Each value of t returns a vector that points from the origin to (x(t), y(t)). This can also be done in three dimensions (just add z(t)k). For all functional purposes, they work like parametric functions. Don't forget to draw arrows for orientation! (That's the direction where t is increasing).
You can graph parametric functions by eliminating the parameter or making a table of values (or using a computer). For functions in 3-space, drawing a rectangular prism makes graphing easier.
Example: r(t)=<t, -t^2+5>
x=t, y=-t^2+5
y=-x^2+5
r(t)=<2cos(t), 2sin(t)>, 0=<t=<2π
x=2cos(t), y=2sin(t)
x^2=4cos^2(t), y^2=4sin^2(t)
x^2+y^2=4
The domain of a vector-valued function is the intersection of the domain of each component.
Circular Helix: r(t)=<acos(t), asin(t), ct> These are not fun to draw.
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