Parametrization of Line Segments?

Question

Consider the line segment AB with endpoints A (-3,5) and B(3,2)

a) Find a parametrization for segment AB using A as the initial point and B as the terminal point

b) Find a parametrization for segment AB from part (a) using B as the initial point and A as the terminal point

There are other parts but they involve graphing. I'll write them regardless and maybe you can provide guidance as to how to graph them?

c) Sketch and describe the graph of the curve parametrized by

x= 1- 3cos^2t, y= -2+4sin^2t, -∞ < t< ∞

d) Consider the graph of the curve parametrized by x= 1- (3/t), y= 2 - (4/t), 1 ≤ t < ∞

Is the graph related to the one drawn in part (c)? Explain.

Any help whatsoever on these would be much appreciated. I have no base knowledge of the subject and am struggling in teaching myself. Please explain as completely as possible so that I can do similar problems on my own. Thanks so much and 10 points to the best answer.

Answer 1

I&#039;m a math teacher.  What&#039;s not clear to me is whether you are using technology, and if so, what.  

For (a) and (b), the general rule to parametrize a line segment is

P(t) = P1 + t(P2-P1), where P1 is the initial point, P2 is the final point, and 0 ≤ t ≤ 1.

a) (-3, 5) + t(6, -3), 0 ≤ t ≤ 1

b) (3, 2) + t(-6, 3),  0 ≤ t ≤ 1

c) Both cosine^2 and sine^2 vary between zero and one, in opposite phase (one is 0 when the other is 1).  At t = 0, the parametrized point would be at (-2, -2), and t = (pi)/2, it would be at (1, 2).  It would just form a line segment, bouncing back and forth between those two points.  

d) At t = 1, this one would start at (-2, -2).  As t ---&gt; ∞, the two &quot;fraction&quot; terms would approach zero, so that in the infinite limit, the endpoint would be (1, 2).  This would also form a line segment between these two points --- the exact same line segment as in (c).