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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 some functions have more than one input, like
$f(x,y) = x^2 + y^2$, while
Every ordinary function with one input and one output is associated
with a
However, some curves can't be written as graphs, since they
don't pass the vertical line test. This simplest example is a circle.
Instead, we usually describe the unit circle by an equation:
$x^2+y^2=1$. This is an example of a
A 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 Every graph can be written as a parametrized curve. For instance, we can just take $x(t)=t$, $y(t)=f(t)$, moving from left to right at constant horizontal speed, and letting the graph do the rest. However, that's not the only way to write the graph as a parametrized curve. The parametrization $x(t)=2t$, $y(t)=f(2t)$ would work just as well. It would trace out the exact same set of points, only twice as fast. The unit circle is usually parametrized as $x(t)=\cos(t)$ and $y(t)=\sin(t)$. This traces the circle counter-clockwise, starting at $(1,0)$, going around once every $2\pi$ seconds. However, we could also start at a different point, go clockwise instead of counterclockwise, or go at a different speed. You should check that all of the following parametrizations also satisfy $x^2+y^2=1$, and so follow the unit circle: $$ x(t) = \sin(t); \qquad y(t)=\cos(t). $$ $$ x(t) = \cos(t); \qquad y(t)=-\sin(t). $$ $$ x(t) = \cos(2t); \qquad y(t)=\sin(2t). $$ The first starts at (0,1) and goes clockwise with period $2\pi$. The second starts at (1,0) and goes clockwise with period $2\pi$. The third starts at (1,0) and goes counter-clockwise with period $\pi$. They're all different parametrizations, but they all trace the same curve.The following video describes the three kinds of functions, and the three kinds of curves: |