Question

Finding a parametrization for a curve given the endpoints of (-1,3) and (3,-2) using slope?

Answer 1

okay so they want u to use a slope so, im gonna guess jut a constant slope and it is a line connecting through these points

Answer 2

lets start at point (-1,3)
what slope do you need to hit point (3,-2)?

Answer 3

hint slope=(y2-y1)/(x2-x1)

Answer 4

-5/4

Answer 5

okay so this means that for 1 unit of x you have -5/4 units in y

Answer 6

we are start at -1,3 so we need to make sure x=-1 and y=3 at starting so at t=0

Answer 7

<-1+t,3+t> <---- is a start now we have to make sure the slope condition is satisfied

for 1 change in x, we have -5/4 change in y so
vector in form
<x,y>
\[<x,y>=<-1+t,3+\frac{-5}{4}t>
\\
=<-1+t,3-\frac{5}{4}t>\]

Answer 8

this is only 1 representation of this vector, there are multiple other ways of saying this same thing, to see if they all mean the same thing, you have to satisfy 2 things, if the direction of travel is the same (i.e same unit vector direction) and that there is a common point between the other representations

Answer 9

because what u have defined here is not just a way to get from point -1,3 to 3,-2, but depending on whatever t you pick you have defined an infinite line