 Introduction: Average Rate of Change The average rate of change of any function is a concept that is not new to you.  You have studied it in relation to a line.  That's right!  The slope is the average rate of change of a line.  For a line, it was unique in the fact that the slope was constant.  It didn't change no matter what two points you calculated it for on the line.  Take a look at the following graph and we will discuss the slope of a function. Demo: Slope of a Secant/Tangent Line (Walter Fendt)
The function is in red.  The blue line connects the two points that we want to find the average rate of change (slope of the blue line).  The two points are (x, f(x)) and (x+h, f(x+h)).  To find the slope, the definition is the change in y over the change of x.  Does this sound familiar!!  Applying this definition we get the following formula:     Notice on the graph that the line we are finding the slope of crosses the graph twice.  Do you remember from geometry what you call a line that crosses a circle twice?  You got it!!  It's a secant line!  When you calculate the average rate of change of a function, you are finding the slope of the secant line between the two points.

As an example, let's find the average rate of change (slope of the secant line) for any point on a given function.  This is finding the general rate of change.  The general rate of change is good for any two points on the function.  Find the general rate of change for f(x) = x2

f(x) = x and f(x + h) = (x + h)2 Therefore, the slope of the secant line between any two points on this function is 2x + h.  To find the specific rate of change between two given values of x, is a simple matter of substitution.  Let's say we are asked to find the average rate of change between the points x1 = 2 and x2 = 4.  Then in our general answer, we will replace x with x1 and h = x2 - x1.  Replacing these values in the formula yields  2(2) + (4 - 2) = 4 + 2 = 6.  Thus, the slope of the secant line connecting the two points of the function is 6.  Note that the answer is a positive number.  That means what?  That's right, you know!  The line is going uphill or increasing as you look at it from left to right.  Be careful that you put the values for determining h in the correct order.  You already know that slope can be positive, negative or zero.

Now using the same function as above, find the average rate of change between x1 = -1 and x2 = -3.  The answer is 2(-1) + ( -3 + 1) = -2 + -2 = -4.  This means that the secant line is going downhill or decreasing as you look at it from left to right. Sample problems
1)   Find the general rate of change for the function f(x) = 2x2 + 1.  Then find the specific rate of change for x1 = 2 to x2 = 5.

2)  Find the general rate of change for the function f(x) = x3.  Then find the specifice rate of change for x1 = 0 to x2 = 2.
Look for the answers worked out somewhere below!   Fly into the next section with Snoopy!   Look down here for the answers to the sample problems!! Specific rate of change = 4(2) +2(5 - 2) = 8 + 6 = 14  Specific rate of change = 3(0) + 3(0)(2) + (2)2 = 4
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