document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=55044>Question 55044</A>: <I><SPAN STYLE=\"word-break: break-all;\"> f(x)=x^2-3 How do I solve this problem I have tried several times and I do not think I have the right answer </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #37271 by </B><A HREF=/tutors/aboutme.mpl?userid=Hook><B>Hook(36)</B></A><img src=\"/images/stars/stars-07.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Hook><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=37271\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> You can't \"solve\" this, per se. I think the problem asked you to \"graph\" <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+x%5E2-3 ALIGN=MIDDLE  ALT=\"f%28x%29+=+x%5E2-3\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
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document.write( "This is a parabola. Here's what I do for graphing parabolae:\r<BR>\n" );
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document.write( "I Find the real zeros (if any)<BR>\n" );
document.write( "II find the apex<BR>\n" );
document.write( "III figure out which way it opens<BR>\n" );
document.write( "IV find any additional points\r<BR>\n" );
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document.write( "Ok...on to the work\r<BR>\n" );
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document.write( "I   Find the real zeros. <BR>\n" );
document.write( "I'll make y=0 to see where the graph crosses the x-axis\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+=+x%5E2+-3 ALIGN=MIDDLE  ALT=\"0+=+x%5E2+-3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "adding three to both sides gives<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3+=+x%5E2 ALIGN=MIDDLE  ALT=\"3+=+x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "and if I take the square root, I get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%283%29=x ALIGN=MIDDLE  ALT=\"sqrt%283%29=x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-sqrt%283%29=x%29 ALIGN=MIDDLE  ALT=\"-sqrt%283%29=x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "(for the record, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%283%29 ALIGN=MIDDLE  ALT=\"sqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is close to 1.71...not easily graphable. \r<BR>\n" );
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document.write( "Ok...not much use.\r<BR>\n" );
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document.write( "II Find the apex. \r<BR>\n" );
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document.write( "the formula for the x-coordinate of the apex of a parabola 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;\" > For us, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+=+0 ALIGN=MIDDLE  ALT=\"b+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > so the x-coordinate must also be zero. Let's plug that in and see what we get:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+0%5E2+-+3 ALIGN=MIDDLE  ALT=\"y+=+0%5E2+-+3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+-3 ALIGN=MIDDLE  ALT=\"y+=+-3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The apex is at (0,-3)\r<BR>\n" );
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document.write( "III which way does it open:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%3E0 ALIGN=MIDDLE  ALT=\"a%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so it must open upwards\r<BR>\n" );
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document.write( "IV Plug in points. \r<BR>\n" );
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document.write( "At this point, we know we have an upward-opening parabola with an apex of (0,-3)<BR>\n" );
document.write( "Let's plug in a few points to get some points to draw. I'll pick a couple on each side of the apex. \r<BR>\n" );
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document.write( "I'll use x = [-4,-2,2,4] as a set of points to graph\r<BR>\n" );
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document.write( "by plugging those into the equation, I get y = [13, 1, 1, 13]\r<BR>\n" );
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document.write( "So, we've come up with five points (really, you only need three..the apex and one on each side of the apex). That should be enough to graph this parabola.\r<BR>\n" );
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document.write( "We've got (x,y) = (-4,13),(-2,1), (0,-3), (2,1), (4,13)\r<BR>\n" );
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document.write( "Plot those points, play connect the dots, and you get this:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph+%28300%2C200%2C-5%2C5%2C-5%2C15%2Cx%5E2-3%29 ALIGN=MIDDLE  ALT=\"graph+%28300%2C200%2C-5%2C5%2C-5%2C15%2Cx%5E2-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Here's probably an easier way to graph this:\r<BR>\n" );
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document.write( "Realize that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+x%5E2-3 ALIGN=MIDDLE  ALT=\"f%28x%29+=+x%5E2-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > is merely a \"standard\" parabola (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=x%5E2 ALIGN=MIDDLE  ALT=\"f%28x%29=x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > shifted down three units. Put a dot on the apex (0,-3). Go right one unit and up one unit (1,-2) and put a dot there. Go back to the apex. Go right two units and up four units to (2, 1). Repeat for the left side. That should be all ya need.  </SPAN>\n" );
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