document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=116018>Question 116018</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Describe the transformations on the following graph of f(x)=e^x.  \r<BR>\n" );
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document.write( "State the placement of the horizontal asymptote and y-intercept after the transformation.  For example, “left 1” or “rotated about the y-axis” are descriptions.\r<BR>\n" );
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document.write( "b) h(x)=e^(-x) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #84388 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=84388\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> b)\r<BR>\n" );
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document.write( "<b>Description of transformation</b>:\r<BR>\n" );
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document.write( "Looking at  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%28x%29=e%5E%28-x%29 ALIGN=MIDDLE  ALT=\"h%28x%29=e%5E%28-x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, notice how the exponent is negated. So let's see what affect this transformation has on  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=e%5Ex ALIGN=MIDDLE  ALT=\"f%28x%29=e%5Ex\"   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=h%28x%29=e%5E%28-x%29 ALIGN=MIDDLE  ALT=\"h%28x%29=e%5E%28-x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given transformation \r<BR>\n" );
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document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%28-2%29=e%5E%28-%28-2%29%29 ALIGN=MIDDLE  ALT=\"h%28-2%29=e%5E%28-%28-2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in x=-2\r<BR>\n" );
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document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%28-2%29=e%5E%282%29 ALIGN=MIDDLE  ALT=\"h%28-2%29=e%5E%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Negate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-%28-2%29 ALIGN=MIDDLE  ALT=\"-%28-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get 2\r<BR>\n" );
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document.write( "Notice if we plug in x=2 into <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=e%5Ex ALIGN=MIDDLE  ALT=\"f%28x%29=e%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" >,  we get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%282%29=e%5E2 ALIGN=MIDDLE  ALT=\"f%282%29=e%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "--------\r<BR>\n" );
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document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%28x%29=e%5E%28-x%29 ALIGN=MIDDLE  ALT=\"h%28x%29=e%5E%28-x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given transformation \r<BR>\n" );
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document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%28-1%29=e%5E%28-%28-1%29%29 ALIGN=MIDDLE  ALT=\"h%28-1%29=e%5E%28-%28-1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in x=-1\r<BR>\n" );
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document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%28-1%29=e%5E%281%29 ALIGN=MIDDLE  ALT=\"h%28-1%29=e%5E%281%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Negate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-%28-1%29 ALIGN=MIDDLE  ALT=\"-%28-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get 1\r<BR>\n" );
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document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%28-1%29=e ALIGN=MIDDLE  ALT=\"h%28-1%29=e\"   DISABLED_style= \"max-width:90%; height:auto;\" > Remove the exponent of 1\r<BR>\n" );
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document.write( "Notice if we plug in x=1 into <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=e%5Ex ALIGN=MIDDLE  ALT=\"f%28x%29=e%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" >,  we get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%281%29=e%5E1=e ALIGN=MIDDLE  ALT=\"f%281%29=e%5E1=e\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So if we take the opposite of x (to get -x), and plug that into g(x), we'll get the same f(x) answer.\r<BR>\n" );
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document.write( "Answer:\r<BR>\n" );
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document.write( "So what this does is simply reflect the entire graph over the y-axis\r<BR>\n" );
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document.write( "Notice if we graph <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29 ALIGN=MIDDLE  ALT=\"f%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <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;\" >, we get\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%2Cexp%28-x%29%29+ ALIGN=MIDDLE  ALT=\"+graph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C+exp%28x%29%2Cexp%28-x%29%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >  Graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=e%5Ex ALIGN=MIDDLE  ALT=\"f%28x%29=e%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" > (red) and  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%28x%29=e%5E%28-x%29 ALIGN=MIDDLE  ALT=\"h%28x%29=e%5E%28-x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (green)\r<BR>\n" );
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document.write( "and we can visually verify the transformation\r<BR>\n" );
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document.write( "<b>Horizontal Asymptote</b>:\r<BR>\n" );
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document.write( "Since we reflected the graph with respect to the y-axis, the horizontal asymptote of  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%28x%29=e%5E%28-x%29 ALIGN=MIDDLE  ALT=\"h%28x%29=e%5E%28-x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the same as  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=e%5Ex ALIGN=MIDDLE  ALT=\"f%28x%29=e%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" > (you can see this from the graph above) \r<BR>\n" );
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document.write( "Answer:\r<BR>\n" );
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document.write( "So the horizontal asymptote is <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;\" >\r<BR>\n" );
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document.write( "Note: you can visually verify this answer by looking at the graph above\r<BR>\n" );
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document.write( "<b>y-intercept in (x, y) form</b>:\r<BR>\n" );
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document.write( "Since the line of symmetry between the two graphs is the line x=0 (ie the y axis), this means that the point that intersects with the y-axis is reflected to itself. So essentially the y-intercept does not change also. Once again, you can visually verify this using the graph above.\r<BR>\n" );
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document.write( "Answer:\r<BR>\n" );
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document.write( "So the y-intercept is (0,1)\r<BR>\n" );
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document.write( "Once again, you can visually verify this answer by looking at the graph above<BR>\n" );
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document.write( " </SPAN>\n" );
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