Using parametric equations to generate linear equations.

Write-up by

Blair T. Dietrich

EMAT 6680

Assignment #10 Problems #5 & 7

This investigation will focus on the use of parametric equations to generate linear equations.

What type of graph does the pair of parametric equations  and  yield?  To investigate this question the equations were graphed for a range of values for k for each selected pair of values for a and b.  Some examples follow:

k = -2                                                  k = -1                                                  k = 0

k = 1                                                    k = 2                                                    k=-2,-1,0,1,2 all superimposed

In this case, a segment of a line is "captured" between the vertical lines x = 0 and x = 2.  As k is changed, the slope of the line segment changes.  In fact, the slope of the line segment equals the value of k.  Also, each of these line segments has a common point of (1,2).  This point seems to be a "pivot" point for the segment as k (the slope) is varied.

Try animating "k" on the GCF file here.

Similarly…

k = -2                                                  k = -1                                                  k = 0

k = 1                                                    k = 2                                                    k=-2,-1,0,1,2 all superimposed

In this case, a segment of a line is "captured" between the vertical lines x = -4 and x = -2.  As k is changed, the slope of the line segment changes.  In fact, the slope of the line segment equals the value of k.  Also, each of these line segments has a common point of (-3,1).  This point seems to be a "pivot" point for the segment as k (the slope) is varied.

Try animating "k" on the GCF file here.

It appears that a and b yield the coordinates of the pivot point.  This point is the point translated from (0,0) in the equation  where k is the slope of the line.  This can be shown by solving the equation  for t and substituting into the other equation:

This equation is clearly the equation  that has been translated horizontally a units and vertically b units.

With this in mind, it seems clear that in order to graph the line segment through (7,5) with a slope of 3, the parametric equations needed would be the following: