Linear functions are functions of the form:
the domain of which belongs to the set of real numbers .
and the codomain also belongs to the set of real numbers .
This formula is the is also call the straight line function, because its graph is a straight line.
The slope m
The constant m of a linear function is the slope or gradient of the function. It tells us how steep the line is, and if it is decaying or growing.
m < 0: The function is decaying
m > 0: The function is growing
m = 0: The function is constant
The constant b
The constant b indicates where the straight line intersects the y-axis. This will be in the point (0,b).
If b is 0, the function is said to be direct proportional, and its graph passes through the origin (0,0).
Calculation of m and b with two points given
If a straight line is passing through two points with the coordinates (x1, y1) and (x2, y2), the slope m can be calculated like this:
when m is known you can calculate b:
If x increases by 1, the change of y will be m.
If x increases by , the change of y will be .
This relationship is given by the formula:
Simple linear regression
If you have two sets of data, and you want to know, if there is a linear relationship between them, you can plot the data in a coordinate system and see if they fit into a straight line, or you can use a calculator to make the linear regression.
Finding the slope with an angle
If you know the angle θ between the x-axis and a line, you can calculate m of the linear function by using tangent: