So what happens when you do not have an equation, rather a graph or a table of values. How do you find the rate of change of a value that is not on the table. Although you cannot find the exact rate of change, you can find the approximate rate of change and to do that, you find the slope between the two values given. An example will help to explain this concept.

Example 1
x
1
3
5
7
y
12
15
20
10
Say you are given a table of values where x1=1, x2=3, x3=5, and x4=7. The corresponding y-vaules are y1=12, y2=15, y3=18, and y4=10. You are asked to find the approximate rate of change when x=2. Since 2 falls between the interval from x1 to x2, you find the average slope of that interval. To do so you divide your change in y by your change in x, so (15-12)/(3-1) or 3/2. So the approximate rate of change when x=2 is 3/2. Suppose you are asked to find the approximate rate of change when x=6. You do the same thing, except use x3 and x4. You get (10-20)/(7-5) which is -5. So the approximate rate of change when x=6 is -5.

Further Note
In approximating slopes at a specific point that falls between two values on a graph, which will be shown in a minute, or in a table, we are actually determining the average slope over that interval. Thus the closer the two x-values you use are to the one you are trying to approximate, the more accurate your approximation will be. That is why I used x1 and x2 for the approximation of x=2 as opposed to x1 and x3, even though x=2 still falls in that interval.

Example 2
graph.JPG
Now let's suppose you were given a graph such as the one above with no equation and are asked to find the rate of change a specific point. Each of the tick marks on the graph is 2 units. You are asked to find the rate of change when x=2. You could estimate the slope but a more accurate way would be to find the y-value when x=1 and the y-value when x=3 and use the slope formula to find the rate of change when x=2. From looking at the graph we approximate y to be 4 when x=1 and y to be -2.5 when x=3. Using the slope formula we find that the approximate rate of change when x=2 is (-2.5 - 4)/(3 - 1) or (-6.5)/(2) which is -3.25.


Find an Example here.
This can be found in Calculus 8th Edition chapter p.2, pages 10-14.