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# Loci in the Plane Help

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## 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.

#### 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 .

Fig. 1.19

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 .

Fig. 1.21

Fig. 1.22

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

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