Proportional means that if x is changed, then y is changed in the same proportion.
The graph of a proportional function is a straight line passing through the origin (0,0).
For example, if one peice of chocolate costs $2, then two pieces of chocolate cost $4, three pieces cost $6 and so forth.
This relationship can be expressed by a proportional/linear function:
The formula expresses that the total price f(x) will increase by 2 dollars every time the quantity of chocolate increases by 1.
The formula for a proportional function
A proportional function is a function of the form:
So f(x) is the result of x multiplied by a number.
Finding the formula for a proportional function
It is possible to find the formula for a proportional function, if two sets of x- and y-values, (x1,y1) and (x2,y2), are given.
First you calculate the constant m, given by:
and then you substitude m into the formula of a proportional function:
Let’s say, only one set of x- and y-values is given to you, (x2,y2). You can still find the function’s formula, because you know that the graph of a proportional function passes through the point (0,0). So the other set of x- and y-values is (x1,y1) = (0,0). You can now calculate the constant m:
So actually you just need to know the coordinates of one point of a line in order to find the formula for a proportional function.
Growth or decay
A proportional function can either be growing or decaying
If f(x) increases, when x increases, it’s a growing function.
If f(x) decreases, when x increases, it’s a decaying function.
The example with the chocolate above is a growing function.
The constant m indicates if it’s a growing or decaying function.
m > 0: The function is growing.
m < 0: The function is decaying.
Recognizing a proportional trend in a data set
If you have two sets of data, the relationship between which follows a proportional trend, and you want to find the best fitting function expressing their relationship, you can plot the data within a coordinate system. Now you draw the best fitting straight line passing through the points. When the straight line is complete, you can find the coordinates of one point on the line and calculate the constant m: