Exercise 3.  Find a parametrization of the curve that is a portion of the parabola    that  

3 (c).  joins the point    to the origin  .  

Solution 3 (c).

See text and/or instructor's solution manual.

    Consider the parameterization in exercise 3 (a)  ,  in this case , and ,  and   is the portion of the parabola that joins the origin to the point .  
    But this orientation is the opposite of what we require!

    Since we need to traverse the curve in the opposite direction, if we substitute    in the above parameterization, then points will move along the curve in the opposite direction, as we desire.

    Thus, we can use the parameterization  ,  then

        ,   and

        .

    Therefore,  

           for     

is the portion of the parabola that joins the point to the origin .  

We are done.   

Aside.  We can let Mathematica double check our work.

        The curve     for   ,   and  

        the initial point is    and the terminal point is  .    

We are done.   

Aside.  There are many other parameterizations.  Here is one of them.

If we use the parameterization  ,  then

        ,   and

        .

    Therefore,  

           for     

is the portion of the parabola that joins the point to the origin .  

Aside.  We can let Mathematica double check our work.

        The curve     for   ,   and  

        the initial point is    and the terminal point is  .    

This solution is complements of the authors.

(c) 2008 John H. Mathews, Russell W. Howell