Here we will explore the concept of a parametric equation. A parametric equation involves the addition of a "third variable" so to speak. More formally we would think of this as a parameter, typically named "t". We let t vary over a defined interval and for each value t we get a coordinate pair (x,y), the collection of which form some curve. It is also possible to obtain a three dimensional cooridinate array (x,y,z) in which each of x,y, and z are given by some function of t.
Lets look at the parametric curve created by the equations (x,y) = ( t + 1, 2t -1) as t varies from [0,4]. The curve comes out looking as such:
it is easy to see why we obtain a segment in this situation. By taking t=0 we see that the corresponding (x,y) coordinate pair is given by ( 0 + 1, 2(0) - 1) = (1, -1). The other endpoint to our segment is given by letting t=4, (5,7)
As a means of investigating different aspects of this parametric equation we could graph for different values of a,b,c,d in the general parametric (x,y) = (at + b, ct + d). It is not difficult to see that the slope of each segment is given by c/a and a change in b results in a translation left or right, and a change in d results in a vertical translation.
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