Parametric Curve: Solving & Graphing

In summary, the conversation is about finding the cartesian product and graphing a parametric curve with the given equations x = e^t and y = e^-t. The participant is asking for help and suggests using identities such as u^{-r}= 1/u^r, -log(p) = log(1/p), and e^log(w)=w to solve the problem. They also mention trying to multiply x and y together as a possible method.
  • #1
Caldus
106
0
Another question...

If I were given these equations:

x = e^t
y = e^-t

Then I have to find the cartesian product for this parametric curve and then I have to sketch the graph of the curve. So here's the cartesian product I came up with:

Solve for t in y, so:

y = e^-t
ln y = ln e^-t
ln y = -t
t = - ln y

Then plug into the x part:

x = e^-ln y

Is that part correct?

I have no idea how to graph this...

Help appreciated.
 
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  • #2
Do you know any identities applicable to it?
 
  • #3
Simple method: u^{-r}= 1/u^r
slightly longer metheod, try multiplying x and y together!

or:

-log(p) = log(1/p)
e^log(w)=w

any of those help?
 

Related to Parametric Curve: Solving & Graphing

1. What is a parametric curve?

A parametric curve is a graph created by a set of equations known as parametric equations. These equations use a parameter, usually denoted by 't', to describe the coordinates of points on the curve. The parameter allows for the independent variation of two or more variables at the same time, resulting in a more complex curve than what can be achieved with a single equation.

2. How do you solve parametric equations?

To solve parametric equations, you need to first isolate the parameter (usually 't') in each equation. Then, substitute the resulting expression for 't' into the other equation. This will give you an equation with only one variable, which can be solved using basic algebraic methods. The resulting solutions can be plugged back into the original equations to find the corresponding coordinates on the curve.

3. What is the difference between a parametric curve and a Cartesian curve?

The main difference between a parametric curve and a Cartesian curve is the number of equations used to describe them. A parametric curve is created by a set of parametric equations, while a Cartesian curve is described by a single equation. Additionally, a parametric curve allows for the independent variation of multiple variables, while a Cartesian curve only has one independent variable (usually 'x').

4. How do you graph a parametric curve?

To graph a parametric curve, you first need to create a table of coordinates by choosing values for the parameter and plugging them into the parametric equations. Then, plot these coordinates on a graph and connect them to create the curve. You can also use a graphing calculator or computer software to graph parametric curves.

5. What are some real-life applications of parametric curves?

Parametric curves have many real-life applications, such as in physics, engineering, and computer graphics. For example, they can be used to model the trajectory of a projectile, the motion of a swinging pendulum, or the shape of a roller coaster. In computer graphics, parametric curves are used to create smooth and realistic animations in video games and movies.

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