INSTRUCTOR: Create a graph of the vector-valued function r of t equals parentheses t squared minus 1 close parentheses I plus parentheses 2t minus 3 close parentheses j for 0 less than or equal to t less than or equal to 3.

In order to create a graph of this, we need to find component values at the values of t0 to 3. So we're going to use some discrete values-- t equals 0, t equals 1, t equals 2, and t equals 3. And I'm going to create a table with three columns. First column is t second column is the i component value. And the third column is the j component value.

So we're going to evaluate the function t squared minus 1 for t equals 0, 1, 2, and 3. Evaluating this for t equals 0 gives us a value of negative 1. Evaluating t squared minus 1 for t equals 1 gives us a 0. Evaluating the same function for t equals 2 gives us 3. And t squared minus 1 when t is equal to 3 gives us a value of 8.

To find the j component values, we'll take 2t minus 3 and evaluate that at 0, 1, 2, and 3. 2t minus 1 when t equals 0 is negative 3. 2t minus 3 when t equals 1 is negative 1. 2t minus 3 when t equals 2 is positive 1. And 2t minus 3 when t is equal to 3 is equal to 3.

So we're going to plot these points, the i and j components. Our points are negative 1, negative 3. Putting that on the coordinate axis. Next, we have the i, j pair of 0, negative 1. So the point 0, negative 1. The pair 3, 1-- plot that point. And the point 8, 3.

Now we can see a shape that appears from this. And beginning at the point negative 1, negative 3-- because that's when t was equal to 0-- continuing on to the point 8, 3, we can connect these dots, these plotted points. And this is a directional graph because t is increasing. And we'll draw an arrow going from the point negative 1, negative 3 to the point 8, 3.