Given two curves, find parametric curve

Question

I am given two graphs x versus t and y versus t and I have to determine the parametric curve.

The two graphs I am given:

Parametric curve (that is the right answer):

So the solutions say that: from the first graph, the values of x cycle through the values from -2 to 2 twice. From the second graph, the values of y do the same thing. Therefore this graph (what I have shown above) satisfies these conditions.

So I am having trouble understanding what this solution means. What does cycle mean? Is there another solution that is plausible in this question?

Answer 1

Parametric curves work on the principle that for each $t$, there are values $x(t)$ and $y(t)$ which correspond to a point on the plane. By plotting these point for all $t$, we get a curve. You got a cycle because $x(-2)=x(2)$ and $y(-2)=y(2)$. Given functions $x(t)$ and $y(t)$, there is a unique parametric curve given by $c(t)=(x(t),y(t))$.

Answer 2

From the graph of $x(t)$ we see that, for $t \in [-2,2]$ The coordinate $x$ assume all values between $-2$ and $2$ two times ( ''cycle twice''). And the same is true for $y(t)$. (Note that from the graphs it seems that $x(-2)=x(2)$ and $y(-2)0y(2)$.

If we plot the points of coordinates $(x(t),y(t))$ we obtain the parametric closed curve $c(t)$ in your third graph.