Question: Find an exponential function given the points (2,3);(5,1/9) .
When you are told that you have an exponential function, then you may assume that it is of the form,
y = k.a^x
where x and y are your independent and dependent variables and k and a are two unknown constant values.
Since you are given two sets of values for x and y (the coordinate points) then you will be able to create a pair of simultaneous equations that you can solve for the unknown constnts.
The point (x,y) = (2,3)
3 = k.a^2 ------------------------------- (1)
The point (x,y) = (5,1/9)
1/9 = k.a^5 --------------------------- (2)
Divide (1) by (2).
27 = a^(-3)
27 = 3^3 = a^(-3)
3 = a^(-1)
a = 1/3 = 3^(-1)
Substituting for a = 3^(-1) into (1),
3 = k.(3^(-1))^2 = k.3^(-2)
k = 27
This gives the equation as: y = 27.3^(-x) = 3^3.3^(-x) = 3^(3-x)
Answer: y = 3^(3-x)