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Investigate the family of curves defined by the parametric equations $ x = t^2 $, $ y = t^3 - ct $. How does the shape change as $ c $ increases? Illustrate by graphing several members of the family.

$x=t^{2}, y=t^{3}-c t .$ We use a graphing device to produce the graphs for various values of $c$ with $-\pi \leq t \leq \pi$. Note that allthe members of the family are symmetric about the $x$ -axis. For $c<0$, the graph does not cross itself, but for $c=0$ it has acusp at (0,0) and for $c>0$ the graph crosses itself at $x=c,$ so the loop grows larger as $c$ increases.

Calculus 2 / BC

Chapter 10

Parametric Equations and Polar Coordinates

Section 1

Curves Defined by Parametric Equations

Parametric Equations

Polar Coordinates

Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

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the problem is investigate the family of curves you find about the parametric equations actually boarded his square. Why go to this Cuban minus city? How does the shape change? As said increases illustrated by graphing several members of the family? Little Cat's a graph here. This curve is sick or too negative one. And this one this they could. Cyril, this is safe. What? You three. This is sick with selling so we can see that one sees last and cereal. The function is an increasing function I see is the greatest and Cyril, the functions of first is secreting on, Then increasing and I see increases. We can see that the local minimum while you is getting smaller and smaller.

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