Average Rate of Change = Slope
As we go through life, things tend to change. We are never stuck in one place. Whether it is how much we grow in one year, how much money our business makes each year, or how fast we drive on average. For all of these instances, we would find the average rate of change. The average rate of change is finding how much something changes over time. This is very much like finding the slope of a line.
If you recall, the slope of a line is found by finding the change in y divided by the change in x.
This can also be written as the slope formula:
The average rate of change and the slope of a line are the same thing. Thinking logically through this formula, we are finding the difference in y divided by the difference in x.
For instance, suppose you have a trip planned. You know you will be traveling through many different areas where the speed limit changes. You will be going 70 mph on one section, then 35 mph on another section. We can find the average speed over the course of the trip by using the slope formula.
Let's take a look at this graph. Time is represented on the horizontal axis, and the speed limit represents the vertical axis. Notice, at the first hour, our speed limit was 30 mph. Here are all the times and speeds represented on this graph.
- Hour 1: 30 mph
- Hour 2: 70 mph
- Hour 3: 55 mph
- Hour 4: 40 mph
- Hour 5: 65 mph
We can represent these values as our ordered pairs, (x,y): (1,30), (2,70), (3,55), (4,40) and (5,65).
At any two points we can find out the average rate just by finding the slope of the line. Let's find the slope of the line between points at hour 1 and hour 4: (1,30) and (4,40). Using the slope formula, we plug in the values from our ordered pair and solve.
This means over the course of three hours our speed changed an average of 3.33 miles every hour. Notice the red line shows the slope or average rate of change as gradual, hence only 3.33 miles per hour.
Now let's find the average from hour 1 to hour 2: (1,30) and (2,70):
Between these two points, our speed changed 40 miles per hour in one hour. The fact that our slope is positive indicates the speed went up or increased. Notice the line goes 'uphill' on this section, which is a positive slope. Also the red line is fairly steep, which indicates a higher average rate increase.
Let's look at the rate change from hour 2 to hour 4: (2,70) and (4,40):
This indicates the speed went down or decreased 15 miles per hour because of the negative sign. Notice the slope on the graph is going 'downhill' on this section, which is a decreasing slope and our mph is negative. The red line is showing a negative or decreasing rate.
As we see here, slope is another version of finding the average rate of change. Average rate of change is finding the difference between the dependent variable (y-term) divided by the difference in the independent variable (x-term). Slope and average rate of change is exactly the same thing. Be sure to keep track of the units in both the numerator and denominator.
More Examples of Average Rate of Change
Let's look at some more examples. Joey's parents are keeping track of Joey's height as they watch him grow. They notice that he had several growth spurts throughout his first 16 years. Here is a graph of the numbers they collected.
From this graph you can see the area where the line is the steepest, the slope is greatest, between the ages of 13 and 14. What is the average rate of change in this interval? At age 13, he was 52 inches and at age 14 he was 60 inches, so using the ordered pairs (13,52) and (14,60), let's use the slope formula to find the average rate of change:
This means Joey's parents had to buy him all new clothes that year!
Here is another example: Mr. Johnson started his construction business in 1990. Each year as he worked on his taxes, he noticed an upward trend in profits. Then in 2008, the bottom fell out of the economy, and he showed a decrease in profit.
Looking at the graph, you can see the biggest loss was in 2008 when the economy crashed. But over the 25 years what was the average rate of change? In 1990, he had a $0 profit, in 2014 he had a $40,000 profit. Using the slope formula, we can find the average rate of change:
So, this means over the course of 24 years, he had an increase of $1,666.67 profit each year. As a business owner, this should be encouraging, even with such a decrease in 2008, he still showed a profit over the 24 years in business.
To see what type of downfall he experienced in 2008, let's find the average rate of change from 2006 to 2008. In 2006, he earned $52,000 in profit. In 2008, his earnings dropped to $10,000, so what was the average rate of change?
This means his profit decreased $21,000 per year over those two years. Notice the slope formula showed a negative slope. This indicates the decrease in profit.
The average rate of change is finding the rate something changes over a period of time. We can look at average rate of change as finding the slope of a series of points. The slope is found by finding the difference in one variable divided by the difference in another variable. The slope formula is used to find the average rate of change.
The y-values are the dependent variables, and the x-values are the independent variables. If the slope is positive, this is an increasing rate of change. If the slope is negative, this is a decreasing rate of change.
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Average Rate of Change: Definition, Formula & Examples Quiz
Instructions: Choose an answer and click 'Next'. You will receive your score and answers at the end.
If the average rate starts out decreasing and continues to decrease, what does the line look like?
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