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Use the graphs of $ x = f(t) $ and $ y = g(t) $ to sketch the parametric curve $ x = f(t) $, $ y = g(t) $. Indicate with arrows the direction in which the cuve is traced as $ t $ increases.

When $t=-1,(x, y)=(1,1) .$ As $t$ increases to $0, x$ and $y$ both decrease to 0As $t$ increases from 0 to $1, x$ increases from 0 to 1 and $y$ decreases from 0 to$-1 .$ As $t$ increases beyond $1, x$ continues to increase and $y$ continues todecrease. For $t<-1, x$ and $y$ are both positive and decreasing. We couldachieve greater accuracy by estimating $x$ - and $y$ -values for selected values of $t$from the given graphs and plotting the corresponding points.

01:28

Wen Z.

Calculus 2 / BC

Chapter 10

Parametric Equations and Polar Coordinates

Section 1

Curves Defined by Parametric Equations

Parametric Equations

Polar Coordinates

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So if we want to sketch this graph in the Cartesian plane, let's first go ahead and maybe just graph a couple of these points. That might be easy for us to find Well, over here. Um, so they give us 11 so we can estimate negative one is probably around here. Negative one is here, so let's just make a little chart. So when t is equal to negative one, it looks like First X is equal to one. And why is going to be equal to one also? Okay, um, when tea is able to zero, it looks like ax zero. It also looks like Why is zero And then when t is equal to one X is one again, and then it looks like why is equal to, uh, negative one? So let's come over here now and just graph those points that we have X Y, uh, I'll just put all those are just gonna be one for the tick marks. So at negative one, we have X is equal to one. Why is it good to one? So that would be right here, and this is T is equal to negative one. Now, when t 00 both X and wire zero. Alright. T is zero. And then when t is he going to one Access one again and then y is negative one. So this is t is he took one. Now we can kind of see So X is going to grow a lot quicker than why it's so when we're going through this or actually I should say access first, going to be decreasing to the left and why is always going to be decreasing So we should know that why should always be going down and then X was going to decrease up until we hit zero and they will start increasing again. So that means we can just go ahead and connect these points and it will kind of look like this quadratic here now for us to figure out what the direction is, we're just going to follow the points. So it goes from negative 1 to 0 and then 0 to 1. So this here is going to be a sketch of what this would look like in the XY plane

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