Quandaries and Queries


 Given a set of ordered pairs, ie  (1,1) (2,4) (3,7), how does one determine the rule f(n) other than by trial and error

n 1 2 3
f(n) 1 4 7

f(n)= 3n-2


I seem to remember an application called the “point slope” method, but it has been 20+ years, and it seems to me that there should be a relatively simple method.

Thanks for you help




Hi Jack,

If you are sure that there is a straight line relationship in the points then there is a relatively easy way to come up with this relationship. I am going to call the ordered pairs (x,y) since that is the notation you will see in most references.

The point slope form that you mention involves knowing one point on the line, I will call it (x1, y1), and the slope, which I will call m. The point slope expression is then

y - y1 = m (x - x1)

In your example if the point (x1, y1) is (2, 4) then the point slope form gives

y - 4 = m (x - 2)

To find the slope you need another point on the line, call it (x2, y2) and then the slope is given by

m = (y2 - y1)/(x2 - x1)

If you use (3, 7) as your second point then the slope is

m = (7 - 4)/(3 - 2) = 3

and hence the line is

y - 4 = 3 (x - 2)
y - 4 = 3x - 6
y = 3x - 2

Try it using (1,1) and (2,4) as the points and check to see that you arrive at the same equation.



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