parameterization

Apr 2010
15
0
Find a parameterization for each of the following curves.

a. The straight line segment from (3,2) to (-1,1).
b. The curve \(\displaystyle y=1-x^2\) from (3,-8) to (-2,-3).

I have no idea how to do this problem. Looks like a basic problem but I have no idea what parameterization means. Any help is appreciated!
 

dwsmith

MHF Hall of Honor
Mar 2010
3,093
582
Florida
Find a parameterization for each of the following curves.

a. The straight line segment from (3,2) to (-1,1).
b. The curve \(\displaystyle y=1-x^2\) from (3,-8) to (-2,-3).

I have no idea how to do this problem. Looks like a basic problem but I have no idea what parameterization means. Any help is appreciated!
a.
\(\displaystyle x(t)=(t-1)\mathbf{i}, 0\leq t \leq 4\) From this equation, the line will move from -1 to 3 on the x axis.
\(\displaystyle y(t)=\left(\frac{t}{4}+1\right)\mathbf{j}, 0\leq t \leq4\) From this equation, the line will move from 1 to 2 on the y axis.
\(\displaystyle r(t)=(t-1)\mathbf{i}+\left(\frac{t}{4}+1\right)\mathbf{j}, 0\leq t \leq4\) \(\displaystyle r(t)=x(t)+y(t)+z(t)\); \(\displaystyle z(t)\) if in 3 dimensions.
 
Apr 2010
15
0
a.
\(\displaystyle x(t)=(t-1)\mathbf{i}, 0\leq t \leq 4\)
\(\displaystyle y(t)=\left(\frac{3t}{4}+2\right)\mathbf{j}, 0\leq t \leq4\)
\(\displaystyle r(t)=(t-1)\mathbf{i}+\left(\frac{3t}{4}+2\right)\mathbf{j}, 0\leq t \leq4\)
hey thanks for the reply! How did you come up with those? Are there some formulas you can show me that would help with these problems? I'm not finding anything in my textbook.
 

dwsmith

MHF Hall of Honor
Mar 2010
3,093
582
Florida
hey thanks for the reply! How did you come up with those? Are there some formulas you can show me that would help with these problems? I'm not finding anything in my textbook.
I had a typo so I edited my last post.
 
Apr 2010
15
0
So what exactly is the question asking for? Sorry but I've never heard of parameterization, nor can I find it in the textbook.

Am I looking for an equation for these two points or..?
 

HallsofIvy

MHF Helper
Apr 2005
20,249
7,909
Find a parameterization for each of the following curves.

a. The straight line segment from (3,2) to (-1,1).
b. The curve \(\displaystyle y=1-x^2\) from (3,-8) to (-2,-3).

I have no idea how to do this problem. Looks like a basic problem but I have no idea what parameterization means. Any help is appreciated!
Then I assume you immediately looked up "parameterization" or "parameter" in the index of your book and now know what it means!

A parameter or a curve is an additional variable, t, say such that both x and y are written in terms of that variable. Any one-dimensional object, such as a curve, can be written in terms of 1 parameter, any two-dimensional object, such as a surface, can be written in terms of two parameters. In effect, that's what "dimension" means.

But as long as y is a function of x, we can use "x" itself as parameter.

If \(\displaystyle y= 1- x^2\) we can take \(\displaystyle x= t\), \(\displaystyle y= 1- t^2\) with \(\displaystyle -3\le t\le t\).