3 kinds of functions; 3 kinds of curves

For most of calculus, the word ``function'' has meant a machine that inputs a number and outputs another number. Something like $f(x)=x^2$. But a function can have more than one input or more than one output. Each kind of function gives a different way to describe a curve in the plane.

Every ordinary function with one input and one output is associated with a graph. We usually call the input variable $x$ and the output $y$, and draw the curve $y=f(x)$. If $f(x)=x^2$, this gives us a parabola. Geometric properties of the curve, like its slope or the area under it, are then related to properties of the function, like its derivative and definite integral.

Some curves don't pass the vertical line test and can't be written as graphs. We usually describe the unit circle by an equation: $x^2+y^2=1$. This is an example of a level set. We define a function of two variables (in this case $f(x,y)=x^2+y^2$) and look at all of the points where $f(x,y)=1$. If we look for different values of $f$ we get different curves. (E.g. $f(x,y)=4$ is a circle of radius 2 instead of 1.)

The third way to get a curve is to track the trajectory of a moving point. If we specify both $x$ and $y$ as functions of a parameter $t$ (time), then we get more than a geometric set. We get a description of how the curve is being traced. This is called a parametrized curve or a parametric curve. (The terms are interchangable.)

Every graph can be written as a parametrized curve. Just take $x(t)=t$, $y(t)=f(t)$. We move from left to right at constant horizontal speed, and let the graph do the rest.

The unit circle is a parametrized curve with $x(t)=\cos(t)$ and $y(t)=\sin(t)$. This traces the circle counter-clockwise, starting at $(1,0)$, going around once every $2\pi$ seconds.