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Next, we plot the solution(s) Once again, let s use rectangular coordinates In this case, the span of absolute values for the input (the independent variable x) is from 0 to 10, while the absolute values of the functions go as high as 277 Let s make each increment on the x axis represent 2 units, and each increment on the y axis represent 50 units With six hash marks going out from 0 in each of the four directions along the axes, that gives us absolute-value spans from 0 to 12 for x and 0 to 300 for y, as shown in Fig 28-2 We then plot the two solution points Finally, we plot the rest We can fill in the graphs by plotting the remaining points indicated in Table 28-2 As before, it s a good idea to draw the points for one graph with a pencil, fill in its curve with a pen, draw the points for the other graph with a pencil, fill in its curve with a pen, wait for the ink to dry, and finally run an eraser over the curves to get rid of the pencil marks In Fig 28-2, the approximate graph for

The earth s ionosphere, at altitudes from about 35 miles to 250 miles, has a great effect on EM waves at frequencies below about 100 MHz

00000000 to 11111111 The result is 16,777,216 (2563) possible colors Any point within the cube represents a unique color Colors in the RGB model can also be represented by six-digit hexadecimal numbers, such as 005CFF In this scheme, the first two digits represent the red (R) intensity in 256 levels ranging from 00 to FF The middle two digits represent the green (G) intensity, and the last two digits represent the blue (B) intensity Some RGB systems use only 16 levels for each primary color (binary 0000 through 1111) This results in 4096 possible colors

y = 2x 2 3x 4 is a solid parabola, and the approximate curve for y = 3x 2 5x + 11 is a dashed parabola

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Here s a challenge!

Radio wave propagation 501 The ionosphere consists of three layers, called the D layer, E layer, and F layer The layers form at different altitudes, with the D layer lowest and the F layer highest (Fig 27-2) The D layer absorbs radio waves at frequencies below about 7 MHz The E and F layers return radio waves to the earth by a process called ionospheric refraction

Refer to the text in this chapter if necessary A good score is at least 18 correct Answers are in the back of the book 1 Suppose inverters are placed in series with both inputs of an AND gate Under what conditions is the output of the resulting black box high (a) If and only if both inverter inputs are high (b) If and only if both inverter inputs are low (c) If and only if one inverter input is high and the other is low (d) Under no conditions (the output is always low) 2 Suppose an AND gate is followed by an inverter Under what conditions is the output of the resulting black box low (a) If and only if both inputs are high (b) If and only if both inputs are low (c) If and only if one input is high and the other is low (d) Under no conditions (the output is always high) 3 What is the binary equivalent of decimal 29 (a) 10101 (b) 11101 (c) 10111 (d) 11011 4 In Boolean algebra, addition represents (a) the logical NOT operation (b) the logical AND operation (c) the logical OR operation (d) the logical NAND operation 5 If the output to a logical inverter is low, it means that (a) both of the two inputs are high (b) both of the two inputs are low

By examining Fig 28-2, describe how the quadratic function y = 2x 2 3x 4 (shown by the solid curve) can be modified to produce a system with no real solutions, assuming that the other quadratic function (shown by the dashed curve) stays the same, and also assuming that the contour of the graph for the modified function stays the same

27-2

We can move the solid parabola straight upward As we do that, the two intersection points get closer together When the solid parabola reaches a certain critical altitude, the intersection points merge, indicating that the system has a single real solution with multiplicity 2 If we move the solid parabola upward beyond the critical altitude, the two parabolas no longer intersect We can intuitively see this by comparing the sharpness of the two curves The solid parabola is not as sharp as the dashed one, so the two curves diverge once we have raised the solid parabola high enough to completely clear the dashed one

How, you ask, can we change a quadratic function to move its parabola straight upward The answer is simple We can increase the value of the stand-alone constant, leaving the rest of the equation unchanged That increases the y-values of all the points without changing their x-values Every point on the parabola is displaced straight upward by the same amount as every other point

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