Comparing parametric equations

[Jbreezy]

Homework Statement

Compare the curves represented by the the parametric equations. How do they differ?
a.) x =t , y = t^-2
b.) x = cost , y = (sect)^2
c.) x = e^t , y = e^(-2t)

Homework Equations

So I drew them on the calculator they all look like umm... how do I describe this picture the x and y-axis ...now picture ...well just picture 1/x^2 that is what they kind look like.

The Attempt at a Solution

I'm just having issue coming up with a reasonable explanation. I'm not sure they all pretty much look the same. . Maybe I can say that the rate at which T changes ? So for equation a.) with one change in t you get one change in x and in y you get smaller and smaller changes in it as t increases. Which is slower then say equation b? I don't know

 

[Mark44]

Homework Statement

Compare the curves represented by the the parametric equations. How do they differ?
a.) x =t , y = t^-2
b.) x = cost , y = (sect)^2
c.) x = e^t , y = e^(-2t)

Homework Equations

So I drew them on the calculator they all look like umm... how do I describe this picture the x and y-axis ...now picture ...well just picture 1/x^2 that is what they kind look like.

The Attempt at a Solution

I'm just having issue coming up with a reasonable explanation. I'm not sure they all pretty much look the same. . Maybe I can say that the rate at which T changes ? So for equation a.) with one change in t you get one change in x and in y you get smaller and smaller changes in it as t increases. Which is slower then say equation b? I don't know

Think about the point on each graph that corresponds to various values of t, say t = 0. Also think about the orientation. As t increases, which direction does a point move along the curve?

 

[pasmith]

Homework Statement

Compare the curves represented by the the parametric equations. How do they differ?
a.) x =t , y = t^-2
b.) x = cost , y = (sect)^2
c.) x = e^t , y = e^(-2t)

Homework Equations

So I drew them on the calculator they all look like umm... how do I describe this picture the x and y-axis ...now picture ...well just picture 1/x^2 that is what they kind look like.

The Attempt at a Solution

I'm just having issue coming up with a reasonable explanation. I'm not sure they all pretty much look the same. . Maybe I can say that the rate at which T changes ? So for equation a.) with one change in t you get one change in x and in y you get smaller and smaller changes in it as t increases. Which is slower then say equation b? I don't know

Note that in (b) you have -1 \leq x(t) \leq 1 (and there's a problem with y when x(t) = 0), but in (a) and (c) x \geq 0.

 

[Jbreezy]

How does a an c differ also? Thanks for response

 

[Mark44]

What's the difference in the graphs of x = t vs. x = e^t, aside from the obvious difference in the shapes?

 

[Jbreezy]

Hmm. Well lol one is exponential? I don't know. x = e^t goes faster?

 

[Mark44]

Are the domains the same?

 

[Jbreezy]

No well x = t is neg to pos inf. and x = e^t is but x will never be 0 here

 

[Mark44]

Seems like that's an important difference between the graphs of a and c.