Parametric Equations

By: Lacy Gainey

Graph the parametric equations

I would recommend using graphing calculator software (such as Graphing Calculator 3.4 or 4), so you can more easily observe the parametric curves.

Notice that the parametric equations produced a circle with a radius of 1.

Up until this point, students may have only been exposed to this equation of a circle: [1 GSP], where the circle is centered at the origin and r is the radius of the circle. 

• How did we get from to ?
  We can use our knowledge of right triangles and trigonometry to derive the parametric equations .
  Observe the diagram below.

In a right triangle, recall that .  In the diagram, we are letting t = θ. t is our parameter.

Observe,

So point P in the diagram is equal to (r cos(t), r sin(t))

 

• Graph the parametric equations , using different values for r and observe what happens.  Use the same value for r in both equations.
  I graphed , for 0 ≤ r ≤ 20.
  

We can see that as r increases, the diameter of the circle increases. As r approaches 0, the diameter of the circle decreases.

• What do you think will happen if we use different values of r for each of the parametric equations?
  

• I chose to graph

and

So when we use different values for each of the parametric equations, it seems that an ellipse is formed.

For further exploration graph , for different values of a and b and observe what happens.

 

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