document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=86032>Question 86032</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I need help with this homework:  use transformations on the graph of y=x^2 to graph y=(x+3)-3.  I do not understand any of the problems related to this in this in the lecture notes. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #62153 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=62153\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> If you want to shift any function, here is the general rule:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x-a%29%2Bk ALIGN=MIDDLE  ALT=\"f%28x-a%29%2Bk\"   DISABLED_style= \"max-width:90%; height:auto;\" > where a represents the units shifted horizontally and k represents the units shifted veritcally\r<BR>\n" );
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document.write( "note: notice the negative \"a\". This means if you have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x-3%29 ALIGN=MIDDLE  ALT=\"f%28x-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > it means shift 3 units in the <b>positive</b> direction. If you have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%2B3%29 ALIGN=MIDDLE  ALT=\"f%28x%2B3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > it means shift 3 units in the <b>negative</b> direction. So just remember that the negative is telling you to shift in the <b>opposite</b> direction.\r<BR>\n" );
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document.write( "<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;\" > Start with the given  function\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%2B3%29=%28x%2B3%29%5E2 ALIGN=MIDDLE  ALT=\"f%28x%2B3%29=%28x%2B3%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >Replace x with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B3 ALIGN=MIDDLE  ALT=\"x%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" > to shift the graph 3 units to the left\r<BR>\n" );
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document.write( "Here is the graph\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28+500%2C+500%2C+-5%2C+5%2C+-5%2C+5%2C%0D%0A+++++++++grid%281%29%2C%0D%0A+++++++++arrow%280%2C0%2C-3%2C0%29%2C%0D%0A+++++++++graph%28+500%2C+500%2C+-5%2C+5%2C+-5%2C+5%2Cx%5E2%2C+%28x%2B3%29%5E2%29%2C%0D%0A+++++++++circle%28-3%2C0%2C0.05%29%2C%0D%0A+++++++++circle%28-3%2C0%2C0.08%29%2C%0D%0A+++++++++circle%280%2C0%2C0.05%29%2C%0D%0A+++++++++circle%280%2C0%2C0.08%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >graph of the original equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=x%5E2 ALIGN=MIDDLE  ALT=\"y=x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > (red) and the shifted equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28x%2B3%29%5E2 ALIGN=MIDDLE  ALT=\"y=%28x%2B3%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >(green)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%2B3%29-3=%28x%2B3%29%5E2-3 ALIGN=MIDDLE  ALT=\"f%28x%2B3%29-3=%28x%2B3%29%5E2-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Now subtract 3 from the whole equation to shift the graph down 3 units\r<BR>\n" );
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document.write( "Here is the graph\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28+500%2C+500%2C+-5%2C+5%2C+-5%2C+5%2C%0D%0A+++++++++grid%281%29%2C%0D%0A+++++++++arrow%280%2C0%2C-3%2C-3%29%2C%0D%0A+++++++++graph%28+500%2C+500%2C+-5%2C+5%2C+-5%2C+5%2Cx%5E2%2C+%28x%2B3%29%5E2-3%29%2C%0D%0A+++++++++circle%28-3%2C-3%2C0.05%29%2C%0D%0A+++++++++circle%28-3%2C-3%2C0.08%29%2C%0D%0A+++++++++circle%280%2C0%2C0.05%29%2C%0D%0A+++++++++circle%280%2C0%2C0.08%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >graph of the original equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=x%5E2 ALIGN=MIDDLE  ALT=\"y=x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > (red) and the shifted equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28x%2B3%29%5E2-3 ALIGN=MIDDLE  ALT=\"y=%28x%2B3%29%5E2-3\"   DISABLED_style= \"max-width:90%; height:auto;\" >(green)\r<BR>\n" );
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document.write( "So in short, to shift the graph 3 units to the left and shift the graph 3 units down,you simply use this composite function:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28x%29=f%28x%2B3%29-3 ALIGN=MIDDLE  ALT=\"g%28x%29=f%28x%2B3%29-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28x%29 ALIGN=MIDDLE  ALT=\"g%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is your new function </SPAN>\n" );
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