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vb.net generate qr code Domain in Visual Studio .NET
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Drawing Bar Code In None Using Barcode drawer for Microsoft Excel Control to generate, create barcode image in Excel applications. Scanning USS Code 39 In VB.NET Using Barcode decoder for .NET framework Control to read, scan read, scan image in .NET framework applications. Example 1 revisited Let s take another look at the relation given by Example 1 in the previous section We described it using the following equation: ECC200 Creator In None Using Barcode drawer for Font Control to generate, create Data Matrix 2d barcode image in Font applications. Make Data Matrix In Java Using Barcode creation for Java Control to generate, create Data Matrix 2d barcode image in Java applications. y=x 1 Figure 113 is a graph of this equation in the Cartesian xy plane It s a straight line with a slope of 1 and a y intercept of 1 If we imagine an infinitely long, movable vertical line sweeping back and forth, it s easy to see that the vertical line never intersects our graph at more than one point Therefore, the relation is a function Example 2 revisited The relation in Example 2 in the previous section has a graph that s a parabola opening upward, as shown in Fig 114 The equation is y = x2 The vertex of the parabola represents the absolute minimum value of the relation, and it coincides with the coordinate origin (0,0) The curve rises symmetrically on either side of the y axis It s not difficult to see that a movable vertical line never intersects the parabola at more than one point This fact tells us that the relation is a function of x y 6 4 2 x 6 4 2 2 4 6
4 6 Movable vertical line
Figure 113 Cartesian graph of the relation y = x 1
The verticalline test reveals that it s a function of x
Relations in TwoSpace
y 6 4 2 x 6 4 2 2 4 6 Movable vertical line 2 4 6
Figure 114 Cartesian graph of the relation y = x2 The
verticalline test reveals that it s a function of x
Example 3 revisited Figure 115 is a graph of the relation we saw in Example 3 in the previous section The equation for that relation was stated as y = (x1/2) In this case, the graph is a parabola that opens to the right The vertex coincides with the coordinate origin, but there is no absolute minimum or maximum for the dependent variable When we construct a movable vertical line in this situation, we find that it doesn t intersect the graph when x < 0 When x = 0, the vertical line intersects the graph at the single point (0,0) When x > 0, the vertical line intersects the graph at two points Therefore, this relation is not a function of x Example 4 revisited Figure 116 is a graph of the relation we saw in Example 4 in the previous section It s the upper half of the parabola of Fig 115, with the point (0,0) included The equation is y = x1/2 The verticalline test tells us that this relation is a function of x No matter where we position the vertical line, it never intersects the graph more than once What s a TwoSpace Function
y 6 4 2 x 6 4 2 2 4 6 Movable vertical line 2 4 6
Figure 115 Cartesian graph of the relation y = (x1/2) The verticalline test reveals that it isn t a function of x
y 6 4 2 x 6 4 2 2 4 6 Movable vertical line 2 4 6
Figure 116 Cartesian graph of the relation y = x1/2 The verticalline test reveals that it s a function of x
Relations in TwoSpace
Are you confused
By now you might wonder, When we have a relation where the independent variable is represented by the polar angle q and the dependent variable is represented by the polar radius r, how can we tell if the relation is a function of q It s easy, but there s a little trick involved We can draw the graph of the relation in a Cartesian plane with q on the horizontal axis and r on the vertical axis We must allow both q and r to attain all possible realnumber values Once we ve drawn the graph of the polar relation the Cartesian way, we can use the Cartesian verticalline test to see whether or not the relation is a function of q Here s a challenge! Consider the relation between an independent variable x and a dependent variable y such that x2 y2 = 1 Sketch a graph of this relation in the Cartesian xy plane Use the verticalline test to determine, on the basis of the graph, whether or not this relation is a function of x Solution
Figure 117 is a graph of this relation It s a geometric figure called a hyperbola The verticalline test tells us that the relation is not a function of x

