The rate of change, ROC, is a ratio that describes the average change between two related points. It can represent the average speed of a car over a certain period of time, or the average growth rate for bacteria in an experiment. It's determined by dividing the change in the vertical direction () by the change in the horizontal direction ().
The Greek letter (Delta) is commonly used to describe a difference, which leads to an alternative notation of the formula.
Determine the rate of change using a table
Some of the - and -values of a function are given in the table.
Determine the rate of change.
To find the ROC, we must determine the change between -coordinates and the change between -coordinates. Let's start by looking at the left column. The difference between each row is constant,
In the right column, the difference between each row is also constant,
Since the rate of change is determined by the ratio of the change in and the change in , we get The rate of change is
Determine if the rate of change is constant
In the tables, the - and -values for two different functions are given.
Determine which, if either, function has a constant rate of change.
To find the rate of change for each function, we must analyze the change between coordinates. We can start by noting that the left columns for both functions are the same. Between rows, they each increase by
Now, let's look at the left table. The first -value is and the second is The difference between and is If we add to we get which is the next -value. In fact, the difference between each -value is
Thus, the ROC of the first function is constant and equals Using the same method, we can note the difference in -coordinates for the second function.
Notice that the difference between -coordinates changes. Thus, the ROC of the second function is not constant.
Determine if the function is linear
A small box contains three golf balls.
The number of golf balls, that fit inside a golf cart depends on how many small boxes, fit in the cart. Determine if is a linear function.
If has a constant rate of change, then it is linear. To begin, we can make a table of values that shows how many boxes fit in the cart. First, if only one box fits, there will be balls in the cart.
If the golf cart can fit two boxes, then it will contain balls.
If the cart contains boxes, there will be golf balls, and so on. Let's extend the table to five boxes.
Now that we have a table that represents we can determine the rate of change. The left column increases by each step and the right column increases by
Thus, and remains constant. This is because each time additional box is added to the cart, the number of balls increases by Thus, is linear, and has a discrete domain.