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We could shrink it by making each division represent, say, 5 units on both axes But then the solutions wouldn t show up well; they d be too close together In situations like this, the best approach is to use rectangular coordinates, but not strict Cartesian coordinates Let s make each increment on the x axis represent 1 unit, and each increment on the y axis represent 5 units When we make the axis increments different in size, we distort the slopes and contours of the lines and curves, but that s not important here Our goal is only to get a good fit for the ranges of the values we determined when we created Table 28-1, and a clear picture of how the graphs intersect To begin, we plot the two solution points

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Finally, we plot the rest Once we ve plotted the solution points as dots on the coordinate grid, we can locate and plot the rest of the points in the table If we do this on paper, we can use pencil and draw the points lightly, so we can erase them later As we draw the points, we must remember the increments we ve chosen for each axis They aren t the same, and it s easy to get them confused It s also important to keep track of which graph goes with which function! We can draw the points for one graph with a pencil, fill in its line or curve with a pen, draw the points for the second graph with a pencil, fill in its line or curve with a pen, and then erase all the penciled-in points when the ink is dry If we ve chosen the axis increments wisely, the solutions will be near the center of the coordinate area, and the other points will be scattered fairly well over the rest of it Fig 28-1 shows the final result

Beams of infrared or visible light can be modulated just as can radio-frequency carriers The frequencies of infrared and visible light are extremely high, allowing modulation by data at rates into the gigahertz range A simple modulated-light transmitter is diagrammed schematically in Fig 26-17 The output of the light-emitting diode (LED) is modulated by audio from the transistor The light is guided into an optical fiber, made from a special mixture of very clear glass In recent years, fiberoptics has begun to replace conventional cable networks on a massive scale Fiberoptics has several advantages over wire cables A fiberoptic cable is cheap, and it s light in weight It is totally immune to interference from outside electromagnetic

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fields A fiberoptic cable will not corrode as metallic wires do Fiberoptic cables are inexpensive to maintain and easy to repair An optical fiber can carry far more signals than a cable, because the carrier frequency is practically infinite The whole radio spectrum, from VLF through microwaves, can be imprinted on a beam of light and sent through a single glass fiber no thicker than a strand of your hair

By examining Fig 28-1, describe how the linear function y = 2x + 1 (shown by the solid line) can be modified to produce a system with no real solutions, assuming that the quadratic function (shown by the dashed curve) stays the same, and assuming the slope of the line stays the same

If we change the linear function so its graph doesn t intersect the parabola, then the resulting two-by-two system will have no real solutions Imagine moving the straight line downward in Fig 28-1 As we do this, the solution points get closer together, eventually merging At the moment the two points become one, the system has a single real solution with multiplicity 2 The linear function is expressed in slope-intercept form, so we can see that the line has a slope of 2 and a y-intercept of 1 Each division on the y axis is 5 units, so we can see that the quadratic function has an absolute minimum of approximately 5 Suppose we change the linear function so the y-intercept is 10 That will put the line completely below the parabola, and will give us the two-by-two system y = 2x 10 and y = x2 + x 5

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