Daniel B. answered • 05/08/21
A retired computer professional to teach math, physics
The requirement that y=abx+k passes through the point (2, 264) means
ab² + k = 264
The requirement that y=abx+k passes through the point (6, 4044) means
ab6 + k = 4044
The requirement that y=abx+k has a horizontal asymptote at y = 12 means
lim(abx+k) = 12 as x -> +∞ or x -> -∞.
Depending on whether b < 1 or b > 1
lim(abx) = 0 at one infinity and
lim(abx) = ∞ at the other.
A y-asymptote is determined by the behavior of the function where the limit is finite.
Therefore we can replace lim(abx) = 0 and get k = 12.
Substituting into the above two equations
ab² + 12 = 264
ab6 + 12 = 4044
ab² = 252
ab6 = 4032
After dividing
ab6/ab² = 4032/252
b4 = 16
b = 2
(Strictly speaking b = -2 is also a solution.
And even more strictly speaking there are two more solutions b = 2i and b = -2i.
However, assuming that you are not to deal with complex numbers,
I assume b = 2.)
Using the equation
ab² = 252
a×2² = 252
a = 63
The final answer is
y = 63×2x + 12
Mirielle M.
I know I'm not the one who asked, but thank you so much for the explanation!! I had to find a exponential function just given a graph with two points (1, -3) & (3,5) and an asymptote y= -4. My professor only showed us how to find the function given a definite y-intercept, but my graph was crossing the y-axis between two whole numbers, so there was no way I could guess the decimal value. I was able to use your dividing technique for my points, and I found that the function was y = 1/3(3)^x - 4. Thanks for giving me a technique to use any time I'm given two points and a horizontal asymptote on the exam!11/09/22