# Graphing Parametric Equations: Solving for t^2, t^4, and t^6

• nameVoid
In summary: It is perhaps difficult to get a really good 3D view of it, but try from different angles and guess what it is.
nameVoid
t^2,t^4,t^6

Trying to graph this

I have the traces
x=y^2 for x>=0 in xy
Also x=z^3 for z>=0 in xz
And z=y^(3/2) for y>=0 in yz

Parametricplot3d in mathematics does nothing to get a picture for this graph and drawing is proofing difficult
In general what is the best way to plot these when it's not obvious plug in for t?

Last edited:
It's not clear what you are trying to do.

Is your function f(x,y,z) = (t^2, t^4, t^6) perhaps? Or something else?

Yes that is the function of x y z
I am trying to graph it

SteamKing said:
It's not clear what you are trying to do.

Is your function f(x,y,z) = (t^2, t^4, t^6) perhaps? Or something else?
I think you mean, vectorially, r(t) = (t^2, t^4, t^6), i.e. x = t^2 etc..
But from the OP,
x=y^2 for x>=0 in xy
Also x=z^3 for z>=0 in xz
suggests r(t) = (t^6, t^4, t^2).
what is the best way to plot these when it's not obvious plug in for t?
It depends what range of t you want to sketch it for.
It's kind of hard to sketch 3D curves. What exactly have you been asked to do?

Sketch the curve
mathematica doesn't help

Last edited:
The book asks for a sketch I assume there is a reasonable way
Plot t values?

Is this what you are looking for?
Code:
ParametricPlot3D[{t^2, t^4, t^6}, {t, 0, 1}]
If it isn't then perhaps you can explain what is incorrect about that.

Is gives a line I am under the impression that the graph is a surface
What's throwing me off is the book taking the trace in each plane
Sint,cost,t cylinder along z

nameVoid said:
Is gives a line I am under the impression that the graph is a surface
What's throwing me off is the book taking the trace in each plane
Sint,cost,t cylinder along z

No: the point-set you describe is a one-dimensional object = a curve in 3d. If you wanted a surface you would need two independent variables, so you would need to have something like three functions x(u,v), y(u,v), z(u,v) in two variables u and v.

In cost,sint,t
z for all real isn't a cylinder?

nameVoid said:
In cost,sint,t z for all real isn't a cylinder?

Nope.

Code:
ParametricPlot3D[{Cos[t], Sin[t], t}, {t, 0, 4 Pi}]

It is perhaps difficult to get a really good 3D view of it, but try from different angles and guess what it is.

## What are parametric equations?

Parametric equations are a set of equations that express the coordinates of a point in terms of one or more independent parameters, usually denoted as t or a. They are commonly used in mathematics to model curves and surfaces.

## How do I graph parametric equations?

To graph parametric equations, you can plot points by substituting different values for t in the equations and then connecting the points to create a curve. Alternatively, you can use a graphing calculator or computer software to graph the equations.

## What is the significance of solving for t^2, t^4, and t^6 in parametric equations?

Solving for t^2, t^4, and t^6 in parametric equations can help identify the shape and characteristics of the curve. For example, t^2 represents the squared term in a quadratic function, t^4 represents the fourth order term in a quartic function, and t^6 represents the sixth order term in a sextic function. By solving for these terms, we can determine the type of curve and its behavior.

## Can parametric equations be converted to Cartesian equations?

Yes, parametric equations can be converted to Cartesian equations by eliminating the parameter t. This can be done by solving one of the equations for t and substituting it into the other equation. The resulting equation will be in terms of x and y, making it a Cartesian equation.

## What are some applications of parametric equations?

Parametric equations have various applications in mathematics, physics, engineering, and computer graphics. They are used to model motion, curves, and surfaces in real-world situations. They are also helpful in analyzing and solving complex equations that cannot be easily expressed in Cartesian form.

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