In this video I go over another example on parametric curves as well as discussing how we can write any function as a pair of parametric equations, which we can then graph out. The example I go over is graphing the function x = y4 - 3y2 which although can be plotted normally, I rewrite it as a pair of parametric equations by setting t = y. Thus the equations become x = t4 - 3t2 and y = t. This makes it possible to track a particle as it follows the path of the curve.
In general any function can be written as a pair of differential equations by first setting a parameter t to equal either x or y. Parametric equations makes it possible to graph very complicated, and near impossible to manually sketch, curves. I go over a few of them and show just how amazing some of the curves that can generated. I also use the Desmos online graphing calculator to illustrate how to graph parametric curves and to trace how the particle moves across them.
This is an incredibly cool video on parametric videos, and the amazing graphs that can be produced has been the driving force for me to pursue deeper into mathematics. This is one of my favorite topics, so make sure to watch this video!
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Graphing Devices
Most graphing calculators and computer graphing programs can be used to graph curves defined by parametric equations.
In fact, it's instructive to watch a parametric curve being drawn by a graphing calculator because the points are plotted in order as the corresponding parameter values increase.
Example: Use a graphing device to graph the curve: x = y4 - 3y2.
Solution:
We can graph this function as is: https://www.desmos.com/calculator/sawqzjrrby
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But we can write this same equation in the form of a pair of parametric equations:
The resulting curve sketched by the parametric equations is the exact same as before: https://www.desmos.com/calculator/zjfzvgxghb
Note that we can play around with the t values and see how the points are plotted.
We can also show a point on the graph and follow the path it takes as t changes.
In general, if we need to graph an equation of the form x = g(y), we can use the parametric equations:
Notice also that curves with equations y = f(x) (the ones we are most familiar with, i.e. graphs of functions) can also be regarded as curves with the following parametric equations.
Graphing devices are particularly useful when sketching complicated curves.
For instance, the curves below would be virtually impossible to produce by hand.
https://www.desmos.com/calculator/amibzzb7qj
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https://www.desmos.com/calculator/pdvwmmlna2
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One of the most important uses of parametric curves is in computer-aided design (CAD).
In later videos I will go over special parametric curves, called Bézier curves, that are used extensively in manufacturing especially in the automotive industry.
These curves are also employed in specifying the shapes of letters and other symbols in laser printers.
So stay tuned!
very cool post thank you!