document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=470687>Question 470687</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The function f is defined by f(x)=x^3 - 4x. The function of g is defined by     g(x)=f(x+h)+k where h and k are constants.  What is the value of hk?\r<BR>\n" );
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document.write( "Points for y=f(x)are (-1,3), (1,-3)<BR>\n" );
document.write( "Points for y=g(x) are (2,1), (4,-5)\r<BR>\n" );
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document.write( "The answer is 6, but I do not understand how they came up with this answer.  Please help! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #322857 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20850)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=322857\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> \r<BR>\n" );
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document.write( "First <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+f%28x%29+ ALIGN=MIDDLE  ALT=\"y+=+f%28x%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+x%5E3+-+4x ALIGN=MIDDLE  ALT=\"f%28x%29+=+x%5E3+-+4x\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+g%28x%29+ ALIGN=MIDDLE  ALT=\"y+=+g%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+=+f%28x+%2B+h%29+%2B+k ALIGN=MIDDLE  ALT=\"g%28x%29+=+f%28x+%2B+h%29+%2B+k\"   DISABLED_style= \"max-width:90%; height:auto;\" >  or \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+f%28x%29+=+x%5E3+-4x ALIGN=MIDDLE  ALT=\"y+=+f%28x%29+=+x%5E3+-4x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+g%28x%29+=+f%28x+%2B+h%29+%2B+k ALIGN=MIDDLE  ALT=\"y+=+g%28x%29+=+f%28x+%2B+h%29+%2B+k\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Looks to me like we have two equations for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" > which means we can solve them as a system. Notice that <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;\" >contains <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;\" >  but with a different input variable, namely <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x+%2B+h%29 ALIGN=MIDDLE  ALT=\"%28x+%2B+h%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
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document.write( "To find <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x+%2B+h%29 ALIGN=MIDDLE  ALT=\"f%28x+%2B+h%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > we replace<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x ALIGN=MIDDLE  ALT=\"+x\"   DISABLED_style= \"max-width:90%; height:auto;\" > with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x+%2B+h%29 ALIGN=MIDDLE  ALT=\"%28x+%2B+h%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > wherever we find an <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > in the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f ALIGN=MIDDLE  ALT=\"f\"   DISABLED_style= \"max-width:90%; height:auto;\" > equation.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+x%5E3+-+4x ALIGN=MIDDLE  ALT=\"f%28x%29+=+x%5E3+-+4x\"   DISABLED_style= \"max-width:90%; height:auto;\" > will become\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x+%2B+h%29+=+%28x+%2B+h%29%5E3+-+4%28x+%2B+h%29 ALIGN=MIDDLE  ALT=\"f%28x+%2B+h%29+=+%28x+%2B+h%29%5E3+-+4%28x+%2B+h%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So now we replace <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x+%2B+h%29 ALIGN=MIDDLE  ALT=\"f%28x+%2B+h%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > in <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;\" > with our new result.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=g%28x%29+=+f%28x+%2B+h%29+%2Bk ALIGN=MIDDLE  ALT=\"g%28x%29+=+f%28x+%2B+h%29+%2Bk\"   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=g%28x%29+=+%28+%28x+%2B+h%29%5E3+-+4%28x+%2B+h%29+%29+%2B+k ALIGN=MIDDLE  ALT=\"g%28x%29+=+%28+%28x+%2B+h%29%5E3+-+4%28x+%2B+h%29+%29+%2B+k\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Also, we are given a pair of points that work with <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;\" > so lets plot them in and get a pair of equations. The first number in each point is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+ ALIGN=MIDDLE  ALT=\"x+\"   DISABLED_style= \"max-width:90%; height:auto;\" >and the second is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Points in g(x) : (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" >) and (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5 ALIGN=MIDDLE  ALT=\"-5\"   DISABLED_style= \"max-width:90%; height:auto;\" >)\r<BR>\n" );
