document.write( "Question 878321: Determine the inequation that would define this region: all points on and below the absolute value function with zeros at x=4 and x=10, and a vertex at (7,2). \n" ); document.write( "

Algebra.Com's Answer #530000 by Edwin McCravy(20060) $\"\"$ $\"About$

You can put this solution on YOUR website!
Determine the inequation that would define this region: all points on and below the absolute value function with zeros at x=4 and x=10, and a vertex at (7,2).
\n" ); document.write( "

\r\n" );
document.write( "Here is the graph of what you want:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "First we get the equation of the boundary line.\r\n" );
document.write( "\r\n" );
document.write( "Absolute value functions have the form\r\n" );
document.write( "\r\n" );
document.write( "  y = a|x-h|+k\r\n" );
document.write( "\r\n" );
document.write( "Where (h,k) is the vertex. If \"a\" is positive the graph will open\r\n" );
document.write( "upward and if it is negative it will open downward. So we expect\r\n" );
document.write( "\"a\" to be negative.\r\n" );
document.write( "\r\n" );
document.write( "Since the vertex is (h,k) = (7,2) we substitute those values:\r\n" );
document.write( "\r\n" );
document.write( "  y = a|x-7|+2\r\n" );
document.write( "\r\n" );
document.write( "To find \"a\" we substitute any point other than the vertex.\r\n" );
document.write( "\r\n" );
document.write( "Since we know it goes through (4,0), we substitute (x,y) = (4,0)\r\n" );
document.write( "\r\n" );
document.write( "  y = a|x-7|+2\r\n" );
document.write( "\r\n" );
document.write( "  0 = a|4-7|+2\r\n" );
document.write( "\r\n" );
document.write( "  0 = a|-3|+2\r\n" );
document.write( "\r\n" );
document.write( "  0 = a(3)+2\r\n" );
document.write( "\r\n" );
document.write( "  0 = 3a+2\r\n" );
document.write( " \r\n" );
document.write( " -2 = 3a\r\n" );
document.write( "\r\n" );
document.write( "  = a\r\n" );
document.write( "\r\n" );
document.write( "Substitute that for 'a' in\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "  y = |x-7|+2\r\n" );
document.write( "\r\n" );
document.write( "Now since the lower area is shaded, and the graph\r\n" );
document.write( "includes the boundary line, we use .\r\n" );
document.write( "\r\n" );
document.write( " \r\n" );
document.write( "\r\n" );
document.write( "Edwin

\n" ); document.write( "

\n" );