document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=109093>Question 109093</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help with the following as functions is a concept I'm struggling to grasp.  <BR>\n" );
document.write( "if f(t+15) = t^2 - 10, find f(t).  <BR>\n" );
document.write( "I believe this has to deal with tranformations and going in reverse order. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #79557 by </B><A HREF=/tutors/aboutme.mpl?userid=bucky><B>bucky(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=bucky><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=79557\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Try this as the answer:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "f(t) = t^2 - 30t + 215<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Here's the way I worked it:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Since f(t+15) has a t^2 term, then f(t) must also have a t^2 term in it so that when you<BR>\n" );
document.write( "substitute (t+15) for t you end up with a t^2 plus additional terms.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Then if f(t) has contains a t^2 term, when you substitute (t+15) for the t in the t^2 term<BR>\n" );
document.write( "you get (t+15)^2 and this squares out to be t^2 + 30t + 225.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Since the answer for f(t+15) contains no term with just \"t\", the +30t must be eliminated.<BR>\n" );
document.write( "This tells you that f(t) must have a -30t term so that when you substitute (t+15) into<BR>\n" );
document.write( "the -30t term of f(t) you get -30t -30*15 = -30t -450.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "So far, we've deduced that f(t) consists of t^2 - 30t. If you substitute (t+15) into this<BR>\n" );
document.write( "partial deduced form you get:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "f(t+15) = (t+15)^2 - 30(t+15) <BR>\n" );
document.write( ".<BR>\n" );
document.write( "and this works out to be:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "f(t+15) = (t^2 + 30t + 225) - 30t - 450<BR>\n" );
document.write( ".<BR>\n" );
document.write( "If you combine the like terms (+30t and -30t = 0) and (+225 and -450 = -225) this reduces<BR>\n" );
document.write( "to:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "f(t+15) = t^2  - 225<BR>\n" );
document.write( ".<BR>\n" );
document.write( "But the problem tells you that f(t+15) should be t^2 - 10.  Therefore f(t) must contain<BR>\n" );
document.write( "a term +215 so that it combines with the -225 to create the -10<BR>\n" );
document.write( ".<BR>\n" );
document.write( "So the deduction is that:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "f(t) = t^2 - 30t + 215<BR>\n" );
document.write( ".<BR>\n" );
document.write( "To check this out, let's find f(t+15) starting with f(t) = t^2 - 30t + 215<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Begin by substituting (t+15) for every t in f(t). When you do you get:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "f(t+15) = (t+15)^2 - 30(t+15) + 215<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Square out and multiply out the terms on the right side and you get:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "f(t+15) = t^2 + 30t + 225 - 30t -450 + 215<BR>\n" );
document.write( ".<BR>\n" );
document.write( "As before, the +30t and the -30t cancel each other out. And now the +225 - 450 and +215<BR>\n" );
document.write( "add to -10.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Therefore, f(t+15) equals t^2 - 10, just as the problem says it should be.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "So the answer is that f(t) = t^2 - 30t + 215<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Hope this helps you to understand the problem. It's hard to explain, and probably is<BR>\n" );
document.write( "confusing as well. If you can't understand what I did, repost the problem and maybe another<BR>\n" );
document.write( "tutor can do a better job of providing a more understandable process for you to use in<BR>\n" );
document.write( "getting the answer.<BR>\n" );
document.write( ". </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );