document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=131786>Question 131786</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the vertex, the line of symmetry, the maximum or minimum value of the quadratic function, and then graph the function.\r<BR>\n" );
document.write( "\n" );
document.write( "f(x) = 4x^2 - 24x + 38\r<BR>\n" );
document.write( "\n" );
document.write( "I believe the vertex points are 3, 2.\r<BR>\n" );
document.write( "\n" );
document.write( "The line of symmetry would then be 3 and then I get stuck.  Is 2 a maximum or minimum value and I need to know that to graph the function.  So, I'm stuck.  Can someone help? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #96213 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=96213\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> So far so good.  The vertex is at (3,2) and the axis of symmetry is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=3 ALIGN=MIDDLE  ALT=\"x=3\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  As to the maximum/minimum thing, all you need to do is determine whether the parabola is concave up (makes a valley), or concave down (makes a hill).  The sign on the lead coefficient tells you directly.  If the a in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2%2Bbx%2Bc ALIGN=MIDDLE  ALT=\"ax%5E2%2Bbx%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" > is negative, the parabola is concave down and the vertex is a maximum, if it is positive, the parabola is concave up and the vertex is a minimum.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "In this case, 4 is positive, so concave up and f(3) = 2 is a minimum.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "As to sketching a graph, you already have one point, the vertex, so plot that.  Then you need to make a table of values for two or three x values on one side of the axis of symmetry.  Once you have done that you will have the same number of values for the other side of the axis because of the symmetrical property.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "x = 4, f(4) = 4*16 - 24*4 + 38 = 102 - 96 = 6  => (4,6)<BR>\n" );
document.write( "x = 5, f(5) = 4*25 - 24*5 + 38 = 138 - 120 = 18 => (5,18)<BR>\n" );
document.write( "x = 6, f(6) = 4*36 - 24*6 + 38 = 182 - 144 = 38 => (6,38)\r<BR>\n" );
document.write( "\n" );
document.write( "And, because of symmetry about <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=3 ALIGN=MIDDLE  ALT=\"x=3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "x = 2, f(2) = f(4) = 6 => (2,6)<BR>\n" );
document.write( "x = 1, f(1) = f(5) = 18 => (1,18)<BR>\n" );
document.write( "x = 0, f(0) = f(6) = 38 => (0,38)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Plot your points and draw a smooth curve.  Done.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28800%2C800%2C-2%2C10%2C-2%2C40%2C%0D%0Agrid%281%29%2C%0D%0Agraph%28800%2C800%2C-2%2C10%2C-2%2C40%2C4x%5E2+-+24x+%2B+38%29%2C%0D%0Alocate%283.2%2C1.8%2CV%283%2C2%29%29%2C%0D%0Acircle%283%2C2%2C.05%29%2C%0D%0Acircle%284%2C6%2C.05%29%2C%0D%0Acircle%285%2C18%2C.05%29%2C%0D%0Acircle%286%2C38%2C.05%29%2C%0D%0Acircle%282%2C6%2C.05%29%2C%0D%0Acircle%281%2C18%2C.05%29%2C%0D%0Acircle%280%2C38%2C.05%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );