Blog Post #4

Rate of Change

A rate of change is a rate that describes how one quantity changes in relation to another quantity. When determining the rate of change we use the rate of change formula, which is:

                                                        y₂ – y₁

Rate of Change      =              ----------

     x₂ – x₁

In order to use this formula, you must have numbers or points to work with. Typically you use the formula by plugging in numbers from a graph or table.

Lets look at an example of how we would use the rate of change formula to find the slope of a line. The diagram below shows how this works.

  

Example)

Lets say that you want to find the rate of change, or in this case the slope, of a linear line:

                                          

The first step would be to pick two different

points on the graph. For this example lets use

(1,1) and (2,2). We take these points and plug

them into the rate of change formula as so:

Rate of change =     2-1

                             2-1

Then we solve it:

                                    2-1   =  1   =   1

                                 2-1       1

The rate of change, or slope, of the line is 1

Keep in mind the numbers wont always work out so perfectly. The rate of change can be a fraction/decimal, a negative number, or there could be no rate of change at all (in a case like that we would have a horizontal line).

*Also, one thing to note about the above example is that we only tested this with two sets of points. We don’t need to test anymore simply because we know it is a linear line, and any points that we plug in will give us the same rate. However in other situations you will need to test multiple points in order to confirm that they all give you the same point.

Now lets look at an example of finding the rate of change based off of a table:

Say you have just driven from Los Angeles to New York, and you want to know what your average rate of speed was.

The table below shows the time spent driving in hours (x), and the distance traveled in miles (y).

Time in hours (x)	Distance in miles (y)
12	400
24	800
36	1200
48	1600
60	2000

To figure out the your average rate of speed we will need to plug at least two numbers into the formula.

y₂ – y₁      

_{---------    =}        800 – 400    =     400      =    33.3 

x₂ – x₁                 24 – 12              12

We get an average rate of change of 33.3, however we need to plug in two more point in order to confirm that this truly is the average rate of speed.

1600 – 400              =    1200        =        33.3

   48 – 12                         36

Since we used multiple points and got the same rate for both, we have confirmed that the average rate of speed is in fact 33.3 mph.