8.7 Parametric Equations: Graphs

Parametric equations offer a unique way to describe curves using a parameter. They're super useful for modeling motion and complex shapes that regular equations struggle with. You'll learn how to plot and interpret these equations.

Converting between parametric and Cartesian forms is a key skill. This lets you switch between different ways of describing the same curve. You'll also explore advanced concepts like vector functions and tangent lines to parametric curves.

Graphing Parametric Equations

Plotting parametric curves

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• Parametric equations express the coordinates of points on a plane curve using an independent variable called a parameter ($t$)
  • $x$-coordinate defined by function $x(t)$
  • $y$-coordinate defined by function $y(t)$
• Plotting points and graphing a curve from parametric equations involves:
  1. Choosing several values for parameter $t$
  2. Calculating corresponding $x$ and $y$ values for each $t$ using given parametric equations
  3. Plotting resulting $(x, y)$ points on Cartesian plane
  4. Connecting plotted points with smooth curve

Interpreting common parametric graphs

• Circles centered at origin represented by parametric equations:
  • $x(t) = r \cos(t)$
  • $y(t) = r \sin(t)$
  • $r$ is radius and $0 \leq t \leq 2\pi$
• Ellipses centered at origin represented by parametric equations:
  • $x(t) = a \cos(t)$
  • $y(t) = b \sin(t)$
  • $a$ and $b$ are lengths of semi-major and semi-minor axes and $0 \leq t \leq 2\pi$
• Projectile motion modeled using parametric equations:
  • $x(t) = v_0 \cos(\theta) t$
  • $y(t) = v_0 \sin(\theta) t - \frac{1}{2}gt^2$
  • $v_0$ is initial velocity, $\theta$ is angle of elevation, $g$ is acceleration due to gravity, and $t$ is time
• Lissajous figures are complex parametric curves resulting from the superposition of two harmonic oscillations

Conversion between parametric and Cartesian forms

• Converting parametric equations to Cartesian form:
  1. Solve one parametric equation for parameter $t$
  2. Substitute expression for $t$ into other parametric equation
  3. Simplify resulting equation to obtain Cartesian form
• Converting Cartesian form to parametric equations:
  1. Introduce parameter $t$
  2. Express $x$ and $y$ in terms of $t$ such that Cartesian equation is satisfied when expressions are substituted back into it
  3. Resulting equations for $x$ and $y$ in terms of $t$ are parametric equations

Advanced Parametric Concepts

• Vector-valued functions provide a compact representation of parametric equations in higher dimensions
• Tangent lines to parametric curves can be found using parametric differentiation techniques
• Arc length of a parametric curve can be calculated using integration methods
• The phase plane is a graphical tool for visualizing the behavior of parametric equations in dynamical systems