AVERAGE RATE OF CHANGE AND SLOPES OF SECANT LINES: The average rate of change of a function f(x) over an interval between two points (a, f(a)) and (b, f(b)) is the slope of the secant line connecting the two points:

 

• For example, to calculate the average rate of change between the points:
 
(0, -2) = (0, f(0)) and (3, 28) = (3, f(3))
 
where f(x) = 3x² + x – 2 we would:

 

This means that the average of all the slopes of lines tangent to the graph of f(x) between the points (0, –2) and (3, f(3)) is 10.

 

If we want the exact slope of a tangent line to this function at the point where x = 2, we would have to use other methods. The average slope can be calculated using two points. The exact slope at one point defies our basic formula for slope since we need to know TWO points, and this will be approached differently.

 

• For example: A man has driven a car 50 miles in one hour. Over the next three hours, he drives 140 miles. What is his average velocity (speed) over that next 3 hours? (We use t and d to represent time in hours and distance in miles).

We use the two points (1, 50) and (4, 190). Notice that 3 additional hours gives us a t value of 4 and the total number of miles is d = 50 + 140 = 190.

The average velocity is the average rate of change of this distance with respect to time. We have:

 SLOPE AND THE EQUATION OF A LINE: The slope of a line connecting two points is a ratio of the “rise” to the “run,” which is a ratio of the vertical distance between the points to the horizontal distance between the two points.

• A line passing through the points (2, 5) and (-3, 1) has a slope of

 

 

Since this is a positive number, the line will appear to slope upwards to the right when graphed.

 

How do we write an equation of this line which will describe ALL points on the line?

 

• The easiest way is to use point-slope form which is

 

 

For our example,

 

 

where m is the slope and we have m = 4/5.

 

Our equation is

 

 

 

since you can use either point.

 

These equations look different, but they can both be rearranged to give us

 

 

which is the one line with slope 4/5 and y-intercept 17/5. Generally speaking, do NOT rewrite this equation unless you have to solve for y to enter it into your calculator or you have specific instructions for rewriting.