I can't figure out how to graph this! Calculus?

Question

I don't know how they are coming up with the table of values for graphs like this. "r(t)=" or "r(t)=". My book doesn't explain. It just says "We can make a able of values, or we can eliminate the parameter...." I just can't visualize this stuff. Please help, I'm lost!

But I have to be able to sketch the graph and I won't have a calculator to help me. that's why I want to figure out the coordinates. Yes they are two separate problems.

Answer 1

q1
x=y^4+1, a U, or rather C-like curve on right side of graph
most left point (1,0)
others (17,+/-2), (82,+/-3)

q2
x=t^3, y=t^2
x^2=y^3, a U-like curve on upper side of graph
lowest point (0,0)
others (+/-1,1), (+/-8,4)

Answer 2

As near as I can tell you have two separate questions. The first is:

r(t) =

This is the vector form of parmetric equations.

r(t)
x = t^4 + 1
y = t

Substituting we have:

x = y^4 + 1
___________

The second one works the same way.