Question

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.
y = 76; y−intercept 389; P(1, 232.5)

Answer 1

Assuming \(a>0\), a horizontal asymptote at y=76 tells you that
\[\lim_{x\to\infty}(ba^{-x}+c)=76~~\iff~~0+c=76\]
A y-intercept of 389 means \(f(x)=389\) when \(x=0\):
\[\begin{align*}f(0)=ba^{-0}+76&=389\\
b+76&=389\\
b&=313
\end{align*}\]
And finally, the given point (1, 232.5) tells you that \(f(1)=232.5\):
\[\begin{align*}f(1)=313a^{-1}+76&=232.5\\
\frac{313}{a}&=156.5\\
a&=2
\end{align*}\]