Parametric

A curve can be written in normal equation involving x and y (in 2D), such as y = ln (x) + 2. However, the curve can be described in parametric equations as well. A pair or a set of parametric equations may look like the equations below:

	System	Normal Equation	Parametric Equations
1.	2D	x = 2
2.	2D
3.	2D
4.	3D

Let's see the process of graphing the parametric equations:

1. 
   If we draw x = 2, we will just draw a vertical line that passes through the point (2,0) regardless with the direction. Somehow, in this parametric equations, we find that the line was built from bottom up as t increases. Click the 'play' button at the bottom left corner to see the forming of this line.
   
   



2. For the second equation , normally we would just draw the ellipse from any starting point regardless of the orientation. In parametric equations , the starting point is fixed relying on the given range of t, same with its orientation. In the graph below, click the 'play' button at the bottom left corner of the graph and we can see the ellipse of this parametric equations starts from the point (x,y) = (2,0) and moving counterclockwise.
   
   



3. For and its parametric equations , the graph of the paramteric equations is drawn as how we draw for the normal equation. Note that the set of parametric equations does not always use the variable t. Using other variables are fine as well.
   
   



4. For a 3D surface, , its parametric equations can be written in one or two variables, depends. In this example, I give a set of parametric equations of x, y and z to be written in two variables s and t, .
   
   

Can you try to build one set of parametric equations from the set of equations {z = ln(y), x = 3} and figure out the graph of the set of equations?


Betty, Created with GeoGebra