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?

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.