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document.write( "So for the first :\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1+=+%282+%2B+h%29%5E3+-+4%282+%2B+h%29+%2B+k ALIGN=MIDDLE  ALT=\"1+=+%282+%2B+h%29%5E3+-+4%282+%2B+h%29+%2B+k\"   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=1+=+%288+%2B+12h+%2B+6h%5E2+%2B+h%5E3%29+-+%288+%2B+4h%29+%2B+k ALIGN=MIDDLE  ALT=\"1+=+%288+%2B+12h+%2B+6h%5E2+%2B+h%5E3%29+-+%288+%2B+4h%29+%2B+k\"   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=1+=+8+%2B+12h+%2B+6h%5E2+%2B+h%5E3+-+8+-+4h+%2B+k ALIGN=MIDDLE  ALT=\"1+=+8+%2B+12h+%2B+6h%5E2+%2B+h%5E3+-+8+-+4h+%2B+k\"   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=1+=+h%5E3+%2B+6h%5E2+%2B+8h+%2B+k ALIGN=MIDDLE  ALT=\"1+=+h%5E3+%2B+6h%5E2+%2B+8h+%2B+k\"   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=1+-+%28h%5E3+%2B+6h%5E2+%2B+8h%29+=+k ALIGN=MIDDLE  ALT=\"1+-+%28h%5E3+%2B+6h%5E2+%2B+8h%29+=+k\"   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%5E3+-+6h%5E2+-+8h+%2B+1+=+k ALIGN=MIDDLE  ALT=\"-+h%5E3+-+6h%5E2+-+8h+%2B+1+=+k\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "and for the second :\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5+=+%284+%2B+h%29%5E3+-+4%284+%2B+h%29+%2B+k ALIGN=MIDDLE  ALT=\"-5+=+%284+%2B+h%29%5E3+-+4%284+%2B+h%29+%2B+k\"   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=-5+=+%2864+%2B+48h+%2B+12h%5E2+%2B+h%5E3%29+-+%2816+%2B+4h%29+%2B+k ALIGN=MIDDLE  ALT=\"-5+=+%2864+%2B+48h+%2B+12h%5E2+%2B+h%5E3%29+-+%2816+%2B+4h%29+%2B+k\"   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=-5+=+64+%2B+48h+%2B+12h%5E2+%2B+h%5E3+-+16+-+4h+%2B+k ALIGN=MIDDLE  ALT=\"-5+=+64+%2B+48h+%2B+12h%5E2+%2B+h%5E3+-+16+-+4h+%2B+k\"   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=-5+=+h%5E3+%2B+12h%5E2+%2B+44h+%2B+48+%2B+k ALIGN=MIDDLE  ALT=\"-5+=+h%5E3+%2B+12h%5E2+%2B+44h+%2B+48+%2B+k\"   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=-5+-+%28h%5E3+%2B+12h%5E2+%2B+44h+%2B+48%29+=+k ALIGN=MIDDLE  ALT=\"-5+-+%28h%5E3+%2B+12h%5E2+%2B+44h+%2B+48%29+=+k\"   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%5E3+-+12h%5E2+-+44h+-+53+=+k ALIGN=MIDDLE  ALT=\"-+h%5E3+-+12h%5E2+-+44h+-+53+=+k\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Notice that both long equations are equal to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k ALIGN=MIDDLE  ALT=\"k\"   DISABLED_style= \"max-width:90%; height:auto;\" >. That means they are also equal to each other.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-+h%5E3+-+6h%5E2+-+8h+%2B+1+=+-+h%5E3+-+12h%5E2+-+44h+-+53 ALIGN=MIDDLE  ALT=\"-+h%5E3+-+6h%5E2+-+8h+%2B+1+=+-+h%5E3+-+12h%5E2+-+44h+-+53\"   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%5E3+-+6h%5E2+-+8h+%2B+1+-+%28-+h%5E3+-+12h%5E2+-+44h+-+53%29+=+0 ALIGN=MIDDLE  ALT=\"-+h%5E3+-+6h%5E2+-+8h+%2B+1+-+%28-+h%5E3+-+12h%5E2+-+44h+-+53%29+=+0\"   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%5E3+-+6h%5E2+-+8h+%2B+1+%2B+h%5E3+%2B+12h%5E2+%2B+44h+%2B+53+=+0 ALIGN=MIDDLE  ALT=\"-+h%5E3+-+6h%5E2+-+8h+%2B+1+%2B+h%5E3+%2B+12h%5E2+%2B+44h+%2B+53+=+0\"   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=6h%5E2+%2B+36h+%2B+54+=+0 ALIGN=MIDDLE  ALT=\"6h%5E2+%2B+36h+%2B+54+=+0\"   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=6%28h%5E2+%2B+6h+%2B+9%29+=+0 ALIGN=MIDDLE  ALT=\"6%28h%5E2+%2B+6h+%2B+9%29+=+0\"   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=%28h+%2B+3%29%28h+%2B+3%29+=+0+%2F+6 ALIGN=MIDDLE  ALT=\"%28h+%2B+3%29%28h+%2B+3%29+=+0+%2F+6\"   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+%2B+3+=+0 ALIGN=MIDDLE  ALT=\"h+%2B+3+=+0\"   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+=+-3 ALIGN=MIDDLE  ALT=\"h+=+-3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+h ALIGN=MIDDLE  ALT=\"+h\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3 ALIGN=MIDDLE  ALT=\"-3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, now we need <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k ALIGN=MIDDLE  ALT=\"k\"   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=g%28x%29+=+%28+%28x+%2B+h%29%5E3+-+4%28x+%2B+h%29+%29+%2B+k ALIGN=MIDDLE  ALT=\"g%28x%29+=+%28+%28x+%2B+h%29%5E3+-+4%28x+%2B+h%29+%29+%2B+k\"   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=y+=+%28+%28x+%2B+h%29%5E3+-+4%28x+%2B+h%29+%29+%2B+k ALIGN=MIDDLE  ALT=\"y+=+%28+%28x+%2B+h%29%5E3+-+4%28x+%2B+h%29+%29+%2B+k\"   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=1+=%28+%282+-+3%29%5E3+-+4%282+-+3%29%29+%2B+k ALIGN=MIDDLE  ALT=\"1+=%28+%282+-+3%29%5E3+-+4%282+-+3%29%29+%2B+k\"   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=1+=+-1%5E3+-+4%28-1%29+%2B+k ALIGN=MIDDLE  ALT=\"1+=+-1%5E3+-+4%28-1%29+%2B+k\"   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=1+=+-1+%2B+4+%2B+k ALIGN=MIDDLE  ALT=\"1+=+-1+%2B+4+%2B+k\"   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=1+=+3+%2B+k ALIGN=MIDDLE  ALT=\"1+=+3+%2B+k\"   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=-2+=+k ALIGN=MIDDLE  ALT=\"-2+=+k\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k+ ALIGN=MIDDLE  ALT=\"k+\"   DISABLED_style= \"max-width:90%; height:auto;\" >is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2 ALIGN=MIDDLE  ALT=\"-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. We already know that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h ALIGN=MIDDLE  ALT=\"h\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3 ALIGN=MIDDLE  ALT=\"-3\"   DISABLED_style= \"max-width:90%; height:auto;\" >. so :\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=hk+=+-3%2A-2=6 ALIGN=MIDDLE  ALT=\"hk+=+-3%2A-2=6\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );