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

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