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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 = 2t - 1 $, $ \quad y = \dfrac{1}{2}t + 1 $

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(a) $$\begin{array}{|c|ccccc|}\hline t & -4 & -2 & 0 & 2 & 4 \\\hline x & -9 & -5 & -1 & 3 & 7 \\\hline y & -1 & 0 & 1 & 2 & 3 \\\hline\end{array}$$(b) $$\begin{array}{l}x=2 t-1 \Rightarrow 2 t=x+1 \Rightarrow t=\frac{1}{2} x+\frac{1}{2}, \text { so } \\y=\frac{1}{2} t+1=\frac{1}{2}\left(\frac{1}{2} x+\frac{1}{2}\right)+1=\frac{1}{4} x+\frac{1}{4}+1 \Rightarrow y=\frac{1}{4} x+\frac{5}{4}\end{array}$$

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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Port, a sketch a curve by using the parametric equations to plot points indicate was an error. The direction in which the curve is retraced, as t increases for part a first t equal to negative 2. We have x equal to negative 5 and y is equal to 0. When t is equal to 0, we have x equal to negative 1 y equal to y e t is equal to 2 x, equal to 3 y equal to 2 and t equal to 4 x, equal to 7 y equal to 3 point now. Let'S sketch the curve is a negative 50 here, negative 1132 and 73 point says: a curve is like this, and its arrow is a direction curve is traced, as t increases, part b eliminated parameter to find a cartesian equation over the curve. So from here we have t is equal to 2 y minus 2 point and then part in this equation. We have x is equal to 4 y minus 5, but this is a cartesian equation.

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