Question 1127178
<br>
The standard form of a logistic function is<br>
{{{f(x) = c/(1+ab^x)}}}<br>
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>
So in this example we know the function is of the form<br>
{{{f(x) = 42/(1+ab^x)}}}<br>
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>
{{{f(0) = 42/(1+ab^0) = 42/(1+a) = 14}}}
{{{1+a = 3}}}
{{{a = 2}}}<br>
Now we know the function is of the form<br>
{{{f(x) = 42/(1+2b^x)}}}<br>
Now, to finish finding the logistic function, find the value of b by using the given data point.<br>
{{{f(1) = 42/(1+2b^1) = 28}}}
{{{42/(1+2b) = 28}}}
{{{42 = 28+56b}}}
{{{14 = 56b}}}
{{{b = 1/4}}}<br>
The logistic function for this problem is<br>
{{{f(x) = 42/(1+2(1/4)^x)}}}<br>
A graph, including horizontal lines at the initial, intermediate, and limiting values of 14, 28, and 42:<br>
{{{graph(400,400,-1,5,-10,50,42/(1+2(.25^x)),14,28,42)}}}