document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1078174>Question 1078174</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Can you please help me to sketch the graph, given is y=ax^2+bx+c where a<0 ; b<0 ; c<0 and ax^2+bx+c has only one solution. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #692642 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=692642\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> You mean that <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;\" > has only one solution.<BR>\n" );
document.write( " <BR>\n" );
document.write( "If a<0 the graph opens down (like a frowny mouth),<BR>\n" );
document.write( "and goes down without a bottom on the left end and on the right end,<BR>\n" );
document.write( "like<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28200%2C200%2C-0.9%2C0.9%2C-0.19%2C0.1%2C-1.3%28x%2B0.2%29%5E2%29 ALIGN=MIDDLE  ALT=\"graph%28200%2C200%2C-0.9%2C0.9%2C-0.19%2C0.1%2C-1.3%28x%2B0.2%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28200%2C200%2C-0.9%2C0.9%2C-0.18%2C0.2%2C-1.3%28x-0.2%29%5E2%2B0.1%29 ALIGN=MIDDLE  ALT=\"graph%28200%2C200%2C-0.9%2C0.9%2C-0.18%2C0.2%2C-1.3%28x-0.2%29%5E2%2B0.1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28200%2C200%2C-0.9%2C0.9%2C-0.19%2C0.1%2C-1.3%28x-0.2%29%5E2-0.1%29 ALIGN=MIDDLE  ALT=\"graph%28200%2C200%2C-0.9%2C0.9%2C-0.19%2C0.1%2C-1.3%28x-0.2%29%5E2-0.1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "That means there are at least some points below the x-axis.<BR>\n" );
document.write( "In this specific case, because b<0 and c<0 too,<BR>\n" );
document.write( "it is also obvious that for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1 ALIGN=MIDDLE  ALT=\"x=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2%2Bbx%2Bc=a%2Bb%2Bc%3C0 ALIGN=MIDDLE  ALT=\"ax%5E2%2Bbx%2Bc=a%2Bb%2Bc%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "If there were also some points above the x-axis<BR>\n" );
document.write( "the graph would have to cross the x-axis twice,<BR>\n" );
document.write( "but the fact that <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;\" > has only one solution<BR>\n" );
document.write( "means that it just touches it once and does not cross it,<BR>\n" );
document.write( "like <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28200%2C200%2C-0.9%2C0.9%2C-0.19%2C0.1%2C-1.3%28x%2B0.2%29%5E2%29 ALIGN=MIDDLE  ALT=\"graph%28200%2C200%2C-0.9%2C0.9%2C-0.19%2C0.1%2C-1.3%28x%2B0.2%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28200%2C200%2C-0.9%2C0.9%2C-0.19%2C0.1%2C-1.1%28x-0.4%29%5E2%29 ALIGN=MIDDLE  ALT=\"graph%28200%2C200%2C-0.9%2C0.9%2C-0.19%2C0.1%2C-1.1%28x-0.4%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28200%2C200%2C-0.9%2C0.9%2C-0.19%2C0.1%2C-x%5E2%29 ALIGN=MIDDLE  ALT=\"graph%28200%2C200%2C-0.9%2C0.9%2C-0.19%2C0.1%2C-x%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "I do not know if it was your teachers intention,<BR>\n" );
document.write( "but the point where the graph touches the x- axis<BR>\n" );
document.write( "must have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-b%2F2a ALIGN=MIDDLE  ALT=\"x=-b%2F2a\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Did you learn in class that the vertex of the graph<BR>\n" );
document.write( "is at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-b%2F2a ALIGN=MIDDLE  ALT=\"x=-b%2F2a\"   DISABLED_style= \"max-width:90%; height:auto;\" > ?<BR>\n" );
document.write( "If both (a and b) are negative,<BR>\n" );
document.write( "that is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-b%2F2a ALIGN=MIDDLE  ALT=\"x=-b%2F2a\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "so the graph can look like  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28200%2C200%2C-0.9%2C0.9%2C-0.19%2C0.1%2C-1.3%28x%2B0.2%29%5E2%29 ALIGN=MIDDLE  ALT=\"graph%28200%2C200%2C-0.9%2C0.9%2C-0.19%2C0.1%2C-1.3%28x%2B0.2%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28200%2C200%2C-0.9%2C0.9%2C-0.19%2C0.1%2C-1.1%28x%2B0.4%29%5E2%29 ALIGN=MIDDLE  ALT=\"graph%28200%2C200%2C-0.9%2C0.9%2C-0.19%2C0.1%2C-1.1%28x%2B0.4%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "touching the x-axis to the left of the origin. </SPAN>\n" );
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