document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1190912>Question 1190912</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Explain the transformations you would need to apply to the graph of y=f(x) to graph the function\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "y=-f(1/4x+6) -7 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #822672 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13216)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=822672\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "For clarity, rewrite the new function as<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-f%28%281%2F4%29x%2B6%29-7 ALIGN=MIDDLE  ALT=\"y=-f%28%281%2F4%29x%2B6%29-7\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Then rewrite the new function in the form<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=a%2Af%28b%28x-c%29%29%2Bd ALIGN=MIDDLE  ALT=\"y=a%2Af%28b%28x-c%29%29%2Bd\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The transformations will be, in order,<br><BR>\n" );
document.write( "(1) translation of c units<BR>\n" );
document.write( "(2) horizontal compression by a factor of b<BR>\n" );
document.write( "(3) vertical stretch by a factor of a<BR>\n" );
document.write( "(4) vertical translation by d<br><BR>\n" );
document.write( "Note the \"(1/4)x+6\" needs to be rewritten as \"(1/4)(x+24)\".<br><BR>\n" );
document.write( "The rest is straightforward....<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-f%28%281%2F4%29x%2B6%29-7 ALIGN=MIDDLE  ALT=\"y=-f%28%281%2F4%29x%2B6%29-7\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28-1%29%28f%28%281%2F4%29%28x-%28-24%29%29%29%29%2B%28-7%29 ALIGN=MIDDLE  ALT=\"y=%28-1%29%28f%28%281%2F4%29%28x-%28-24%29%29%29%29%2B%28-7%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "That gives us a=-1, b=1/4, c=-24, and d=-7<br><BR>\n" );
document.write( "The transformations are then, in order,<BR>\n" );
document.write( "(1) translation 24 units left<BR>\n" );
document.write( "(2) horizontal compression by a factor of 1/4<BR>\n" );
document.write( "(3) vertical stretch by a factor of -1 (i.e., reflect over the x-axis)<BR>\n" );
document.write( "(4) vertical translation down 7<br><BR>\n" );
document.write( "-----------------------------------------------------------------------<br><BR>\n" );
document.write( "In reply to the student's comment....<br><BR>\n" );
document.write( "A horizontal compression by a factor of a is a horizontal stretch by a factor of 1/a.  So in this example, the compression by a factor of 1/4 is a stretch by a factor of 4.<br><BR>\n" );
document.write( "That is the correct interpretation.<br><BR>\n" );
document.write( "As an example, consider f(x)=sin((1/4)x).  The compression factor is 1/4 which means there is a horizontal stretch by a factor of 4.<br><BR>\n" );
document.write( "This is correct, because the function sin((1/4)x) completes a cycle every 8pi (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%3C=%281%2F4%29x%3C-2pi ALIGN=MIDDLE  ALT=\"0%3C=%281%2F4%29x%3C-2pi\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%3C=x%3C=8pi ALIGN=MIDDLE  ALT=\"0%3C=x%3C=8pi\"   DISABLED_style= \"max-width:90%; height:auto;\" >), while the function sin(x) completes a cycle every 2pi -- the graph is stretched horizontally by a factor of 4.<br><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );