document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=148405>Question 148405</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 24. From the list of plotted points<BR>\n" );
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document.write( "(-5,5) (3,5)<BR>\n" );
document.write( "(-4, 3/2) (2, 3/2)<BR>\n" );
document.write( "(-3,-1) (1,-1)<BR>\n" );
document.write( "(-2, -5/2) (0, -5/2)<BR>\n" );
document.write( "(-1,-3) <BR>\n" );
document.write( "Form an equation after plotting these points and then answer the questions below:\r<BR>\n" );
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document.write( "Is the degree of the polynomial odd or even?\r<BR>\n" );
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document.write( "Is the leading coefficient of the polynomial positive or negative? \r<BR>\n" );
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document.write( "Number of zeros<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #108758 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=108758\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> If we plot the points we get\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src=\"http://i150.photobucket.com/albums/s91/jim_thompson5910/points.jpg\" alt=\"Photobucket - Video and Image Hosting\">\r<BR>\n" );
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document.write( "Now draw a curve through these points to get the equation:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src=\"http://i150.photobucket.com/albums/s91/jim_thompson5910/graphthroughpoints.jpg\" alt=\"Photobucket - Video and Image Hosting\">\r<BR>\n" );
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document.write( "From the graph, we can see that the degree is even (since both ends of the graph go in the same y direction). Also, we can see that the leading coefficient is positive (since the graph opens upward). Finally, we can see that there are two zeros (since there are two x-intercepts).\r<BR>\n" );
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document.write( "Since we have all of this info, we don't need the equation. However, if you do need the equation, this is how you would find it:\r<BR>\n" );
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document.write( "From the graph, we can see that the vertex is (-1,-3). \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=a%28x-h%29%5E2%2Bk ALIGN=MIDDLE  ALT=\"y=a%28x-h%29%5E2%2Bk\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the general vertex equation.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=a%28x--1%29%5E2-3 ALIGN=MIDDLE  ALT=\"y=a%28x--1%29%5E2-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=-1 ALIGN=MIDDLE  ALT=\"h=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=-3 ALIGN=MIDDLE  ALT=\"k=-3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=a%28x%2B1%29%5E2-3 ALIGN=MIDDLE  ALT=\"y=a%28x%2B1%29%5E2-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify.\r<BR>\n" );
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document.write( "Now let's plug in another point. Let's plug in (1,-1)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1=a%281%2B1%29%5E2-3 ALIGN=MIDDLE  ALT=\"-1=a%281%2B1%29%5E2-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <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;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-1 ALIGN=MIDDLE  ALT=\"y=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1=a%282%29%5E2-3 ALIGN=MIDDLE  ALT=\"-1=a%282%29%5E2-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1=4a-3 ALIGN=MIDDLE  ALT=\"-1=4a-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Square 2 to get 4.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0=4a-3%2B1 ALIGN=MIDDLE  ALT=\"0=4a-3%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" > to both sides.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4a=-3%2B1 ALIGN=MIDDLE  ALT=\"-4a=-3%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4a ALIGN=MIDDLE  ALT=\"4a\"   DISABLED_style= \"max-width:90%; height:auto;\" > from both sides.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4a=-2 ALIGN=MIDDLE  ALT=\"-4a=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms on the right side.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=%28-2%29%2F%28-4%29 ALIGN=MIDDLE  ALT=\"a=%28-2%29%2F%28-4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4 ALIGN=MIDDLE  ALT=\"-4\"   DISABLED_style= \"max-width:90%; height:auto;\" > to isolate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=1%2F2 ALIGN=MIDDLE  ALT=\"a=1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Reduce.\r<BR>\n" );
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document.write( "So this means that the equation is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%281%2F2%29%28x%2B1%29%5E2-3 ALIGN=MIDDLE  ALT=\"y=%281%2F2%29%28x%2B1%29%5E2-3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%281%2F2%29%28x%5E2%2B2x%2B1%29-3 ALIGN=MIDDLE  ALT=\"y=%281%2F2%29%28x%5E2%2B2x%2B1%29-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > FOIL\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%281%2F2%29x%5E2%2B%281%2F2%292x%2B%281%2F2%291-3 ALIGN=MIDDLE  ALT=\"y=%281%2F2%29x%5E2%2B%281%2F2%292x%2B%281%2F2%291-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Distribute\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%281%2F2%29x%5E2%2Bx%2B1%2F2-3 ALIGN=MIDDLE  ALT=\"y=%281%2F2%29x%5E2%2Bx%2B1%2F2-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%281%2F2%29x%5E2%2Bx-5%2F2 ALIGN=MIDDLE  ALT=\"y=%281%2F2%29x%5E2%2Bx-5%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms.\r<BR>\n" );
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document.write( "So the equation that goes through the points is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%281%2F2%29x%5E2%2Bx-5%2F2 ALIGN=MIDDLE  ALT=\"y=%281%2F2%29x%5E2%2Bx-5%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "From the equation, we can see that the degree of the polynomial is even. Also, we can see that the leading coefficient is positive since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=1%2F2 ALIGN=MIDDLE  ALT=\"a=1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Finally, if we use the quadratic formula, we'll find that the polynomial <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%281%2F2%29x%5E2%2Bx-5%2F2 ALIGN=MIDDLE  ALT=\"y=%281%2F2%29x%5E2%2Bx-5%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > has 2 zeros.\r<BR>\n" );
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