Y=206; y-int 278; P(2,238)?

Question

Find an exponential function of the form f(x )

Find an exponential function of the form f(x )=ba^-x+c that has the given horizontal asymptote and y-intercept and passes through point P.

Answer 1

(The form given is f(x) = b a^(-x) + c)

The horizontal asymptote is the line to which your function's graph tends as x →∞, assuming that a > 1. As x →∞, a^(-x) → 0 (again, assuming a > 1) so c=206. So now your model is

f(x) = b a^(-x) + 206

The y-intercept is the point where the function intersects the y-axis. This is given as the point (0,278). (Remember, points on the y-axis have x-coordinate of 0.) So

278 = f(0) = b a^(-0) + 206 = b + 206

Therefore, b=72. So your model is now

f(x) = 72 a^(-x) + 206

It must pass through P, so f(2) = 238. So

238 = 72 a^(-2) + 206
32 = 72 a^(-2)
4 = 9 a^(-2)
4/9 = a^(-2)
Invert both sides:
9/4 = a^2
3/2 = a

Note that a satisfies our assumption that a > 1.

Hence,

f(x) = 72 (3/2)^(-x) + 206

which can also be written

f(x) = 72 (2/3)^x + 206