Graph the function defined by the parametric equations , and .
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.
Example 2
Graph the function described by the given parametric equations.
x(t) = t, y(t) = cos t for -π≤ t ≤ π
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.
Example 3
Exercises. Graph the function described by the given parametric equations.
x(t) = 4 + 2t, y(t) = 1 + t
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.
Example 4
Exercises. Graph the function described by the given parametric equations.
x(t) = cos t, y(t) = sin t for -π ≤ t ≤ π
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).
Example 5
Gillian was asked to graph the function <f(t), g(t)> with f(t) and g(t) given by the following graphs:
She came up with this:
What did Gillian do wrong?
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).
Example 6
Graph the function < f(t), g(t)> where x = f(t) and y = g(t) are given by the graphs.
Answer
We'll show the table of values, as well as the final graph.
Example 7
Graph the function < f(t), g(t)> where x = f(t) and y = g(t) are given by the graphs.
Answer
As always, we start with a table, then move onto the graph.
Example 8
Graph the function < f(t), g(t)> where x = f(t) and y = g(t) are given by the graphs.
Answer
This should sound like a broken record by now. Start with a table, then move onto a graph.
, and 
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