document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1127178>Question 1127178</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve for the logistic function with initial condition = 14, limit to growth = 42 and passing through (1, 28)\r<BR>\n" );
document.write( "\n" );
document.write( "Thank you! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #743541 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13203)</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=743541\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "The standard form of a logistic function is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+c%2F%281%2Bab%5Ex%29 ALIGN=MIDDLE  ALT=\"f%28x%29+=+c%2F%281%2Bab%5Ex%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The \"ab^x\" in the denominator is a decreasing exponential function; as x gets very large that exponential goes to zero, so the denominator goes to 1.  That means the numerator c is the limiting value of the function.<br><BR>\n" );
document.write( "So in this example we know the function is of the form<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+42%2F%281%2Bab%5Ex%29 ALIGN=MIDDLE  ALT=\"f%28x%29+=+42%2F%281%2Bab%5Ex%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The initial value of the function is when x is 0.  When x is 0, the denominator of the function is just 1+a.  Use the given initial value to find the value of a.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%280%29+=+42%2F%281%2Bab%5E0%29+=+42%2F%281%2Ba%29+=+14 ALIGN=MIDDLE  ALT=\"f%280%29+=+42%2F%281%2Bab%5E0%29+=+42%2F%281%2Ba%29+=+14\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Ba+=+3 ALIGN=MIDDLE  ALT=\"1%2Ba+=+3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+2 ALIGN=MIDDLE  ALT=\"a+=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Now we know the function is of the form<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+42%2F%281%2B2b%5Ex%29 ALIGN=MIDDLE  ALT=\"f%28x%29+=+42%2F%281%2B2b%5Ex%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Now, to finish finding the logistic function, find the value of b by using the given data point.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%281%29+=+42%2F%281%2B2b%5E1%29+=+28 ALIGN=MIDDLE  ALT=\"f%281%29+=+42%2F%281%2B2b%5E1%29+=+28\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=42%2F%281%2B2b%29+=+28 ALIGN=MIDDLE  ALT=\"42%2F%281%2B2b%29+=+28\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=42+=+28%2B56b ALIGN=MIDDLE  ALT=\"42+=+28%2B56b\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=14+=+56b ALIGN=MIDDLE  ALT=\"14+=+56b\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+=+1%2F4 ALIGN=MIDDLE  ALT=\"b+=+1%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The logistic function for this problem is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+42%2F%281%2B2%281%2F4%29%5Ex%29 ALIGN=MIDDLE  ALT=\"f%28x%29+=+42%2F%281%2B2%281%2F4%29%5Ex%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "A graph, including horizontal lines at the initial, intermediate, and limiting values of 14, 28, and 42:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28400%2C400%2C-1%2C5%2C-10%2C50%2C42%2F%281%2B2%28.25%5Ex%29%29%2C14%2C28%2C42%29 ALIGN=MIDDLE  ALT=\"graph%28400%2C400%2C-1%2C5%2C-10%2C50%2C42%2F%281%2B2%28.25%5Ex%29%29%2C14%2C28%2C42%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );