Draw the graph of your function $\displaystyle y = x^2 $Then draw a straight line through the function. Do this on graph paper if you need to so that you can draw a line through points that you can see the value of. I have attached a graph that I made to show you what I mean by this.
SLOPE-AVERAGE RATE OF CHANGE.doc
It is a parabola with its vertex at the origin. "a" & "b" are x-values somewhere on the graph. f(a) and f(b) are the y-values associated with those x-values.
The average rate of change of a function is the slope of a secant line of the function. A secant line is a straight line that goes through the graph of the function intersecting at two separate points.
The slope of the secant line is:
$\displaystyle slope = \frac{change.in.y.values)}{change.in.x.values}$
OR
$\displaystyle m = \frac{f(b) - f(a)}{b-a}$
So, choose any value of x, let's say x=2. Now, to find the y-value associated with that x-value, plug the x-value into your function.
$\displaystyle y = x^2$
which can also be written as
$\displaystyle f(x) = x^2$
So,
$\displaystyle y= (2)^2
$
or
$\displaystyle f(2) = (2)^2 = 4$
So when x=2 on the graph of this function, y=4
Does that make sense?
So,
a = 2
f(2) = 4
Now pick another value on your function, let's say x = -1
Plug -1 into $\displaystyle y = x^2$, you get $\displaystyle y = (-1)^2 = 1$
So,
b = -1
f(b) = 1
So from your problem:
$\displaystyle \frac{\delta y}{\delta x} = \frac{f(b)-f(a)}{b-a} = \frac{-1-2}{1-4}$
$\displaystyle = \frac{-3}{-3} = 1$
So, the average slope of your function is m = 1
Does that help you understand now?