Part 1: Vector-Valued Functions

Now that we have introduced and developed the concept of a vector, we are ready to use vectors to define functions. To begin with, a vector-valued function is a function whose inputs are a parameter t and whose outputs are vectors r( t) .

In 2 dimensions, a vector-valued function is of the form

r( t) = á f( t) ,g( t) ñ

Moreover, the set of position vectors of the form r( t) = á f( t) ,g( t) ñ for t in [ a,b] forms a curve whose orientation is in the direction in which the parameter is increasing. The point (f(a),g(a)) is called the initial point of the curve, and the point (f(b),g(b))  is the curve's terminal point.

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The function r( t) = á f(t) ,g( t) ñ , t in [ a,b] , is called a parametrization of the curve, and in order to sketch the curve, we often let

x = f( t) ,        y = g( t) ,        t  in [ a,b]

and then eliminate the parameter t in hopes of obtaining a familiar equation in x and y.      

EXAMPLE 1    Sketch the curve parametrized by r(t) = á 1-t²,2t²-t⁴ ñ for t in [ 0,1] .       

Solution: To begin with, x = 1-t², which means that t² = 1-x. Since y = 2t²-t⁴, that means that

y = 2( 1-x) -( 1-x) 2

(1)

which simplifies to y = 1-x². Moreover, the initial and terminal points are

initial: ( t = 0) x = 1-02 = 1, y = 2·0-04 = 0 terminal: ( t = 1) x = 1-12 = 0, y = 2·1-14 = 1

That is, the graph of r( t) = á1-t²,2t²-t⁴ ñ for t in [ 0,1] is the section of the graph of y = 1-x² between the initial point (1,0) and the terminal point ( 0,1) .

       

Check your Reading: How would we simplify (1) to y = 1-x²?