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B) Now move the cursor as close to x ϭ 4 on the right side of 4 as possible. What is the y coordinate of this point? (C) Change Ymax to a value greater than the y coordinate in (A) and Ymin to a value less than the y coordinate in (B). (D) Redraw the graph in the window from part (C) and discuss the result. 200 Ϫ10 10 Ϫ200 Z Figure 7 Depending on the model of graphing calculator you have, your efforts in ExploreDiscuss 3 may have produced a graph similar to Figure 7. The nearly vertical line is produced by connecting the last point on the left of x ϭ 4 with the first point on the right of x ϭ 4.

Then try different values for Xmin and Xmax until the graph looks more like a circle. Use the TRACE command to find two points on this circle with the same x coordinate and different y coordinates. (C) Is it possible to graph a single equation of the form y ϭ (expression in x) on your graphing calculator and obtain a graph that is not the graph of a function? Explain your answer. If we want the graph of a circle to actually appear to be circular, we must choose window variables so that a unit length on the x axis is the same number of pixels as a unit length on the y axis.

K(v) ϭ vϩ8 v2 ϩ 16 54. N(x) ϭ 1x Ϫ 3 xϩ2 53. M(x) ϭ 55. s(t) ϭ 2 1x ϩ 4 xϪ1 1 3 Ϫ 1t 56. r (t) ϭ 2 1 1t Ϫ 4 In Problems 57–60, find a function f that makes all three equations true. ] 57. f (1) ϭ 2(1) Ϫ 3 58. f (1) ϭ 5(1)2 Ϫ 6 f (2) ϭ 2(2) Ϫ 3 f (2) ϭ 5(2)2 Ϫ 6 f (3) ϭ 2(3) Ϫ 3 f (3) ϭ 5(3)2 Ϫ 6 59. f (1) ϭ 4(1)2 Ϫ 2(1) ϩ 9 f (2) ϭ 4(2)2 Ϫ 2(2) ϩ 9 f (3) ϭ 4(3)2 Ϫ 2(3) ϩ 9 60. f (1) ϭ Ϫ8 ϩ 5(1) Ϫ 2(1)2 f (2) ϭ Ϫ8 ϩ 5(2) Ϫ 2(2)2 f (3) ϭ Ϫ8 ϩ 5(3) Ϫ 2(3)2 61. If F(s) ϭ 3s ϩ 15, find F(2 ϩ h) Ϫ F(2) .

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Algebraic Number Theory - Papers Contributed for the Kyoto International Symposium 1976 by S. Iyanaga (Editor)


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