OBJECTIVE To recognize changes in the average rate of change of a function and to see how such changes affect the graph of the function.

In an episode of the popular television show The Simpsons, Homer reads from his US of A Today newspaper and says, "Here's good news! According to this eyecatching article, SAT scores are declining at a slower rate."

In this statement Homer is talking about the rate of change of a rate of change: SAT scores are changing (declining), but the rate at which they are declining is itself changing (slowing down).* In the real world, rates of change are changing all the time. When you drive, your speed (rate of change of distance) increases when you accelerate and decreases when you decelerate. As teenagers grow, the rate at which their height increases slows down and eventually stops as they become adults. In this Exploration we investigate how a changing rate of change affects the graph of a function.

* References to mathematical ideas are frequent on The Simpsons. Professor Sarah Greenwald of Appalachian State University and Professor Andrew Nestler of Santa Monica College maintain a website devoted to "the mathematics of The Simpsons": www.mathsci.appstate.edu/~sjg/simpsonsmath/.

I. Changes in Tuition Fees

1. The table gives the average annual cost of tuition at public $4$-year colleges in the United States. The first section in the table gives the tuition in actual current dollars; the second section adjusts these numbers for inflation to constant 2006 dollars.
   1. Fill in the "Rate of change" columns by finding the change in tuition and fees in dollars per year, over the preceding year.
   2. Fill in the "Annual percentage change" columns by expressing the rate of change as a percentage of the preceding year's tuition (to the nearest percent).

   Current dollars

   Academic year Tuition Rate of change Annual % change $99-00$ $\$3362$ — — $00-01$ $\$3508$ $\$146$ $4.3$% $01-02$ $\$3766$ $02-03$ $\$4098$ $03-04$ $\$4645$ $04-05$ $\$5126$ $05-06$ $\$5492$ $06-07$ $\$5836$

   Source: The College Board, New York, NY.

   Inflation-adjusted dollars

   Tuition Rate of change Annual % change $\$4102$ — — $\$4139$ $\$37$ $0.9$% $\$4326$ $\$4624$ $\$5131$ $\$5516$ $\$5702$ $\$5836$

   Source: The College Board, New York, NY.

2. From the table we see that the cost of tuition has changed from year to year over this ten-year period, in both actual and inflation-adjusted dollars.
   1. Did tuition increase every year over this period?
   2. Did the rate of change increase every year over this period? If not, describe how the rate of change of tuition changed over this period.
3. Consider the actual current dollar data shown in the table.
   1. Over what period was the rate of change of tuition increasing?
   2. Over what period was the rate of change of tuition decreasing?
4. Repeat Question 3 for the inflation-adjusted data.
5. 1. Which do you think is a better way of measuring the change in tuition, actual dollars or inflation-adjusted dollars?
   2. Which do you think is a better way of expressing the rate of change of tuition, dollars per year or the percentage change per year?

II. Rates of Change and the Shapes of Graphs

The graphs in Figure 1 show the temperatures in Springfield over a $12$-hour period on two different days, starting at midnight. From the graphs we see that a warm front was moving in overnight, causing the temperature to rise.

1. Complete the following tables by finding the average rates of change of temperature over consecutive $2$-hour intervals. Read the temperature from each graph as accurately as you can.

   Temperature on Day 1

   Time interval (h) $[0, 2]$ $[2, 4]$ $[4, 6]$ $[6, 8]$ $[8, 10]$ $[10, 12]$ Average rate of change on interval $\frac{11 - 5}{2 - 0} = 3.0$

   Temperature on Day 2

   Time interval (h) $[0, 2]$ $[2, 4]$ $[4, 6]$ $[6, 8]$ $[8, 10]$ $[10, 12]$ Average rate of change on interval $\frac{5.5 - 5}{2 - 0} = 0.25$

2. It is obvious from the graphs that the temperature increases on both days. Is the average rate of change of temperature increasing or decreasing on Day 1? On Day 2?
3. Which of the basic shapes in Figure 2 below best describe each of the graphs in Figure 1?
   
   
   

   Figure 2: Basic shapes

4. Use your answers to Questions 2 and 3 to complete the following table.

   	Day 1	Day 2
   Temperature	Increasing
   Average rate of change	Decreasing
   Shape of graph

5. On Day 3 a cold front moves into Springfield. The following table shows the temperatures on that day.
   1. Plot the data and connect the points with a smooth curve.

      Time	Temperature (°C)
      $0$	$24$
      $2$	$23$
      $4$	$21$
      $6$	$18$
      $8$	$14$
      $10$	$9$
      $12$	$2$

      
      
   2. Complete the following table by finding the average rates of change of temperature over consecutive $2$-hour intervals.
      

      Temperature on Day 3

      Interval time (h) $[0, 2]$ $[2, 4]$ $[4, 6]$ $[6, 8]$ $[8, 10]$ $[10, 12]$ Average rate of change on interval $\frac{23 - 24}{2 - 0} = -0.5$

   3. The temperature is decreasing in this $12$-hour period. Is the average rate of change of temperature increasing or decreasing?
   4. Which of the basic shapes in Figure 2 best describes the graph for Day 3?
6. On Day 4 the temperature also decreases from $24$° C to $2$° C between midnight and noon. But this time, the rate of change of temperature is increasing. Sketch a rough graph that describes this situation. (You may find it helpful to consider the basic shapes in Figure 2.)
7. Use your answers to Questions 5 and 6 to complete the following table.

   	Day 3	Day 4
   Temperature
   Average rate of change
   Shape of graph

8. Six different functions are graphed. For each function, determine whether the function is increasing or decreasing and whether the average rate of change is increasing or decreasing.
   
   
   

   Function:		Increasing
   Rate of change:		Decreasing

   
   
   

   Function:
   Rate of change:

   
   
   

   Function:
   Rate of change:

   
   
   

   Function:
   Rate of change:

   
   
   

   Function:
   Rate of change:

   
   
   

   Function:
   Rate of change:

III. SAT Scores

We began this exploration with a discussion of the change in the rate of change of SAT scores. Let's examine what really happened to SAT scores. The table shows the combined verbal and mathematical SAT scores between 1988 and 2002.

Year	$1988$	$1990$	$1992$	$1994$	$1996$	$1998$	$2000$	$2002$
SAT score	$1006$	$1001$	$1001$	$1003$	$1013$	$1017$	$1019$	$1020$

Let's examine SAT scores from 1994 to 2002.

1. Did SAT scores decrease or increase in this period?
2. Did the rate of change in SAT scores increase or decrease in this period?
3. Did Homer Simpson's US of A Today newspaper report the facts accurately?