document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=519298>Question 519298</A>: <I><SPAN STYLE=\"word-break: break-all;\"> identify the vertex,axis of symmetry and intercepts (if they exist) of the function:  x^2-4x+2 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #345525 by </B><A HREF=/tutors/aboutme.mpl?userid=oberobic><B>oberobic(2304)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=oberobic><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=345525\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> y = x^2 -4x + 2<BR>\n" );
document.write( ".<BR>\n" );
document.write( "A graph will help visualize the problem and the solution.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28500%2C500%2C-5%2C5%2C-5%2C5%2Cx%5E2-4%2Ax%2B2%29+ ALIGN=MIDDLE  ALT=\"+graph%28500%2C500%2C-5%2C5%2C-5%2C5%2Cx%5E2-4%2Ax%2B2%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Note that y = x^2 -4x +2 is n the form:  y = ax^2 + bx +c<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Axis of symmetry is the value of x, where<BR>\n" );
document.write( ".<BR>\n" );
document.write( "x = -b/2a is the known formula<BR>\n" );
document.write( ".<BR>\n" );
document.write( "substitute b= -4 and a=1<BR>\n" );
document.write( ".<BR>\n" );
document.write( "x = -(-4)/(2*1) = 4/2 = 2<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Vertex is the (x,y) coordinate, where the value of x from above is substituted<BR>\n" );
document.write( ".<BR>\n" );
document.write( "y = x^2 -4x +2<BR>\n" );
document.write( "y = (2)^2 -4(2) + 2<BR>\n" );
document.write( "y = 4 -8 +2<BR>\n" );
document.write( "y = -2<BR>\n" );
document.write( "so the vertex is<BR>\n" );
document.write( "(2,-2)<BR>\n" );
document.write( ".<BR>\n" );
document.write( "The y-intercept occurs when x=0<BR>\n" );
document.write( "y = 2, which is the point (0,2)<BR>\n" );
document.write( ".<BR>\n" );
document.write( "To find the x-intercepts you have to solve:<BR>\n" );
document.write( "0 = x^2 -4x +2<BR>\n" );
document.write( ".<BR>\n" );
document.write( "This equation does not factor, so to find the intercepts use the quadratic equation<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<TABLE  BORDER=1>\n" );
document.write( "\n" );
document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=quadratic.solver>SOLVE quadratic equation with variable</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT>Quadratic equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2%2Bbx%2Bc=0 ALIGN=MIDDLE  ALT=\"ax%5E2%2Bbx%2Bc=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (in our case <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1x%5E2%2B-4x%2B2+=+0 ALIGN=MIDDLE  ALT=\"1x%5E2%2B-4x%2B2+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >) has the following solutons:<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  For these solutions to exist, the <U>discriminant</U> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" > should not be a negative number.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  First, we need to compute the discriminant <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" >: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac=%28-4%29%5E2-4%2A1%2A2=8 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%28-4%29%5E2-4%2A1%2A2=8\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=8 is greater than zero. That means that there are two solutions: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x%5B12%5D+=+%28--4%2B-sqrt%28+8+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28--4%2B-sqrt%28+8+%29%29%2F2%5Ca\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B1%5D+=+%28-%28-4%29%2Bsqrt%28+8+%29%29%2F2%5C1+=+3.41421356237309 ALIGN=MIDDLE  ALT=\"x%5B1%5D+=+%28-%28-4%29%2Bsqrt%28+8+%29%29%2F2%5C1+=+3.41421356237309\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B2%5D+=+%28-%28-4%29-sqrt%28+8+%29%29%2F2%5C1+=+0.585786437626905 ALIGN=MIDDLE  ALT=\"x%5B2%5D+=+%28-%28-4%29-sqrt%28+8+%29%29%2F2%5C1+=+0.585786437626905\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <BR>\n" );
document.write( "    Quadratic expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1x%5E2%2B-4x%2B2 ALIGN=MIDDLE  ALT=\"1x%5E2%2B-4x%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" > can be factored:<BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1x%5E2%2B-4x%2B2+=+1%28x-3.41421356237309%29%2A%28x-0.585786437626905%29 ALIGN=MIDDLE  ALT=\"1x%5E2%2B-4x%2B2+=+1%28x-3.41421356237309%29%2A%28x-0.585786437626905%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: 3.41421356237309, 0.585786437626905.\n" );
document.write( "Here's your graph:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-4%2Ax%2B2+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-4%2Ax%2B2+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
document.write( "</TABLE>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );