Parametric equations are powerful tools to model projectile motions and to graph things that are not functions like circle or ellipses. The x and y coordinates are defined as two separate functions with a common independent variable often labelled “t”.
We often see Lissajous figures in old sci-fi movies because they are so cool. As you play with them I think you will find them as fascinating as I do. They created those figures by graphing points as a function of a third variable (the parameter). We can do the same and in the process come to understand Parametric Functions.
George is in New York and Martha is in Washington. They leave at the same time and follow the same road to meet each other on the way. The distance between New York and Washington is 229 miles. George has a fast horse and averages 16 miles/hr. Martha has a slow carriage and averages 7 miles/hr. How far will George have gone when they meet?
Spreadsheets make it very easy to switch axes and add graphs. They enable students to play with what may have been difficult and abstract concepts like the inverse of a function. You may want to approach the inverse of a function by challenging students to fill in a table of values with a rule that creates a mirror of that function. So you can approach the inverse of a function either as the interchange of axes or as a symmetry issue. Either one works well on spreadsheets.
Most museums with math exhibits have a Pascal’s triangle made up of pegs with balls falling down between them and bouncing off of them. One of the things they want to show is probability and the Normal or Bell curve produced by these balls as they fall down most of us are familiar with. This is the curve produced if we flip a honest coin a large number of times and ask what are the chances of getting all heads, of all heads but one and one tail, of getting all but 2 heads etc. We ask what does a Normal distribution look like and why does this extremely simple pattern produce it?