How to derive sigmoid function from e by scaling & translating?

Question

The Sigmoid function is like this: $\frac{1}{1 + e^{-x}}$

Can it be derived by simply scaling and translating the graph of $e^{-x}$ ?

It looks to me as thought you could:

1). Translate it up, by 1

2). Scale it vertically by -1

However, when I draw this out it doesn't look like the sigmoid function?

Answer 1

$\displaystyle\frac{1}{1+e^{-x}}$ is bounded while any translation or scaling of $e^{-x}$ is not.