Sketching Vector Functions Examples

Sketch the vector function f(t) = <t², t³> 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.

Sketch the vector function f(t) = <t², t³> 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:

Sketch the vector function f(t) = <t², t³>.

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.