document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=155784>Question 155784</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Graph the follwing functions:<BR>\n" );
document.write( " <BR>\n" );
document.write( "e) y= |-5x| and y= 5|x| </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #114733 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=114733\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> First, lets graph <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=abs%28-5x%29 ALIGN=MIDDLE  ALT=\"y=abs%28-5x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Remember, we can break <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28-5x%29 ALIGN=MIDDLE  ALT=\"abs%28-5x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > into <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5x ALIGN=MIDDLE  ALT=\"-5x\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-%28-5x%29=5x ALIGN=MIDDLE  ALT=\"-%28-5x%29=5x\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So first graph <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-5x ALIGN=MIDDLE  ALT=\"y=-5x\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28500%2C500%2C-10%2C10%2C-8%2C12%2C-5x%29 ALIGN=MIDDLE  ALT=\"graph%28500%2C500%2C-10%2C10%2C-8%2C12%2C-5x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-5x ALIGN=MIDDLE  ALT=\"y=-5x\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now graph <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=5x ALIGN=MIDDLE  ALT=\"y=5x\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28500%2C500%2C-10%2C10%2C-8%2C12%2C5x%29 ALIGN=MIDDLE  ALT=\"graph%28500%2C500%2C-10%2C10%2C-8%2C12%2C5x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=5x ALIGN=MIDDLE  ALT=\"y=5x\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Together, they look like this:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28500%2C500%2C-10%2C10%2C-8%2C12%2C-5x%2C5x%29 ALIGN=MIDDLE  ALT=\"graph%28500%2C500%2C-10%2C10%2C-8%2C12%2C-5x%2C5x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-5x ALIGN=MIDDLE  ALT=\"y=-5x\"   DISABLED_style= \"max-width:90%; height:auto;\" > (red) and graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=5x ALIGN=MIDDLE  ALT=\"y=5x\"   DISABLED_style= \"max-width:90%; height:auto;\" > (green)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now because the sign outside the absolute value is positive (ie there is no sign), this tells us to only draw the upper portion like this:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28500%2C500%2C-10%2C10%2C-8%2C12%2Cabs%28-5x%29%29 ALIGN=MIDDLE  ALT=\"graph%28500%2C500%2C-10%2C10%2C-8%2C12%2Cabs%28-5x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=abs%28-5x%29 ALIGN=MIDDLE  ALT=\"y=abs%28-5x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now let's graph <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=5%2Aabs%28x%29 ALIGN=MIDDLE  ALT=\"y=5%2Aabs%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. To do this, first graph <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28x%29 ALIGN=MIDDLE  ALT=\"abs%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (which is simply a \"v\" shaped graph):\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28500%2C500%2C-10%2C10%2C-8%2C12%2Cabs%28x%29%29 ALIGN=MIDDLE  ALT=\"graph%28500%2C500%2C-10%2C10%2C-8%2C12%2Cabs%28x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=abs%28x%29 ALIGN=MIDDLE  ALT=\"y=abs%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now vertically stretch the graph by a factor of 5 to get the graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=5%2Aabs%28x%29 ALIGN=MIDDLE  ALT=\"y=5%2Aabs%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28500%2C500%2C-10%2C10%2C-8%2C12%2C5%2Aabs%28x%29%29 ALIGN=MIDDLE  ALT=\"graph%28500%2C500%2C-10%2C10%2C-8%2C12%2C5%2Aabs%28x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=5%2Aabs%28x%29 ALIGN=MIDDLE  ALT=\"y=5%2Aabs%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Notice that if you graph the two equations together you get\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28500%2C500%2C-10%2C10%2C-8%2C12%2Cabs%28-5x%29%2C5%2Aabs%28x%29%29 ALIGN=MIDDLE  ALT=\"graph%28500%2C500%2C-10%2C10%2C-8%2C12%2Cabs%28-5x%29%2C5%2Aabs%28x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >  Graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=abs%28-5x%29 ALIGN=MIDDLE  ALT=\"y=abs%28-5x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (red) Graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=5%2Aabs%28x%29 ALIGN=MIDDLE  ALT=\"y=5%2Aabs%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (green)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So this means that the two graphs are identical (since one is right on top of the other)<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );