**Exploring Parametric curves**

**by**

**Julie Anne Laycock**

For various values of **a** and **b** investigate the equations:

In this animation, I have designated **a** and **b** to change from 0 to 7. We can see that as **a** and **b** are the same value and as they increase from 0 to 7 a circle is being formed.
In this case the circle is formed starting at 1 on the x-axis and moving counterclockwise crossing the x-axis again at -1 and the y-axis at 1 and -1.The circle becomes a complete circle when** a** and **b **somewhere between 6 and 7. If we continue to have **a** and **b** the same and increase their value, the circle will continue to follow the counterclockwise path and overlap.

In this animation, I have designated **a** and** b** to change from -7 to 0. We can see that as **a** and **b **are the same value and as they increase from -7 to 0 a circle is being formed. In this case the circle is formed starting at 1 on the x-axis and moving clockwise crossing the x-axis again at -1 and the y-axis at 1 and -1. The circle becomes a complete circle when** a** and **b **somewhere between -6 and -7. If we continue to have **a** and **b** the same and decrease their value, the circle will continue to follow the clockwise path and overlap.

Now lets look at what happens when **a** and **b** are different. In this case I have **a** at 7 and **b** at 8.

When **a** and **b** were the same value greater than 7 we would expect a circle but when we change the value of **b** to 8 we can see the graph has changed. The graph looks like it starts to form a circle but moves towards forming an ellipse.

Now lets look at what happens when we keep **a** at 7 and change** b** from 0 to 100.

As we can see the graph makes a curve that runs horizontally on the x-axis from -1 to 1 and the curve also doesn't go higher or lower than 1 or -1 on the y-axis. It appears that the curve will continue making the same curve but would move closer and closer together.

Now lets look what happens when we keep **b** at 7 and change and change **a** from 0 to 100.

As we can see the graph makes a curve that runs vertically on the y-axis from -1 to 1 and the curve also doesn't go left or right than 1 or -1 on the x-axis. This curve looks the same as the one above only now its running on the y-axis versus the x-axis.

Taking a closer look at the graphs we can make a couple of interesting observations. We can see when **b** is double the value of **a** we will get the graph on the left every time. Also, when **b** is half the value of **a** we will see a graph like the one on the right.