Find expotential function from two points Clever people on this place, I'm having trouble with this, and I'm not able to see why what I'm doing is wrong... Here are two points:
$(3,1)$, $(-1,16)$
And this is what my calculations are:
First, I'll find $a$:
$
a = (x_2-x_1)\sqrt{\frac{y2}{y1}}= (-1 + (-3) )\sqrt{\frac{16}{1}}\\\Leftrightarrow \\a = -4\sqrt{\frac{16}{1}}n \\\Leftrightarrow \\a = -4\sqrt{\frac{16}{1}} \Rightarrow a=−16 $
Then, I can find $b$:
$b = (\frac{y1}{a^x1}))= \frac{1}{-16^3}\\ 
\Rightarrow b=−0.000244$
This is wrong, my book says that the answer is $f(x) = 8(2^{-x})$
What's wrong, and what should be changed here?
I'm not the biggest math professor, but i hope you can help me aswell.
What i need to know is how the final function can be, as it says in the book - in this case, it's $f(x) = a (b^x)$

You know that $f(3)=1$ and $f(-1)=16$, so as $f(x)=a\times b^x$, you have:
\begin{align*}
ab^3&=1\\
ab^{-1}&=16
\end{align*}
Now we can cancel the $a$s by dividing:
$$b^4=\frac{ab^3}{ab^{-1}}=\frac{1}{16}$$
So one choice of $b$ is $b=\frac{1}{2}$, and then you can check that to satisfy the two equations you must take $a=8$. However, taking $b=-\frac{1}{2}$ and $a=-8$ also works, and there are two more choices where $a$ and $b$ are complex numbers.
