Help me understand linear function. today in my high school lesson I learned for linear function. I know that each linear function has the form of $f(x)=kx+n$ where $k,x,n ∈ \mathbb R$. Now let's say be have those two functions:
$f_1(x)=kx+n_1\\
f_2(x)=kx+ n_2$
Since $k$ is same in both functions, we know that it represents those function whose graph will be parallel, but why and how to prove that those functions will be parallel?
 A: I always make a drawing if possible and it is possible in this case, since you are working in the plane. Drawing two parallel lines, what you see is that the vertical distance is always constant (I included an example of such a drawing). 
The vertical lines correspond with points on the $x$-axis, so let us take such a point and call it $z$. We want to show that the vertical distance does not depend on this point. The vertical distance corresponding to the vertical line at the point $z$ is given by 
$$|f_1(z) - f_2(z)| = |(kz + n_1) - (kz + n_2)| = |n_1 - n_2|$$
and this does not depend on the point we consider! 
This correspond to vertically translating the function $f_1$ over a distance $|n_1 - n_2|$ and we see that the shifted function coincides with $f_2$. Indeed, assuming that $n_1 \geq n_2$, we have that 
$$f_1(x) - (n_1 - n_2) = (kx + n_1) - (n_1 - n_2) = kx + n_2 = f_2(x)$$
so the translated function coincides with the second function, hence they have to be parallel. 
These are all things which you can quickly deduce by just drawing a picture.
A: At a high school level, I would say that a fair definition for parallel would be saying that the distance (i.e., the difference) between the lines remains constant. This is straightforward to prove:
$f_1(x)-f_2(x)=(kx + n_1) - (kx + n_2) = (k-k)x + (n_1-n_2) = n_1-n_2$,
therefore the difference $f_1(x)-f_2(x)$ does not depend on $x$, i.e. it is the same for all $x \in \mathbb{R}$.
Comment 1: Precisely speaking, a linear function is a function of the form $f(x)=kx$. Functions of the form you defined ($f(x)=kx + b$) are better called affine functions. This is because linearity is a well defined term in mathematics which implies (among other things) that $f(0)=0$, i.e. that the function passes through the origin.
Comment 2: You used the letter $n$ to denote a real number, which is not wrong, but it's very unusual. Usually, the letter $n$ is reserved to denote integer numbers or even natural numbers only (i.e. non-negative integer numbers). Unless you cannot really think of any other symbol to denote a real number, do not use the letter $n$, to avoid inducing confusion on the reader.
A: Suppose $n_1 \ne n_2$.  If there was a value $x=a$ where $f_1(a) = f_2(a)$ then $ka+n_1 = ka+n_2$.  But then $n_1=n_2$ which is a contradiction.
