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$19-21=$ Use the graphs of $x=f(t)$ and $y=g(t)$ to sketch theparametric curve $x=f(t), y=g(t) .$ Indicate with arrows thedirection in which the curve is traced as $t$ increases.
At $t=0,(x, y)=(0,1)$ asd, as $t$ increases from 0 to $1, y$ decreases from 1 to 0 and $x$ is positive. At $t=1,(x, y)=(0,0)$ again. so the loop is completed. For $t>1, x$ and $y$ both become large negative. This enables us to draw a rough sketch. We could achieve greater accuracy by estimating $x$ and $y$ values for selected values of $t$ from the given graphs and plotting the corresponding points.
Calculus 2 / BC
PARAMETRIC EQUATIONS AND POLAR COORDINATES
Harvey Mudd College
University of Michigan - Ann Arbor
University of Nottingham
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we're going to use the graphs of the functions X as a function of T and wise, a bunch empty to graft the Parametric graph of X and y. So first, we're going to find a view values of T, where we can get values of exit like pretty usually from the graphs. And we can do that. It's he is equal to negative 10 and one on both graphs. So when T is equal to negative, one X is equal to zero. When T is equal to zero, X is equal zero and when he is equal to one, X equals zero on the graph of wise amongst in empty, when T is equal to negative one y 00 when t is equal to zero, why is equal to one? And when T is equal to one? Why is equal to zero? So we'll use these points. A graph points on the Parametric graph. When's he is equal to negative one action while you're both equal to zero when C is equal, zero X is equal to zero and why is equal to one? And when t is equal to one, X is equal to zero and Y is equal to zero. So that's actually only two distant points. Me, I'm gonna have some kind of loop around here. So now we're gonna look at the shape of these two graphs to get the shape of the Parametric graph. So when t is less than negative, one X is having a sharp decrease from being positive to being zero. So we're gonna have something that's pretty quickly moving this direction on this graph and on the graph of y is a function T. We're having a sharp but not quite a sharp possess increase to zero. So we're going to have something that looks like this with ex decreasing more quickly than why decreasing less quickly. Excuse me? That should be more like this. Okay, Now, when t is between negative one and zero, ex briefly becomes negative, but circles back pretty quickly and why pretty steadily increases from 0 to 1. So it's going to look something like this when tea is from 0 to 1. T excuse me. Ex briefly becomes positive again. Then much the same way became negative. So we're gonna have something similar there. And why steadily decrease is back to being zero again. Similarly, the other side, this is pretty symmetric and lastly asked. He increases from one onwards to Infinity X has a pretty sharp and decrease downwards. So it's going to decrease pretty quickly and why has a less sharp but still sharp decrease? So it's going to look like this somewhat and asked, He is increasing. We're moving this direction along the graph and there we go. This is the graph of X and y as parametric equations.
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