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Graphs of Equations in .NET framework
Graphs of Equations Read QR In Visual Studio .NET Using Barcode Control SDK for .NET Control to generate, create, read, scan barcode image in Visual Studio .NET applications. Quick Response Code Creation In VS .NET Using Barcode creator for .NET Control to generate, create QR image in .NET applications. Consider the following equation involving the variables x and y: 2y 3x = 6 (i) Scan QR Code 2d Barcode In VS .NET Using Barcode scanner for .NET framework Control to read, scan read, scan image in .NET applications. Paint Bar Code In Visual Studio .NET Using Barcode generator for VS .NET Control to generate, create bar code image in .NET applications. Notice that the point (2, 6) satis es the equation; that is, when the xcoordinate 2 is substituted for x and the ycoordinate 6 is substituted for y, the lefthand side, 2y 3x, assumes the value of the righthand side, 6. The graph of (i) consists of all points (a, b) that satisfy the equation when a is substituted for x and b is substituted for y. We tabulate some points that satisfy (i) in Fig. 31(a), and indicate these points in Fig. 31(b). It is apparent that these points all lie on a straight line. In fact, it will be shown later that the graph of (i) actually is a straight line. Read Bar Code In .NET Framework Using Barcode scanner for Visual Studio .NET Control to read, scan read, scan image in .NET applications. QR Code Creator In C# Using Barcode creation for VS .NET Control to generate, create Denso QR Bar Code image in .NET applications. Fig. 31 In general, the graph of an equation involving x and y as its only variables consists of all points (x, y) satisfying the equation. Encoding QR Code In .NET Using Barcode drawer for ASP.NET Control to generate, create QR image in ASP.NET applications. Painting QR Code In VB.NET Using Barcode generator for .NET Control to generate, create QR Code JIS X 0510 image in .NET applications. Copyright 2008, 1997, 1985 by The McGrawHill Companies, Inc. Click here for terms of use.
Making Bar Code In Visual Studio .NET Using Barcode encoder for VS .NET Control to generate, create bar code image in Visual Studio .NET applications. Printing Bar Code In Visual Studio .NET Using Barcode encoder for Visual Studio .NET Control to generate, create bar code image in .NET applications. GRAPHS OF EQUATIONS
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ECC200 Creator In Visual C# Using Barcode generator for .NET Control to generate, create Data Matrix image in .NET applications. Barcode Creation In VS .NET Using Barcode encoder for ASP.NET Control to generate, create bar code image in ASP.NET applications. (a) Some points on the graph of y = x 2 are computed in Fig. 32(a) and shown in Fig. 32(b). These points suggest that the graph looks like what would be obtained by lling in the dashed curve. This graph is of the type known as a parabola. GTIN  128 Printer In ObjectiveC Using Barcode printer for iPad Control to generate, create GTIN  128 image in iPad applications. Painting Code39 In None Using Barcode drawer for Font Control to generate, create Code 39 Extended image in Font applications. Fig. 32 Draw Universal Product Code Version A In Java Using Barcode creator for Java Control to generate, create UPCA Supplement 5 image in Java applications. Printing 2D Barcode In Visual Basic .NET Using Barcode maker for .NET Control to generate, create Matrix Barcode image in VS .NET applications. (b) The graph of the equation xy = 1 is called a hyperbola. As shown in Fig. 33(b), the graph splits into two separate pieces. The points on the hyperbola get closer and closer to the axes as they move farther and farther from the origin. Fig. 33 CHAP. 3] GRAPHS OF EQUATIONS
(c) The graph of the equation y2 x2 + =1 9 4 is a closed curve, called an ellipse (see Fig. 34). Fig. 34 Circles For a point P(x, y) to lie on the circle with center C(a, b) and radius r, the distance PC must be r (Fig. 35). Now by (2.1), PC = (x a)2 + (y b)2 The standard equation, PC = r 2 , of the circle with center (a, b) and radius r is then (x a)2 + (y b)2 = r 2 For a circle centered at the origin, (3.1) becomes simply x 2 + y2 = r 2 (3.2) (3.1) Fig. 35 GRAPHS OF EQUATIONS
[CHAP. 3
EXAMPLES
(a) The circle with center (1, 2) and radius 3 has the equation (x 1)2 + (y 2)2 = 9 (b) The circle with center ( 1, 4) and radius 6 has the equation (x + 1)2 + (y 4)2 = 36 (c) The graph of the equation (x 3)2 + (y 7)2 = 16 is a circle with center (3, 7) and radius 4. (d) The graph of the equation x 2 + (y + 2)2 = 1 is a circle with center (0, 2) and radius 1. Sometimes the equation of a circle will appear in a disguised form. For example, the equation x 2 + y2 6x + 2y + 6 = 0 is equivalent to (x 3)2 + (y + 1)2 = 4 algebra Use the formulas (u + v)2 = u2 + 2uv + v 2 and (u v)2 = u2 2uv + v 2 to expand the lefthand side of (iii). (ii) (iii) If an equation such as (ii) is given, there is a simple method for recovering the equivalent standard equation of the form (iii) and thus nding the center and the radius of the circle. This method depends on completing the squares; that is, replacing the quantities x 2 + Ax and y2 + By by the equal quantities x+ A 2 A2 4 B2 4 EXAMPLE Let us nd the graph of the equation
x 2 + y2 + 4x 2y + 1 = 0 Completing the squares, replace x 2 + 4x by (x + 2)2 4 and y2 2y by (y 1)2 1, (x + 2)2 4 + (y 1)2 1 + 1 = 0 (x + 2)2 + (y 1)2 = 4 This is the equation of a circle with center ( 2, 1) and radius 2.

