You've probably seen the average rate of change before, when it's called the slope of a line. The difference here is that you are looking at a curve, not a straight line. We can talk about the slope of a line through two points on the curve, that is, through x = 1 and x = 2 given by the interval [1,2]. FYI, this is called the secant line. Now, this is probably more detail than you wanted...
We have a bunch of different ways to calculate the average rate of change.
The most straightforward for a function f(x) is to write
[tex] \dfrac{f(b)-f(a)}{b-a} [/tex],
and b can't equal a if we're not dealing with calculus.
So for the first function, we take our two points, 1 and 2, and look at the function definition and evaluate:
[tex]\dfrac{f(2)-f(1)}{2-1}=6[/tex]
For the second function:
[tex]\dfrac{f(2)-f(1)}{2-1}=f(2)-f(1)=(\dfrac{1}{4})^2 + 4-[(\dfrac{1}{4})^1 + 4]= \dfrac{-3}{16} [/tex]
For the third function, we evaluate based on the graph:
[tex]\dfrac{f(2)-f(1)}{2-1}=0-(-2)=2[/tex]
Now, we move onto the question. None of these have the same average rates of change, so we eliminate A and D. Comparing the average rates of change, it is true that function 2 has the lowest, so we choose B.
