Sketch the vector function
f(t) = <t^{2}, t^{3}>for 0≤ t ≤ 5.

We can start by making a table of values for x and y, going from t = 0 to t = 5:

When we plot these points, we can see that they're lying along a curve:

We connect the dots with that curve:

and there's our graph.

Example 2

Sketch the vector function
f(t) = <t^{2}, t^{3}>
for -5≤ t ≤ 5.

We'll still have the part of the graph we found in the previous example, but now we need to figure out what happens for -5≤ t ≤ 0. Make another table of values:

Plotting these points and connecting them with a curve, we see that the graph of f(t) for -5≤ t ≤ 5 looks like this:

Example 3

Sketch the vector function
f(t) = <t^{2}, t^{3}>.

Since no bounds for t were given, this vector function is like the one in the previous exercise, but it keeps going. We can indicate this by drawing arrows on the
ends of the graph.