By H. Piaggio
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Additional resources for An Elementary Treatise on Diff. Eqns. and Their Applns.
Sample text
Thus we find that the graph of y = (x - 1)2 is the graph of y = x2 shifted one unit to the right. Similarly, in (e) we find that the graph of y = (x + 1)2 is the graph of y = x2 shifted one unit to the left. Some values are given in Table 2. , turned upside down). x= 0 for the function y 50 ClI. I PRELIMINARIES TABLE 1 TABLE 2 x2 (x - 1)2 25�36 16 �25 9 :---:6:! ---36 -5 -4 -3 -2 -1 0 1 2 3 45 In general, we have the following rules. Let y (i) (ii) (iii) (iv) 1)2 = f(x). To obtain the graph of y = f(x) + e, shift the graph of y = f(x) up e units if e 2: 0 and down l ei units if e < 0.
2, 1), (3, 2), (2, 3) 16. (1, 1), (1, 3), (1, 5) 13. (0, - 5), (1, - 1), (2, 3) 15. (2, 7), ( - 3, (7, 10) 17. (5, -2), (0, -2), ( -2, -2) 4), In Problems 18-32, find the slope of a line having the given angle of inclination. Use a calculator or table when necessary. 18. 22. 26. 30. 5° 161° In Problems 33. 1 37. 25 41. 5 19. 23. 27. 31. 5° 110° 20. 60° 24. 10° 28. 5° 32. 73° 25. 5° 21. 135° 29. 37° find the angle of inclination of a line having the given slope. 34. v'3 38. -37 42. 8 35. - llv'3 39.
Choosing (x1, y1) :::: ( - 1, -2), we obtain another point-slope equation of the line: y - < 2) = Hx - < - 1)], or y + 2 �(x + 1). This line is sketched in Figure 1. • As Example 1 shows, there are many point-slope equations of a line. In fact, there are an infinite number of them-one for each point on the line. A more com monly used equation of a line is given below. First, we define the y-intercept of a line to be the y-coordinate of the point at which the line intersects the y-axis. Find the y-intercept of the line y + 2 �(x + 1).
An Elementary Treatise on Diff. Eqns. and Their Applns. by H. Piaggio
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