document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=924774>Question 924774</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The exponential function y=4^x has been compressed vertically by a factor of 1/3, stretched horizontally by a factor of 5, and reflected in the x-axis. Its asymptote is the line y=3. Its y-intercept is 2 2/3. Write the equation of the transformed function. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #561078 by </B><A HREF=/tutors/aboutme.mpl?userid=Edwin+McCravy><B>Edwin McCravy(20060)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Edwin+McCravy><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/cgi-bin/show-face.mpl?user=Edwin+McCravy><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=561078\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%22=%22%22 ALIGN=MIDDLE  ALT=\"%22%22=%22%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%5Ex ALIGN=MIDDLE  ALT=\"4%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "That has this red graph:\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C720%2C-5%2C15%2C-18%2C18%2C+graph%28400%2C720%2C-5%2C15%2C-18%2C18%2C4%5Ex%29+%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "Now we'll compress the red graph vertically by 1/3 multiplying the right<BR>\n" );
document.write( "side by 1/3.  That's the green graph.  Notice that it is lower by 1/3 than <BR>\n" );
document.write( "the red graph.  It may not look so at the top, but remember that the red<BR>\n" );
document.write( "graph is cut off at the top and actual goes three times as high as the<BR>\n" );
document.write( "green graph:\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%22=%22%22 ALIGN=MIDDLE  ALT=\"%22%22=%22%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=expr%281%2F3%294%5Ex ALIGN=MIDDLE  ALT=\"expr%281%2F3%294%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C720%2C-5%2C15%2C-18%2C18%2C+graph%28400%2C720%2C-5%2C15%2C-18%2C18%2C4%5Ex%2C%284%5Ex%29%2F3%29+%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "Now we'll stretch the green graph horizontally by a factor of 5 by replacing<BR>\n" );
document.write( "x in the right sides by x/5.  That's the blue graph below. Notice that it<BR>\n" );
document.write( "sticks 5 times as far to the right as the green graph. Its equation is:\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=matrix%282%2C3%2C%22%22%2C%22%22%2C%22%22%2Cy%2C%22%22=%22%22%2Cexpr%281%2F3%294%5E%28x%2F5%29+%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C720%2C-5%2C15%2C-18%2C18%2C+graph%28400%2C720%2C-5%2C15%2C-18%2C18%2C4%5Ex%2C%284%5Ex%29%2F3%2C%284%5E%28x%2F5%29%29%2F3%29+%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
document.write( "\n" );
document.write( "Next we refect the blue graph in the x-axis by multiplying the right side<BR>\n" );
document.write( "by -1.  That's the purple graph below.  It equation is:\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=matrix%282%2C3%2C%22%22%2C%22%22%2C%22%22%2Cy%2C%22%22=%22%22%2Cexpr%28-1%2F3%294%5E%28x%2F5%29+%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C720%2C-5%2C15%2C-18%2C18%2C+graph%28400%2C720%2C-5%2C15%2C-18%2C18%2C4%5Ex%2C%284%5Ex%29%2F3%2C%284%5E%28x%2F5%29%29%2F3%2C%28-4%5E%28x%2F5%29%29%2F3%29%29+%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
document.write( "\n" );
document.write( "Next we make the horizontal asymptote of the purple graph become y=3 instead<BR>\n" );
document.write( "of the x-axis, y=0.  The graph is getting too cluttered, so I'll erase all the<BR>\n" );
document.write( "graphs except the purple one.  The purple graph shifted upward so that the<BR>\n" );
document.write( "horizontal asymptote is y=3 is obtained by adding 3 to the right side of the<BR>\n" );
document.write( "equation.  That graph is the greenish-grey one.  The red dotted line is the<BR>\n" );
document.write( "new horizontal asymptote y=3.  The equation of the greenish-grey graph is:\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=matrix%282%2C3%2C%22%22%2C%22%22%2C%22%22%2Cy%2C%22%22=%22%22%2Cexpr%28-1%2F3%294%5E%28x%2F5%29%2B3+%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C720%2C-5%2C15%2C-18%2C18%2C+graph%28400%2C720%2C-5%2C15%2C-18%2C18%2C%0D%0A%0D%0A3%2Asqrt%28sin%289x%29%29%2Fsqrt%28sin%289x%29%29%0D%0A%0D%0A%2C20%2C20%2C%28-4%5E%28x%2F5%29%29%2F3%2C%28-4%5E%28x%2F5%29%29%2F3%2B3%29+%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "Finally it is to have the y-intercept of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%262%2F3 ALIGN=MIDDLE  ALT=\"2%262%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
document.write( "\n" );
document.write( "Let's see what the y-intercept of the greenish-grey graph is.<BR>\n" );
document.write( "We do that by substituting 0 for x:\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=matrix%282%2C3%2C%22%22%2C%22%22%2C%22%22%2Cy%2C%22%22=%22%22%2Cexpr%28-1%2F3%294%5E%280%2F5%29%2B3+%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=matrix%282%2C3%2C%22%22%2C%22%22%2C%22%22%2Cy%2C%22%22=%22%22%2Cexpr%28-1%2F3%294%5E0%2B3+%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=matrix%282%2C3%2C%22%22%2C%22%22%2C%22%22%2Cy%2C%22%22=%22%22%2Cexpr%28-1%2F3%29%281%29%2B3+%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=matrix%282%2C3%2C%22%22%2C%22%22%2C%22%22%2Cy%2C%22%22=%22%22%2C-1%2F3%2B3+%29 ALIGN=MIDDLE  ALT=\"matrix%282%2C3%2C%22%22%2C%22%22%2C%22%22%2Cy%2C%22%22=%22%22%2C-1%2F3%2B3+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=matrix%282%2C3%2C%22%22%2C%22%22%2C%22%22%2Cy%2C%22%22=%22%22%2C2%262%2F3+%29 ALIGN=MIDDLE  ALT=\"matrix%282%2C3%2C%22%22%2C%22%22%2C%22%22%2Cy%2C%22%22=%22%22%2C2%262%2F3+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "Wow!  It already has that y-intercept!  So we don't need to<BR>\n" );
document.write( "stretch, compress, reflect or shift it.  The greenish-grey graph is<BR>\n" );
document.write( "the graph of the equation we were looking for.\r<BR>\n" );
document.write( "\n" );
document.write( "The answer is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=matrix%282%2C3%2C%22%22%2C%22%22%2C%22%22%2Cy%2C%22%22=%22%22%2Cexpr%28-1%2F3%294%5E%28x%2F5%29%2B3+%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "Edwin</pre> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );