How do you determine the formula for the graph of an exponential function given (0,2) and (3,1)?

1 Answer
Sep 25, 2016

I found: #y(x)=2e^(-0.231x)#

Explanation:

Let us start with the general form of an exponential function:
#y(x)=Ae^(kx)#
where #A# and #k# are two constants we need to evaluate.
We use the coordinates of the points into the general expression to find the two constants:

First : #x=0 and y=2# getting:
#y(0)=Ae^(0k)# that must be equal to #2#
or:
#Ae^(0k)=2#
#Ae^(0)=2#
#A*1=2#
so that #A=2#

Second : #x=3 and y=1# getting:
#y(3)=Ae^(3k)# that must be equal to #1#;
we know from the previous step that #A=2# so:
#y(3)=2e^(3k)=1#
and so:
#2e^(3k)=1#
#e^(3k)=1/2#
take the natural log of both sides:
#ln(e^(3k))=ln(1/2)#
rearranging:
#3k=ln(1/2)#
and:
#k=-0.231#

Finally our function will be:

#y(x)=2e^(-0.231x)#

Graphically:
graph{2e^(-0.231x) [-10, 10, -5, 5]}