Question

I don’t understand how I’m supposed to find the rule

Answer 1

You will find that this "rule" is a linear function; its graph is a straight line.

The trick here is to determine the slope and y intercept of the str. line that goes through these points (2,3), (4,4) and (6,5), IF such a line exists.

The slope of the line segment connecting (2,3) and (4,4) is 
         4-3
m = -------- = 1/2
         4-2
                                                                                       5-4
The slope of the line connecting (4,4) and (6,5) is m = ------ = 1/2
                                                                                        6-4

Since both slopes are the same, this IS a linear function.

Let's find its equation (rule):

Use the point-slope formula for the str. line:

m = slope = 1/2, and the point could be any of the 3 given points.  Let's use (2,3).  Then,

y - 3 = (1/2)(x - 2), or 2y - 6 = x - 2, or 2y = x - 2 + 6, or 2y = x + 4.

This could be written in "standard form" as -x + 2y =4, or we could solve for y and end up with 

y = (1/2)x + 2 (which, as you can see, has a slope of 1/2).