Parameter

Consider the following function from $\mathbb{R}$ to $\mathbb{R}$.

$$f\_\alpha (x) = x^2 + \alpha \times e^{-x}$$

This function has one variable, named $x$ and one parameter $\alpha \in \mathbb{R}$. A parameter is a number that changes the behaviour of a function, but is not a variable.

You can study the function $f(x, \alpha) = x^2 + \alpha \times e^{-x}$, where $\alpha$ and $x$ are now both variables to understand how the parameter changes the previous function.

You can now take the derivative of $f$ with respect to $\alpha$ and deduce various facts about $f_\alpha$.