By A. C. Burdette (Auth.)
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Additional info for An Introduction to Analytic Geometry and Calculus
3-5. GRAPHING EQUATIONS 29 Fig. 3-4 2 2 Example 3-10. Graph x - y = 4. (a) Intercepts: If x = 0, y = ±2i. y-axis. This is consistent with (c) below, where it is seen that x = 0 is in the excluded portion of the plane. lfy = 09x= ±2. (b) Symmetry: Theorems 3-1, 3-2, 3-3 all indicate symmetry. Hence this curve is symmetric to both axes and the origin.! (c) Extent: From Example 3-8, |JC| < 2 is excluded. (d) Additional points: The results of (a), (b), (c) make it possible to sketch the required graph from very few additional points.
If we were to represent the function/ discussed above, in a similar manner, the input bin would contain only the numbers —3, —2, —1, 0, 1, 2, 3. We shall deal exclusively with functions for which the "rule" is expressible as one or more equations. In this case we may not always state the domain. However, the student should not lose sight of the fact that the domain is an essential part of the definition and is, when not restricted otherwise, implied by the rule. Thus, if we speak of the function g defined by g(x) = V 1 - *> the domain x ^ 1 is implied automatically because these are the only values of x for which the rule gives real values for g(x).
Solving for x, we obtain x= ±Jl5-y2, and by precisely the same reasoning we conclude that the curve is bounded by the lines y = ± 5 . Thus the curve is contained in a square ten units on a side, the sides being parallel to the coordinate axes and symmetric to them. Example 3-8. Discuss the extent of x2 — y2 = 4. We solve for x and obtain x = ±yjy 2 + 4, 28 3. NONLINEAR EQUATIONS AND GRAPHS Since y2 + 4 is positive for all real y, this imposes no restriction on y. Hence the curve is infinite in extent in the j-direction.
An Introduction to Analytic Geometry and Calculus by A. C. Burdette (Auth.)