Åk 6–9

4.2 The Function of a Straight Line
If you know two points’ coordinates, you can draw the line in a coordinate system. You can even determine the formula for the straight line’s function.  Two parts which are important in this case:

(Translation note: in the Swedish system the coefficients and constants in a straight line (y = kx + m) are normally k and m while in the English system (y = mx + c) they are m and c. Note that the m differs between the two systems.)

• m  is the gradient and indicates the functions slope.
• c is the point where the function intersect the y-axis.

A formula for a straight line’s function is: Gradient m

The gradient determines which direction a straight lines slopes, ”the number in front of x”. If two functions have the same “number in front of x”, in other words the same gradient, then they have the same slope and are parallel. We take an example:   If you have a line, can you determine the gradient by studying the line? We will try:

Here you see a graph and we will find the gradient, the slope of the line. What happens if we go one step to the right on the x-axis, in other words if x becomes 1 larger? Here you see that for every step to the right we go on the x-axis, the y becomes 3 steps bigger. So y becomes 3 steps larger, when x become 1 step bigger. It is 3x for every y.  Then you know that the gradient is 3. ”The number in front of x” should be 3. You can figure this out if you know the two points’ coordinates: Here, you know the points (0, 1) and (1, 4). We find the lines slope between these points. We begin at the point (0, 1):

x-value: Changed from 0 to 1, in other words, the change is +1.
y-value: Changed from 1 to 4, in other words, the change is +3.

The gradient or the slope can then be calculated by taking:

Gradient m: In the example above, we have:

 m = change in y = + 3 = + 3  change in x + 1

The gradient is +3, which we also figured out when we moved using arrows in the coordinate system.  The intercept point with the y-axis

Now that you know how to determine the direction of a linear function, we will learn how to determine the intersect point along the y-axis. If you look at the first picture in this chapter, you can see that two of the lines have the same slope with the same gradient of 2. These, however, do not have the same intercept point with the y-axis, and now we will learn how to determine this position. Where does the line intersect the y-axis?  The value which y has determines the function’s constant term. In the example above, the first line intersects the y-axis when y = 0 and the second one when
y
= -5. This is the function’s constant term and it determines the point of interception with the y-axis.

c = constant We decide which intersect point our line has with the y-axis: By looking at where the line intersects the y-axis, we can determine the function’s constant term, the c-value. We see that the y-value is +1 when it intersects the y-axis. We can now determine both the slope and the position:

m = 3
c = 1

The function is then y = 3x + 1

Mathematicians have created several formulas to describe a linear function.  We will use the one we showed in the beginning of the page: m  is the gradient and indicates the functions slope.
c is the point where the function intersect the y-axis.  