riktningskoefficient

The slope $k$ of a line $y=kx+m$ is a number which tells you how much the function $y$ changes for a given change in $x$. Graphically, it represents the steepness of the line. The greater the number $k$, the steeper the line will be. Conversely, a smaller value of $k$ will result in a flatter line.

In English, there is the expression "rise over run". It is a mnemonic to help you remember the formula for the slope, namely:

$\displaystyle k=\frac{\Delta y}{\Delta x}$

You "run" $\Delta x$ units in the $x$-direction, and then correspondingly (according to the functional relationship) "rise" $\Delta y$ units in the $y$-direction. 

Example:

A line $\displaystyle y=kx+m$ goes through the points $(0,1)$ and $(1,2)$. Determine the function $y$.

Since only a single line can go through two specific points at the same time, two points on the graph are enough to uniquely determine the line. We start by plugging in the first coordinate into our functional relationship:

$y(0)=k\cdot 0 +m=1 \implies m=1$

Now we know the line can be written as $y=kx+1$. To determine the slope $k$, we plug in the other point on the graph:

$y(1)=k\cdot 1 + 1 = 2 \implies k=1$

Putting together what we know, we can determine the line to be:

$\displaystyle \implies y=x+1$