Loci in the Plane Help

Example 1

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

Solution 1

It is convenient to make a table of values:

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:

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