate Systems for Graphing

1. Cartesian (Rectangular) Coordinates: y = f(x)

The most familiar system, where every point is defined by an x (horizontal) and y (vertical) value.

Key Concepts:

• Domain: All valid x-values (e.g., for √x, domain is x ≥ 0)
• Range: All possible y-values
• Intercepts:
  • x-intercept: Set y = 0 and solve for x (roots/zeros)
  • y-intercept: Set x = 0 and solve for y
• Asymptotes:
  • Vertical: Where function is undefined (e.g., x = 2 for 1/(x-2))
  • Horizontal: Behavior as x → ±∞ (e.g., y = 0 for 1/x)

Plotting Steps:

1. Determine domain and range
2. Find intercepts
3. Identify asymptotes
4. Calculate key points (e.g., vertex for quadratics)
5. Sketch the curve, respecting symmetry and end behavior

Example: f(x) = x² – 4

• Domain: All real numbers
• x-intercepts: x = ±2
• y-intercept: y = –4
• Vertex: (0, –4)
• Parabola opening upward

2. Parametric Equations: x = f(t), y = g(t)

Both x and y are defined in terms of a third variable (parameter t), ideal for curves that aren’t functions (e.g., circles, ellipses).

Key Concepts:

• Parameter range: Defines the portion of the curve plotted
• Direction of motion: As t increases, the curve is traced in a specific direction
• Eliminating the parameter: Solve one equation for t and substitute into the other to get Cartesian form

Example: Circle of radius 3

• x = 3cos(t), y = 3sin(t), 0 ≤ t ≤ 2π
• Eliminating t: x² + y² = 9

3. Polar Coordinates: r = f(θ)

Points are defined by a distance from origin (r) and an angle from positive x-axis (θ).

Key Concepts:

• Symmetry tests:
  • x-axis: Replace θ with –θ
  • y-axis: Replace θ with π – θ
  • origin: Replace r with –r or θ with θ + π
• Common curves:
  • Circles: r = a
  • Cardioids: r = a(1 ± cosθ)
  • Roses: r = a cos(nθ) (n petals if n odd, 2n if n even)

Example: Four-petal rose

• r = 3 cos(2θ)
• Petals at θ = 0, π/2, π, 3π/2

Advanced Graphing Techniques

Finding Derivatives and Integrals Graphically

• Derivative f'(x): Slope of tangent line at each point
  • f'(x) > 0 → function increasing
  • f'(x) below 0 → function decreasing
  • f'(x) = 0 → local max/min
• Integral ∫f(x)dx: Area under the curve between two points
  • Positive area above x-axis, negative below

Analyzing Transformations

Functions can be transformed via:

• Vertical shift: f(x) + k
• Horizontal shift: f(x – h)
• Vertical stretch: a·f(x)
• Horizontal stretch: f(bx)
• Reflection: –f(x) (over x-axis), f(–x) (over y-axis)

Example: f(x) = (x – 2)² + 3 is x² shifted right 2, up 3.

Using Technology Effectively

Modern graphing tools offer:

• Trace: Move along curve to see coordinates
• Zoom/Window: Adjust view to focus on key features
• Intersection: Find where two graphs cross
• Numerical calculus: Compute derivatives/integrals at points

Pro Tips & Common Mistakes

• Set appropriate window: A default [-10,10] window may miss key features—zoom in on intercepts or asymptotes.
• Check for discontinuities: Functions like 1/x have breaks—don’t connect across asymptotes.
• Use radians for trig functions: Most calculators default to radians for calculus.
• Verify with algebra: Don’t trust the graph blindly—confirm intercepts and asymptotes analytically.
• Label axes and key points: A well-annotated graph is far more useful than a bare plot.
• Consider domain restrictions: Square roots, logs, and denominators limit valid x-values.

Practical Applications

• Physics: Parametric equations model projectile motion (x = v₀t cosθ, y = v₀t sinθ – ½gt²)
• Engineering: Polar plots show antenna radiation patterns or stress distributions
• Economics: Supply/demand curves (Cartesian) illustrate market equilibrium
• Biology: Logistic growth curves model population dynamics
• Computer Graphics: Parametric equations generate smooth curves in vector graphics