Points, Vectors, and Functions Exercises

Answer

Start out with a table of values, to have an idea of what's going on:

Plot and connect the points we have:

From t = 0 to t = π we have a quarter circle. That means it will take until t = 4π to find the complete circle:

As t goes from 4π to 8π, the etch-a-sketch tip will go around the circle again.

Answer

Tables, then graphs.

• We start out with a partial table.

We don't have to find many values of this before recognizing it's the graph of y = cos x.

There doesn't seem to be any reason to have this function in parametric form when we could write y = cos x instead.

Answer

• No bounds for t were given, so we'll start with a table using some basic values of t.

When we graph and connect these points, we have a line:

From graphing this much we can see how the rest of the graph behaves. Therefore, we didn't need any bounds on t.

Answer

• Here's a partial table:

Graph these points:

As t goes out to π we'll end up completing a semicircle and landing at the point (-1, 0).

As t goes down to -π we'll maneuver through the fourth and third quadrants and once again end up landing at (-1, 0).

So, as t goes from -π to π our function completes a clockwise tour of the unit circle, starting at (-1, 0).

Answer

Gillian switched the x and y components. Her graph shows the function <g(t), f(t)>. As an example, look at whathappens when t = 2. Her graph has the point (-1, 1) but it should have the point(f(2), g(2)) = (1, -1).

Answer

We'll show the table of values, as well as the final graph.

Answer

As always, we start with a table, then move onto the graph.

Answer

This should sound like a broken record by now. Start with a table, then move onto a graph.