
Linear Functions:
You are already familiar with the concept of "average rate of change".
When working with straight lines (linear functions) you saw the "average rate of change" to be:
The word "slope" may also be referred to as "gradient", "incline" or "pitch", and be expressed as:
A special circumstance exists when working with straight lines (linear functions), in that the "average rate of change" (the slope) is constant. No matter where you check the slope on a straight line, you will get the same answer.


NonLinear Functions:
When working with nonlinear functions, the "average rate of change" is not constant.
The process of computing the "average rate of change", however, remains the same as was used with straight lines: two points are chosen, and is computed.
FYI: You will learn in later courses that the "average rate of change" in nonlinear functions is actually the slope of the secant line passing through the two chosen points. A secant line cuts a graph in two points. 

When you find the "average rate of change" you are finding the rate at which (how fast) the function's yvalues (output) are changing as compared to the function's xvalues (input).
When working with functions (of all types), the "average rate of change" is expressed using function notation.
Average Rate of Change
For the function y = f (x) between x = a and x = b, the

While this new formula may look strange, it is really just a rewrite of .
Remember that y = f (x).
So, when working with points (x_{1}, y_{1}) and (x_{2}, y_{2}), we can also write them as
the points .
Then our slope formula can be expressed as .
If we rename x_{1} to be a, and x_{2} to be b, we will have the new formula.
The points are , and the
.
Finding average rate of change from a table.
Function f (x) is shown in the table at the right.
Find the average rate of change over the interval 1 < x < 3.
Solution:
If the interval is 1 < x < 3, then you are examining the points (1,4) and (3,16). From the first point, let a = 1, and f ( a) = 4. From the second point, let b = 3 and f ( b) = 16.
Substitute into the formula:


The average rate of change is 6 over 1, or just 6.
The yvalues change 6 units every time the xvalues change 1 unit, on this interval. 
Finding average rate of change from a graph.
Function g (x) is shown in the graph at the right.
Find the average rate of change over the interval
1 < x < 4.
Solution:
If the interval is 1 < x < 4, then you are examining the points (1,1) and (4,2), as seen on the graph. From the first point, let a = 1, and g ( a) = 1. From the second point, let b = 4 and g ( b) = 2.
Substitute into the formula:


The average rate of change is 1 over 3, or just 1/3.
The yvalues change 1 unit every time the xvalues change 3 units, on this interval. 
Finding average rate of change from a word problem.
A ball thrown in the air has a height of h(t) =  16t² + 50t + 3 feet
after t seconds.
a) What are the units of measurement for the average rate of change
of h?
b) Find the average rate of change of h between t = 0 and t = 2?


Solution:
a) In the formula, , the numerator (top) is measured in feet and the denominator (bottom) is measured in seconds. This ratio is measured in feet per second, which will be the velocity of the ball.
b) Start by finding h(t) when t = 0 and t = 2, by plugging the t values into h(t).
h(2) = 16(2)² + 50(2) + 3 = 39
h(0) = 16(0)² + 50(0) + 3 = 3
Now, use the average rate of change formula:

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