document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=171206>Question 171206</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Can you PLEASE show me how to graph a quadratic function using its properties?<BR>\n" );
document.write( "f(x)=x^2-4x-5  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #126604 by </B><A HREF=/tutors/aboutme.mpl?userid=jojo14344><B>jojo14344(1513)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jojo14344><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=126604\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=x%5E2-4x-5+ ALIGN=MIDDLE  ALT=\"f%28x%29=x%5E2-4x-5+\"   DISABLED_style= \"max-width:90%; height:auto;\" > ---> <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;\" ><BR>\n" );
document.write( "Okay, first we go for the Vertex-form of quadratic: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=a%28x-h%29%5E2%2Bk ALIGN=MIDDLE  ALT=\"f%28x%29=a%28x-h%29%5E2%2Bk\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "where, h & k are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2Cy ALIGN=MIDDLE  ALT=\"x%2Cy\"   DISABLED_style= \"max-width:90%; height:auto;\" > points on the vertex.<BR>\n" );
document.write( "We'll do this by completing the square: We take <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F2 ALIGN=MIDDLE  ALT=\"1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > of \"b\" in the quadratic eq'n. and squared it. In this case it, \"b\"=4, so half of it --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F2%29%284%29=2 ALIGN=MIDDLE  ALT=\"%281%2F2%29%284%29=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then squared, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5E2=highlight%284%29 ALIGN=MIDDLE  ALT=\"2%5E2=highlight%284%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "So we need to add \"4\" to the eqn:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=%28x%5E2-4x%2B4%29highlight%28-4%29-5 ALIGN=MIDDLE  ALT=\"f%28x%29=%28x%5E2-4x%2B4%29highlight%28-4%29-5\"   DISABLED_style= \"max-width:90%; height:auto;\" > ----> remember to subtract 4 too so the expression does not change.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=%28x-2%29%5E2-9 ALIGN=MIDDLE  ALT=\"f%28x%29=%28x-2%29%5E2-9\"   DISABLED_style= \"max-width:90%; height:auto;\" > ----> follows our vertex form now, where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=2 ALIGN=MIDDLE  ALT=\"h=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > & <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=-9 ALIGN=MIDDLE  ALT=\"k=-9\"   DISABLED_style= \"max-width:90%; height:auto;\" >. We'll see in the graph:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-10%2C10%2C-11%2C10%2Cgrid%281%29%2Cgraph%28300%2C300%2C-10%2C10%2C-11%2C10%2Cx%5E2-4x-5%29%2Cblue%28circle%282%2C-9%2C.20%29%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > ----> see points (2,-9). This will be the vertex.<BR>\n" );
document.write( "Now we'll see the x-intercepts by solving the eqn:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-4x-5=0 ALIGN=MIDDLE  ALT=\"x%5E2-4x-5=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "By Quadratic Formula, where ----><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28a=1%2Cb=-4%2Cc=-5%29 ALIGN=MIDDLE  ALT=\"system%28a=1%2Cb=-4%2Cc=-5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%28-b%2B-sqrt%28b%5E2-4ac%29%29%2F%282a%29 ALIGN=MIDDLE  ALT=\"x=%28-b%2B-sqrt%28b%5E2-4ac%29%29%2F%282a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%28-%28-4%29%2B-sqrt%28-4%5E2-4%2A1%2A-5%29%29%2F%282%2A1%29 ALIGN=MIDDLE  ALT=\"x=%28-%28-4%29%2B-sqrt%28-4%5E2-4%2A1%2A-5%29%29%2F%282%2A1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%284%2B-sqrt%2816%2B20%29%29%2F2=%284%2B-sqrt%2836%29%29%2F2=%284%2B-6%29%2F2 ALIGN=MIDDLE  ALT=\"x=%284%2B-sqrt%2816%2B20%29%29%2F2=%284%2B-sqrt%2836%29%29%2F2=%284%2B-6%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%284%2B6%29%2F2=10%2F2=highlight%285%29 ALIGN=MIDDLE  ALT=\"x=%284%2B6%29%2F2=10%2F2=highlight%285%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%284-6%29%2F2=-2%2F2=highlight%28-1%29 ALIGN=MIDDLE  ALT=\"x=%284-6%29%2F2=-2%2F2=highlight%28-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "X-intercepts----><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28x=5%2Cx=-1%29 ALIGN=MIDDLE  ALT=\"system%28x=5%2Cx=-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "See the graph again:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-5%2C7%2C-11%2C10%2Cgrid%281%29%2Cgraph%28300%2C300%2C-5%2C7%2C-11%2C10%2Cx%5E2-4x-5%29%2Cblue%28circle%282%2C-9%2C.20%29%29%2Cgreen%28circle%285%2C0%2C.20%29%29%2Cgreen%28circle%28-1%2C0%2C.20%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Thank you,<BR>\n" );
document.write( "Jojo\r<BR>\n" );
document.write( "<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );