If a function is increasing at a rate of  and the points 
(0,2) and (3,4) are both on the graph of f(x). What will be 
the y value when x is 3 if the rate is changed to  and 
the point (0,2) remains on the new line.

--------------------------------------------------------

Here is what we must know in order to do the problem:

If the rate of change of a function is constant 

then

#1. The function is a linear function of the form f(x) = mx + b.

and

#2. The slope m is the constant rate of change of the linear 
   function f(x).

and

#3. The ordered pair (0,b) is a zero of f(x) and is called the 
    y-intercept, as it is represented by the point on the graph 
    of f(x), which is a non-vertical straight line, where that 
    line crosses the y-axis. 
   
Since we are told that:

>>"...a function is increasing at a rate of 2/3..."<<

we know by #2 above that m = 

and since we are told that

>>"...(0,2)...[IS]...on the graph of f(x)"<<

we know by #3 above that b = 2

then by #1, we know the function is

f(x) = mx + b  or

f(x) = x + 2

So we didn't need the information that (3,4) is on the line. 
It was extra.

------------------------------------

>>"...the rate is changed to 4/3..."<<


Now we're going to change the rate (i.e,, the slope m from 
to , so the new m = and since

>>"...the point (0,2) remains on the new line..."<<

We know that the new function, call it g(x), has b = 2, so

g(x) = mx + b
g(x) = x + 2  

So we are asked:

>>"...What will be the [NEW] y value when x is 3..."<<

So we substitute 3 for x

g(x) = x + 2
 
g(3) = (3) + 2

g(3) = 4 + 2

g(3) = 6.

Since the y-value IS the function value, then the
final answer is 6.

Edwin