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By A. C. Burdette (Auth.)
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Additional resources for Analytic Geometry
DEFINITION 3-4. \x\ = x, \x\ = —x, x > 0; x < 0. This merely says \x\ is always positive or zero and equal to the numerical value of x. Thus, if x = 2, \x\ = 2; if x = — 2, \x\ = 2 also. The results in the preceding examples may be expressed in terms of excluded values of x and y. Thus, in Example 3-6 we could say there are no excluded values of y but values of x > 4 are excluded. Example 3-7 has |JC| > 5 and \y\ > 5 excluded; Example 3-8 has |JC| < 2 excluded. This is often the simplest method of describing the extent of a curve.
This is consistent with (c) below, where it is seen that x = 0 is in the excluded portion of the plane. lfy = 0,x= ± 2 . t In order to emphasize the role of excluded values, that portion of the plane from which the curve is excluded is shaded. 3-6. GRAPHING EQUATIONS 57 (-5,3) Figure 3-4 (b) Symmetry: Theorems 3-1-3-3 all indicate symmetry. f (c) Extent: From Example 3-8, |JC| < 2 is excluded. (d) Horizontal and vertical asymptotes: Section 3-5 gives no horizontal or vertical asymptotes. (e) Additional points: The results of (a)-(d) make it possible to sketch the required graph from very few additional points.
When this is impossible, or impractical, there are other means of discovering horizontal and vertical asymptotes when they exist. However, in this brief treatment, we shall not discuss these methods. If we can find the asymptotes easily, we will make use of them; if not, we will get along without them, perhaps by plotting more points. 3-6. Graphing Equations If we combine the general remarks of Section 2-2 with the special results of Sections 3-2-3-5, we have a fairly sound working basis for drawing the graphs of many equations.
Analytic Geometry by A. C. Burdette (Auth.)