document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=116009>Question 116009</A>: <I><SPAN STYLE=\"word-break: break-all;\"> CAN SOMEONE HELP ME SOLVE THIS GRAPHING?<BR>\n" );
document.write( " 1. DESCRIBE THE TRANSFORMATIONSON ON A GRAPH OF f(x)=e^x. State the placement of the horizonal asympotote and y-intercept after the transformation. For example,\"left 1\" or \"rotated about the y-axis\" are descriptions<BR>\n" );
document.write( "   a) g(x)=e^x+3<BR>\n" );
document.write( "     description of transformation:<BR>\n" );
document.write( "     Horizonal asymptote:<BR>\n" );
document.write( "     y-intercept in (x,y) form:<BR>\n" );
document.write( "    b)H(x)=e^-x<BR>\n" );
document.write( "     description of transformation:<BR>\n" );
document.write( "     Horizonal asymptote:<BR>\n" );
document.write( "     y-intercept in (x,y)form<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 #84400 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=84400\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> a)\r<BR>\n" );
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document.write( "<b>Description of transformation</b>:\r<BR>\n" );
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document.write( "Remember, <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;\" > is the same as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So this means  <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;\" >.\r<BR>\n" );
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document.write( "So when we say <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5Ex%2B3 ALIGN=MIDDLE  ALT=\"e%5Ex%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we're also saying <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%2B3 ALIGN=MIDDLE  ALT=\"y%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" > (replace <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;\" > with y). So this means we're adding 3 to each y value which graphically shows us that we're shifting each y value up 3 units. \r<BR>\n" );
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document.write( "Answer:\r<BR>\n" );
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document.write( "So the transformation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28x%29=e%5Ex%2B3 ALIGN=MIDDLE  ALT=\"g%28x%29=e%5Ex%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" > simply shifts the entire curve up 3 units.\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%28x%29%2B3%29+ ALIGN=MIDDLE  ALT=\"+graph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C+exp%28x%29%2Cexp%28x%29%2B3%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%5Ex%2B3 ALIGN=MIDDLE  ALT=\"h%28x%29=e%5Ex%2B3\"   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( "Now if we found the asymptote of  <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;\" >, we would find that the 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;\" >. Since we're translating each point on  <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;\" > up 3 units, we're also translating the horizontal asymptote up 3 units. \r<BR>\n" );
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document.write( "Answer:\r<BR>\n" );
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document.write( "So the new horizontal asymptote is <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;\" >.\r<BR>\n" );
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document.write( "Also, 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( "If we let x=0 and plug it into <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;\" >, we get\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=e%5E0 ALIGN=MIDDLE  ALT=\"y=e%5E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in x=0\r<BR>\n" );
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document.write( "<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;\" > Raise e to the zeroth power to get one. Remember any number x to the zeroth power is always one (ie <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E0=1 ALIGN=MIDDLE  ALT=\"x%5E0=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >)\r<BR>\n" );
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document.write( "So for <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;\" > the y-intercept is (0,1)\r<BR>\n" );
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document.write( "Now if <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;\" > translates each y value up 3 units, then simply add 3 to the y-coordinate of the y-intercept to get\r<BR>\n" );
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document.write( "(0,1+3)---->(0,4)\r<BR>\n" );
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document.write( "Answer:\r<BR>\n" );
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document.write( "So the new y-intercept is (0,4)\r<BR>\n" );
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document.write( "Once again, you can visually verify this answer by looking at the graph above\r<BR>\n" );
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document.write( "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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