document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209647>Question 1209647</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Suppose f(x) is a quadratic function such that f(1) = -24, f(4) = 10, and f(3) = 60.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Determine the value of f(-1). </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849721 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=849721\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to determine f(-1):\r<BR>\n" );
document.write( "\n" );
document.write( "1. **General form of a quadratic function:**\r<BR>\n" );
document.write( "\n" );
document.write( "   f(x) = ax² + bx + c\r<BR>\n" );
document.write( "\n" );
document.write( "2. **Set up a system of equations using the given information:**\r<BR>\n" );
document.write( "\n" );
document.write( "   * f(1) = a(1)² + b(1) + c = a + b + c = -24<BR>\n" );
document.write( "   * f(4) = a(4)² + b(4) + c = 16a + 4b + c = 10<BR>\n" );
document.write( "   * f(3) = a(3)² + b(3) + c = 9a + 3b + c = 60\r<BR>\n" );
document.write( "\n" );
document.write( "3. **Solve the system of equations:**\r<BR>\n" );
document.write( "\n" );
document.write( "   There are several ways to solve this system. One approach is to use elimination or substitution. Here's one way:\r<BR>\n" );
document.write( "\n" );
document.write( "   * Subtract the first equation from the second:<BR>\n" );
document.write( "     15a + 3b = 34\r<BR>\n" );
document.write( "\n" );
document.write( "   * Subtract the first equation from the third:<BR>\n" );
document.write( "     8a + 2b = 84, which simplifies to 4a + b = 42\r<BR>\n" );
document.write( "\n" );
document.write( "   * Solve for b in terms of a from the simplified third equation:<BR>\n" );
document.write( "     b = 42 - 4a\r<BR>\n" );
document.write( "\n" );
document.write( "   * Substitute this expression for b into the equation from subtracting the first two:<BR>\n" );
document.write( "     15a + 3(42 - 4a) = 34<BR>\n" );
document.write( "     15a + 126 - 12a = 34<BR>\n" );
document.write( "     3a = -92<BR>\n" );
document.write( "     a = -92/3\r<BR>\n" );
document.write( "\n" );
document.write( "   * Substitute a back into the equation for b:<BR>\n" );
document.write( "     b = 42 - 4(-92/3)<BR>\n" );
document.write( "     b = 42 + 368/3<BR>\n" );
document.write( "     b = (126 + 368)/3<BR>\n" );
document.write( "     b = 494/3\r<BR>\n" );
document.write( "\n" );
document.write( "   * Substitute a and b back into the first equation to solve for c:<BR>\n" );
document.write( "     (-92/3) + (494/3) + c = -24<BR>\n" );
document.write( "     402/3 + c = -24<BR>\n" );
document.write( "     134 + c = -24<BR>\n" );
document.write( "     c = -158\r<BR>\n" );
document.write( "\n" );
document.write( "4. **Write the quadratic function:**\r<BR>\n" );
document.write( "\n" );
document.write( "   f(x) = (-92/3)x² + (494/3)x - 158\r<BR>\n" );
document.write( "\n" );
document.write( "5. **Calculate f(-1):**\r<BR>\n" );
document.write( "\n" );
document.write( "   f(-1) = (-92/3)(-1)² + (494/3)(-1) - 158<BR>\n" );
document.write( "   f(-1) = (-92/3) - (494/3) - 158<BR>\n" );
document.write( "   f(-1) = -586/3 - 158<BR>\n" );
document.write( "   f(-1) = -586/3 - 474/3<BR>\n" );
document.write( "   f(-1) = -1060/3<BR>\n" );
document.write( "   f(-1) = -353.3333...\r<BR>\n" );
document.write( "\n" );
document.write( "Therefore, f(-1) = -1060/3 or approximately -353.33.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );