12.2 Graphs of Parametric Equations

Parametric equations offer a unique way to describe curves, using a parameter to define both x and y coordinates. This approach allows for more complex shapes and behaviors than traditional function notation, opening up new possibilities in graphing and analysis.

Understanding parametric equations involves graphing techniques, identifying key features, and analyzing curve behavior. From tangent lines to asymptotes, these tools provide insights into the nature of curves, enhancing our ability to model real-world phenomena mathematically.

Understanding Parametric Equations and Their Graphs

Graphing parametric equations

• Parametric equations express x and y coordinates as functions of parameter t $x = f(t)$, $y = g(t)$ (circle: $x = \cos t$, $y = \sin t$)
• Plot points by creating table of t, x, y values choosing appropriate range and interval for t
• Connect plotted points forming curve with arrows indicating increasing t values
• Identify domain restrictions on t (t ≥ 0 for spiral)

Features of parametric curves

• x-intercepts: t-values where $y = 0$ (ellipse: $t = 0$, $\pi$)
• y-intercepts: t-values where $x = 0$ (ellipse: $t = \pi/2$, $3\pi/2$)
• Symmetry: check about x-axis, y-axis, origin test even/odd properties of $f(t)$, $g(t)$
• Periodic behavior: curve repeats after certain t interval (circle: period $2\pi$)
• Maximum/minimum points: t-values where $dx/dt = 0$ or $dy/dt = 0$ (cycloid: max at $t = 2\pi n$)

Tangent lines to parametric curves

• Calculate derivatives: $dx/dt$ and $dy/dt$ using chain rule
• Tangent line slope: $dy/dx = (dy/dt) / (dx/dt)$
• Determine curve point for specific t-value
• Write tangent line equation: $y - y_0 = m(x - x_0)$
• Special cases:
  • Vertical tangent: $dx/dt = 0$ (cardioid: $t = \pi$)
  • Horizontal tangent: $dy/dt = 0$ (lemniscate: $t = \pi/4$)

Behavior of parametric curves

• Evaluate $\lim_{t \to \infty} x(t)$ and $\lim_{t \to \infty} y(t)$
• Identify asymptotes:
  • Horizontal: $\lim_{t \to \infty} y(t)$ (catenary: $y = a\cosh(x/a)$)
  • Vertical: $\lim_{t \to a} x(t)$ (a is finite value)
• End behavior: curve direction as t increases/decreases indefinitely
• Analyze continuity at specific t-values checking for removable or jump discontinuities
• Investigate near critical points examining curve as t approaches values where derivatives undefined (cusp: $t = 0$ in astroid)