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(a) Sketch the curve by using the parametric equations to plot points. Indicat with an arrow the direction in which the curve is traced as $ t $ increases.(b) Eliminate the parameter to find a Cartesian equation of the curve.

$ x = 3t + 2 $, $ \quad y = 2t + 3 $

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(a) Graph(b) $y=\frac{2}{3} x+\frac{5}{3}$

02:44

Wen Zheng

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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16:57

In mathematics, a graph is a representation of a set of objects where some pairs of the objects are in some sense "related". The objects correspond to mathematical abstractions called "vertices" or "nodes", and the relations between them are represented by mathematical abstractions called "edges" or "arcs". The basic notion of a graph was developed by the 17th-century French mathematician Pierre de Fermat, and the term "graph" was coined by the 19th-century mathematician James Joseph Sylvester. The more general mathematical concept of a graph "in which any kind of relation between elements of the set is expressed as an edge, is called a network" (Kolmogorov, "1956, p. 111"). In other words, an undirected graph is a graph in which the edges have no direction associated with them. The most familiar examples of graphs are the graphs of equations. In general, the vertices of a graph can represent concepts and the edges can represent real-valued functions on the concepts, so one can speak of the graph as a function's graph or of the edge as a function's edge.

01:59

Polar coordinates are a two-dimensional coordinate system that specifies a point in terms of distance from a reference direction (the pole) and angle from a reference direction (the polar axis).

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(a) Sketch the curve by us…

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02:56

02:53

03:32

2. (a) Sketch the curve by…

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06:37

Consider the parametric eq…

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01:57

01:22

So they first want us to go ahead and grab this. Um, so let's just go ahead and plug in some points, so they kind of get a nice sketch of it. Let's do so I'll make a chart here, So let's do like, I don't know, negative to negative 101 and two. So we're just going to replace these into these equations, and then the output should become our coordinate points. So if I plug in negative to, uh, that would get negative six plus four, So that would be negative. Four negative one is going to be just negative. One zero is too. Uh, one is five and then two is eight. And then over here, if we plug into why shall start with zero, because I'll probably the easiest to do so would be three. And then we're going to go up by two and down by two. So this would be one negative one. And then over here, this is going to be five and then seven. So this first point is going to be negative. Four negative one. So 1234 a negative one, and then we have negative 11 and then at zero, we have 212123 and then we have five I So 1234512345 And then lastly, we have 87 So we need to go three to the right. 123 and then two up. So you can see how this looks like. It forms a pretty straight line. So we just go ahead and connect all the bar points like this. And then the other thing they wanted us to do was to indicate direction. So remember, this is t is equal to negative two. This over here is T 0 to 0. T is equal to two, so you can see that it increases in this direction. Or I should say the direction goes up to the right like that There. Okay, Now, what they wanted us to do after that was to get this from Parametric into a Cartesian equation. So what I'm going to first do is solve for y uh sorry. Solve for X in terms of t. And then just plug that in. So xz 23 t plus two. So we subtract two over divided by three seconds expires to over three. Is he good? T Now we're going to take y is equal to two t plus three. Take that, plug it in here. And that's going to give two times X minus two or three plus three. Uh, so we can go ahead and distribute that three and the two. So that would give us two thirds. Two thirds X minus four, third plus three. And then, uh, three minus four thirds should give us five thirds, So this is going to be two thirds X plus five third. So this is going to be for part B. So that is our equation in the Cartesian form. And if we were to kind of compare it with what we have are here, um, five thirds, Yeah, kind of looks like we're passing through it because it's like, right below two. And then all of the points that we got from our table over here, the slopes for all of these would follow that two thirds, so that looks pretty good

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