\r\n" );
document.write( "We draw the graph of y=4/x, by getting and plotting some points:\r\n" );
document.write( "\r\n" );
document.write( " x|y\r\n" );
document.write( "--|--\r\n" );
document.write( "±1|±4\r\n" );
document.write( "±2|±2\r\n" );
document.write( "±3|±1.3\r\n" );
document.write( "±4|±1\r\n" );
document.write( "\r\n" );
document.write( "
\r\n" );
document.write( "\r\n" );
document.write( "Now the lines y=mx+4 will all have y intercept (0,4),\r\n" );
document.write( "but will have different slopes. Here are some lines\r\n" );
document.write( "that have the equation y=mx+4 where m has different\r\n" );
document.write( "values\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "
\r\n" );
document.write( "\r\n" );
document.write( "Some of those lines intersect the graph in 0 point, 1 points or 2 points.\r\n" );
document.write( "\r\n" );
document.write( "The brownish-green line does not intersect it at all\r\n" );
document.write( "The green and horizontal intersect the graph in 1 point\r\n" );
document.write( "The purple and the light blue lines intersect the curve\r\n" );
document.write( "in two points each.\r\n" );
document.write( "\r\n" );
document.write( "If we solve the system of equations:\r\n" );
document.write( "\r\n" );
document.write( "
\r\n" );
document.write( "\r\n" );
document.write( "by substitution, we get\r\n" );
document.write( "\r\n" );
document.write( "
\r\n" );
document.write( "\r\n" );
document.write( "
\r\n" );
document.write( "\r\n" );
document.write( "
\r\n" );
document.write( "\r\n" );
document.write( "What determines whether the line y=mx+4 intersects\r\n" );
document.write( "the curve 0, 1, or 2 times, is whether this\r\n" );
document.write( "quadratic equation has 0,1, or 2 real solutions.\r\n" );
document.write( "\r\n" );
document.write( "And what determines whether a quadratic equation has\r\n" );
document.write( "0, 1, or 2 real solutions is the DISCRIMINANT 
.\r\n" );
document.write( "\r\n" );
document.write( " \r\n" );
document.write( "\r\n" );
document.write( "a=m, b=4, c=-4 so the \r\n" );
document.write( "\r\n" );
document.write( "discriminant = = 
= 
\r\n" );
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document.write( "Case 1: If the discriminant is negative, there will be NO real solutions.\r\n" );
document.write( "\r\n" );
document.write( "So we set the discriminant less than 0\r\n" );
document.write( "\r\n" );
document.write( "
\r\n" );
document.write( " 
\r\n" );
document.write( " 
\r\n" );
document.write( "\r\n" );
document.write( "So when m is less than -4, there will be NO solutions.\r\n" );
document.write( "\r\n" );
document.write( "So we set the discriminant less than 0\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( " \r\n" );
document.write( " 
\r\n" );
document.write( "\r\n" );
document.write( "So when m is less than -1, there will be NO solutions.\r\n" );
document.write( "and therefore the line will NOT intersect the curve at all.\r\n" );
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document.write( "\r\n" );
document.write( "Case 2: If the discriminant is EQUAL to zero, there will be ONE \r\n" );
document.write( "real solution, and therefore the line will intersect the curve ONE time.\r\n" );
document.write( "\r\n" );
document.write( "So we set the discriminant EQUAL to 0\r\n" );
document.write( "\r\n" );
document.write( "
\r\n" );
document.write( " 
\r\n" );
document.write( " 
\r\n" );
document.write( "\r\n" );
document.write( "So when m is equal to -1, there will be ONE real solutions.\r\n" );
document.write( "and therefore the line will intersect the curve ONE time.\r\n" );
document.write( "\r\n" );
document.write( "Case 3: If the discriminant is positive, there will be TWO real solutions.\r\n" );
document.write( "\r\n" );
document.write( "So we set the discriminant GREATER than 0\r\n" );
document.write( "\r\n" );
document.write( "
\r\n" );
document.write( " 
\r\n" );
document.write( " 
\r\n" );
document.write( "\r\n" );
document.write( "So when m is greater than -1, there will be TWO solutions,\r\n" );
document.write( "and therefore the line will intersect the curve at TWICE.\r\n" );
document.write( "\r\n" );
document.write( "Answer:\r\n" );
document.write( "\r\n" );
document.write( "If the slope m < -1, the line y=mx+4 will intersect the curve 0 times.\r\n" );
document.write( "If the slope m = -1, the line y=mx+4 will intersect the curve 1 time.\r\n" );
document.write( "If the slope m > -1, the line y=mx+4 will intersect the curve 2 times.\r\n" );
document.write( " \r\n" );
document.write( "Edwin
\n" );
document.write( "![\"\"](\"/images/stars/stars-13.gif\")
