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We have actually been employing the idea of parametric functions all along but have not named it. We called it, "Keeping track of two quantities' values simultaneously." We did that in the Cities A and B activity, where we kept track of a car's distances from City A and from City B simultaneously.
We also employed the idea of paramatric functions in every graphing activity in which we kept track simultaneously of a variable's value and the value of a function defined in terms of that variable's value. We did this when we
The graphic below shows the graphs of sin(2x) and cos(x) on the same coordinate system. As an exercise (in the spirit of City A and City B), do this:
Functions are typically defined as a relationship between two variables, such as y =f(x) where f(x) = 3x2-2x+1. We then graph the function by plotting all points that have the coordinates (x,f(x)).
Another approach is to defining a function is to think of making the coordinate's of a point on a function's graph themselves to be functions of a common variable. This is often done in physics, for example, when you are interested in tracking an object's position and its velocity simultaneously. In that case, a point (x,y) actually has coordinates
(position at time t, velocity at time t).
Thus, in this example, the coordinates of the function's graph are each a function of time. We say that we are defining a function parametrically when the function is defined by making the coordinates of points on its graph themselves functions of a common variable.
Predict the graph of 
. Check yourself by clicking here.