How do you find the exponential function that contains both of these points (2,12.6) and (5, 42.525)?

1 Answer
Jun 18, 2015

Answer:

Suppose #f(x) = k*a^x# for some #k in RR# and #a > 0#, #a != 1#

Then #a = root(3)(42.525/12.6) = root(3)(3.375) = 1.5#

and #k = 12.6/(a^2) = 12.6/2.25 = 5.6#

So #f(x) = 5.6(1.5)^x#

Explanation:

Suppose #f(x) = k*a^x# for some #k in RR# and #a > 0#, #a != 1#

Then #f(5)/f(2) = (k*a^5)/(k*a^2) = a^3#

So #a = root(3)(f(5)/f(2)) = root(3)(42.525/12.6) = root(3)(3.375) = 1.5#

#f(2) = k*a^2#, so #k = f(2)/(a^2) = 12.6/(1.5^2) = 12.6 / 2.25 = 5.6#

So #f(x) = 5.6(1.5)^x#