find and interpret the average rate of change

Question

In 2006, the national federal deficit was 248 billion.  By 2013, it was 680 billion.  Find and interpret the average rate of change (slope).

Answer 1

First we have to think of the information as two points.  We'll let the x-axis be year, and the y-axis be the deficit in dollars.

 

 

So our first point, P1, is at (2006,248 billion).

Our second point, P2, is at (2013, 680 billion).

 

Now we can plug the numbers into the equation for slope, which is

m = (y2 - y1)/(x2 - x1).

m=(680 billion - 248 billion) / (2013 - 2006)

m= 432 billion / 7,

 

which comes out to about 61.7 billion dollars per year.

 

Why did we choose time (in years) for the x axis and not the y axis?  It comes down to units.  We want to find the rate in dollars per year, which is written as dollars/year (dollars divided by years).  So for our answer to be right, the units for money need to be in the numerator in the slope equation, and the units for time need to be in the denominator.