document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=177947>Question 177947</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Starting with the graph of y = e^x, find the equation of the graph that results from reflecting about the line y=5. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #132902 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=132902\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> In order to reflect about the line y=5, we first need to reflect about the line y=0 (which is much easier). To do that, simply shift the graph <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=e%5Ex ALIGN=MIDDLE  ALT=\"y=e%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" > 5 units down (to effectively move the line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=5 ALIGN=MIDDLE  ALT=\"y=5\"   DISABLED_style= \"max-width:90%; height:auto;\" > to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=0 ALIGN=MIDDLE  ALT=\"y=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >). Algebraically, you just subtract 5 from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5Ex ALIGN=MIDDLE  ALT=\"e%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=e%5Ex-5 ALIGN=MIDDLE  ALT=\"y=e%5Ex-5\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now to reflect over the line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=0 ALIGN=MIDDLE  ALT=\"y=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (the x-axis), simply replace \"y\" with \"-y\" to get\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-y=e%5Ex-5 ALIGN=MIDDLE  ALT=\"-y=e%5Ex-5\"   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=-e%5Ex%2B5 ALIGN=MIDDLE  ALT=\"y=-e%5Ex%2B5\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply both sides by -1 to make y positive.\r<BR>\n" );
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document.write( "Now since we shifted <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=e%5Ex ALIGN=MIDDLE  ALT=\"y=e%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" > down 5 units (ie subtracted 5), we need to shift <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-e%5Ex%2B5 ALIGN=MIDDLE  ALT=\"y=-e%5Ex%2B5\"   DISABLED_style= \"max-width:90%; height:auto;\" > back up (by adding 5). So add 5 to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-e%5Ex%2B5 ALIGN=MIDDLE  ALT=\"y=-e%5Ex%2B5\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-e%5Ex%2B5%2B5 ALIGN=MIDDLE  ALT=\"y=-e%5Ex%2B5%2B5\"   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=-e%5Ex%2B10 ALIGN=MIDDLE  ALT=\"y=-e%5Ex%2B10\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add\r<BR>\n" );
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document.write( "Answer:\r<BR>\n" );
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document.write( "So after reflecting <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=e%5Ex ALIGN=MIDDLE  ALT=\"y=e%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" > over the line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=5 ALIGN=MIDDLE  ALT=\"y=5\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we get the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-e%5Ex%2B10 ALIGN=MIDDLE  ALT=\"y=-e%5Ex%2B10\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Here's a graph to visually verify the answer\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C+exp%28x%29%2C+-exp%28x%29%2B10%2C5%29+ ALIGN=MIDDLE  ALT=\"+graph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C+exp%28x%29%2C+-exp%28x%29%2B10%2C5%29+\"   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=e%5Ex ALIGN=MIDDLE  ALT=\"y=e%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" > (red) and the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-e%5Ex%2B10 ALIGN=MIDDLE  ALT=\"y=-e%5Ex%2B10\"   DISABLED_style= \"max-width:90%; height:auto;\" > (green) which is a reflection over the line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=5 ALIGN=MIDDLE  ALT=\"y=5\"   DISABLED_style= \"max-width:90%; height:auto;\" > (blue) </SPAN>\n" );
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