A parametric curve in the plane is a pair of functions, called the parametric equations of a curve,

where the two continuous functions define ordered pairs (x, y). The extent of the curve will depend on the range of t.

Here is an example of the parametric curve defined by

Problem 3.

For various **a** and **b**, investigate

First, let's look at the graph when **a**
=1 and **b** = 1.

x = cos (t)

y = sin (t)

The graph is a circle with a radius of 1.

Next, let's examine what happens when
we set **a** =1 and **b** = -5.

x = cos (t)

y = -5 sin (t)

Now we have an ellipse instead of a circle.
Looking at this graph, it appears that the **b** value alters
the vertical component of the graph and that the ellipse crosses
the y-axis at the positive and negative **b** values.

Let's look at a few more values of **b**
to confirm this speculation.

x = cos (t)

y = -3 sin (t)

x = cos (t)

y = -sin (t)

x = cos (t)

y = 3 sin(t)

Looking at those graphs confirms that
the **b** value alters the vertical component of the graph
and that the graph crosses the y-axis at the positive and negative
**b **values. Another interesting thing noticed is that the
graph is the same for a positive and negative values of **b**.

Next, let's set **b **=1 and observe
what happens when we change the value of **a**. First, let's
set **a** = -5.

x = -5 cos (t)

y = sin (t)

Once again we have an ellipse. Looking
at this graph, it appears that the **a** value alters the horizontal
component of the graph and that the ellipse crosses the x-axis
at the positive and negative **a** values.

Let's look at a few more values of **a**
to confirm this speculation.

x = -3 cos (t)

y = sin (t)

x = - cos (t)

y = sin (t)

x = 3 cos (t)

y = sin (t)

Looking at those graphs confirms that
the **a** value alters the horizontal component of the graph
and that the graph crosses the x-axis at the positive and negative
**a** values. We also notice that the graph is the same for
a positive and negative values of **a**.

Now that we know the **a** value alters
the horizontal component of the graph and the **b** value alters
the vertical component of the graph, let's change both values
at the same time and see what happens.

First, let's set **a** and **b**
to the same value.

x = 3 cos (t)

y = 3 sin (t)

x = 5 cos (t)

y = 5 sin (t)

We notice from these graphs that if**
a** and **b** are the same value then the graph is a circle
with a radius equal to the value of **a** and **b**.

Next, let's examine what happens when
we set **a** = 0 and change the value of **b**.

x = 0 cos (t) = 0

y = 5 sin (t)

x = 0 cos 9t) = 0

y = -2 sin (t)

Looking at these graphs we see that when
**a** = 0 the graph becomes a vertical line with length 2**b**.

Next, let's examine what happens when
we set **b** = 0 and change the value of **a**.

x = 5 cos (t)

y = 0 sin (t) = 0

x = -3 cos (t)

y = 0 sin (t) = 0

Looking at these graphs we notice that
when **b** = 0 the graph collapses to a horizontal line with
length 2**a**.