Question 1127693
.


            Below find my solution for case a).


find the formula for an exponential function that passes through the two points given.
a) (-1,1/2) and (4,512)


<pre>
General exponential function formula


    y = {{{ab^x}}}.        (1)


At  x= -1,  y= {{{1/2}}}  formula (1) becomes   {{{1/2}}} = {{{a*b^(-1)}}}    (2).


At  x= 4,  y= 512  formula (1) becomes   512 = {{{a*b^4}}}    (3).


Divide  (3)  by (2).  You will get


    {{{512/((1/2))}}} = {{{b^4/b^(-1)}}},   or, equivalently,

    1024 = {{{b^5}}},

    {{{4^5}}} = {{{b^5}}}.


It implies  b = 4,  so half of the problem is just solved.


Now substitute the found value b= 4 into equation (3). You will get

  
    512 = {{{a*4^4}}},  

    
which implies  a = {{{512/4^4}}} = {{{512/256}}} = 2.


<U>Answer</U>.  The function is  y = {{{2*4^x}}}.
</pre>

Solved.