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Loci in the Plane Help

By — McGraw-Hill Professional
Updated on Aug 31, 2011

Introduction to Loci in the Plane

The most interesting sets of points to graph are collections of points that are defined by an equation. We call such a graph the locus of the equation. We cannot give all the theory of loci here, but instead consider a few examples. See [SCH2] for more on this matter.

Examples

Example 1

Sketch the graph of {( x , y ): y = x 2 }.

Solution 1

It is convenient to make a table of values:

x

y = x 2

−3

9

−2

4

−1

1

0

0

1

1

2

4

3

9

We plot these points on a single set of axes ( Fig. 1.19 ). Supposing that the curve we seek to draw is a smooth interpolation of these points (calculus will later show us that this supposition is correct), we find that our curve is as shown in Fig. 1.20 . This curve is called a parabola .

Basics 1.6 Loci in the Plane

Fig. 1.19

Basics 1.6 Loci in the Plane

Fig. 1.20

Example 2

Sketch the graph of the curve {( x , y ): y = x 3 }.

Solution 2

It is convenient to make a table of values:

x

y = x 3

−3

−27

−2

−8

−1

−1

0

0

1

1

2

8

3

27

We plot these points on a single set of axes (Fig. 1.21). Supposing that the curve we seek to draw is a smooth interpolation of these points (calculus will later show us that this supposition is correct), we find that our curve is as shown in Fig. 1.22. This curve is called a cubic .

Basics 1.6 Loci in the Plane

Fig. 1.21

Basics 1.6 Loci in the Plane

Fig. 1.22

You Try It: Sketch the graph of the locus ∣ x ∣ = ∣ y ∣.

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