\r\n" );
document.write( "In most business, increasing proces of their product can have a negative effect\r\n" );
document.write( "on the number of customers of the business. A bus company in a small town has\r\n" );
document.write( "an average number of riders of 800 per day. The bus company charges $2.25 for\r\n" );
document.write( "a ride. They conducted a survey of their customers and found that they will\r\n" );
document.write( "lose approximately 40 customers per day for each $.25 increase in fare.\r\n" );
document.write( "\r\n" );
document.write( "Given the description above, graph the function, identify the graph of the\r\n" );
document.write( "function, and give the domain and range as shown in the graph, and also the\r\n" );
document.write( "domain and the range of the entire function excluding unrealistic values.\r\n" );
document.write( " \r\n" );
document.write( "These instructions are incomplete. You didn't tell us what you want\r\n" );
document.write( "to express in terms of what, so I can only guess that you want to \r\n" );
document.write( "express the bus company's daily income as a function of the number \r\n" );
document.write( "of 25 cent increases in the fare. If that was not what you wanted, \r\n" );
document.write( "then you'll have to post again with the complete instructions. \r\n" );
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document.write( "Income = (new higher fare)×(new lower number of passengers)\r\n" );
document.write( "\r\n" );
document.write( "Let x = the number of price increases of $0.25 each in the fare\r\n" );
document.write( "Then .25x will be the increase in the fare\r\n" );
document.write( "Then the new higher fare will be \" 2.25 + .25x \" dollars\r\n" );
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document.write( "Then \" 40x \" will be the decrease in the number of passengers.\r\n" );
document.write( "Then \" 800 - 40x \" will be the new lower number of passenger \r\n" );
document.write( "\r\n" );
document.write( "Income = (new higher fare)×(new lower number of passengers)\r\n" );
document.write( "\r\n" );
document.write( "Let y = income\r\n" );
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document.write( "y = (2.25 + .25x)(800 - 40x)\r\n" );
document.write( "\r\n" );
document.write( "Use FOIL\r\n" );
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document.write( "y = 1800 - 90x + 200x - 10x²\r\n" );
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document.write( "y = -10x² + 110x + 1800 or in functional notation,\r\n" );
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document.write( "y = f(x) = -10x² + 110x + 1800\r\n" );
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document.write( "This plots as this parabola.\r\n" );
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document.write( "y = daily income\r\n" );
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document.write( "0 x = no. of 25c increases \r\n" );
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document.write( "The smallest value in the domain is when there are\r\n" );
document.write( "NO 25c increases, i.e., when x = 0.\r\n" );
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document.write( "The largest value in the domain will be when the fare\r\n" );
document.write( "is so high that there will be NO passengers, which is when y = 0\r\n" );
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document.write( "So we set y = 0 and solve for x\r\n" );
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document.write( "y = -10x² + 110x + 1800\r\n" );
document.write( "\r\n" );
document.write( "y = 0\r\n" );
document.write( "\r\n" );
document.write( "-10x² + 110x + 1800 = 0\r\n" );
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document.write( "Divide every term by -10 to simplify\r\n" );
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document.write( " x² - 11x - 180 = 0\r\n" );
document.write( "Factor:\r\n" );
document.write( " (x + 9)(x - 20) = 0\r\n" );
document.write( "Use zero-factor property, set each factor = 0\r\n" );
document.write( "\r\n" );
document.write( "x + 9 = 0 gives x = -9 which we discard as this \r\n" );
document.write( "is outside domain\r\n" );
document.write( "\r\n" );
document.write( "x - 20 = 0 gives x = 20, which is the largest\r\n" );
document.write( "value in the domain. So domain = \r\n" );
document.write( "\r\n" );
document.write( "{x | 0 < x < 20}, or in interval notation, [0, 20]\r\n" );
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document.write( "For the range, the smallest value of y is, of course, 0.\r\n" );
document.write( "\r\n" );
document.write( "To find the largest value of y in the range, we will need to find\r\n" );
document.write( "the vertex of the parabola. \r\n" );
document.write( "\r\n" );
document.write( "The vertex of y = f(x) = Ax² + Bx + C is the point\r\n" );
document.write( "\r\n" );
document.write( "( -B/(2A), f(-B/(2A)) )\r\n" );
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document.write( "y = f(x) = -10x² + 110x + 1800\r\n" );
document.write( "\r\n" );
document.write( "A = -10, B = 110, C = 1800\r\n" );
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document.write( "-B/(2A) = -(110)/[2(-10)] = 5.5\r\n" );
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document.write( "f(-B/(2A)) = 2102.5\r\n" );
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document.write( "So the vertex is the point (5.5, 2102.5)\r\n" );
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document.write( "Therefore the largest value of y is 2102.5, or $2102.50\r\n" );
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document.write( "So the range = {y | 0 < y < $2102.50} or [0, 2102.50] in\r\n" );
document.write( "interval notation.\r\n" );
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document.write( "$2102.50 is the income per day when 5.5 25c increases are\r\n" );
document.write( "made in the fare.\r\n" );
document.write( "\r\n" );
document.write( "The bus company makes the most money per day when there\r\n" );
document.write( "are 5.5 25 cent increases, or an increase of $1.375, or a \r\n" );
document.write( "fare of $3.625, but since people cannot cut pennies in\r\n" );
document.write( "two they either have to charge $3.62 or $3.63.\r\n" );
document.write( "\r\n" );
document.write( "Edwin\r\n" );
document.write( "AnlytcPhil@aol.com
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