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Algebra 1

4.8.2 Average Rate of Change and Slope

Algebra 14.8.2 Average Rate of Change and Slope
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Activity

The table and graph show a more complete picture of the temperature changes on the same winter day. The function T T gives the temperature, in degrees Fahrenheit, h h hours since noon.

A scatter plot showing temperature in °F versus hours since noon. Points start at 20°F at 0 hours, peak at 25°F at 4 hours, then drop to 8°F at 10 hours. There is a data point for every hour since noon.
h h 0 1 2 3 4 5 6 7 8 9 10 11 12
T ( h ) T ( h ) 18 19 20 20 25 23 17 15 11 11 8 6 7

In questions 1 – 3, find the average rate of change for the following intervals. Be prepared to show your reasoning.

1. between noon and 1 p.m.

2. between noon and 4 p.m.

3. between noon and midnight

In questions 4 – 6, use the slope formula to find the slope of the line that connects the following points.

4. ( 0 , 18 ) ( 0 , 18 ) and ( 1 , 19 ) ( 1 , 19 )

5. ( 0 , 18 ) ( 0 , 18 ) and ( 4 , 25 ) ( 4 , 25 )

6. ( 0 , 18 ) ( 0 , 18 ) and ( 12 , 7 ) ( 12 , 7 )

7. Compare your answers to questions 1 – 3 with your answers to questions 4 – 6. Explain what this tells you about the average rate of change and the slope of a line.

For questions 8 and 9, think back to Mai and Tyler’s disagreement about temperature to solve these problems.

8. Use the average rate of change to show which time period—4 p.m. to 6 p.m. or 6 p.m. to 10 p.m.—experienced a faster temperature drop.

9. Use the slope formula to show which line has the greater slope—the line connecting the points that represent the temperature at 4 p.m. and 6 p.m. or the line connecting the points that represent the temperature at 6 p.m. and 10 p.m.

10. Compare your answers to question 8 and question 9.

Are you ready for more?

Extending Your Thinking

Refer to graph and table of the previous function T T to answer the questions:

1. Over what interval did the temperature decrease the most rapidly?

2. Over what interval did the temperature increase the most rapidly?

Self Check

The temperature t , in degrees Fahrenheit, of a cup of coffee that was poured and then forgotten for m minutes is represented in the table below

Time in minutes 0 15 25 40
Temperature 185°F 155°F 140°F 120°F

What was the average rate of change from when the cup was poured to 40 minutes later? 

  1. -1.625 degrees per minute
  2. -1.333 degrees per minute
  3. -0.615 degrees per minute
  4. -2 degrees per minute

Additional Resources

Finding Rates of Change

Jay was racing bikes with a friend on the way to the park. A table and a graph of the times are shown below.

Time (minutes) Distance (miles)
0 0
5 0.84
10 1.86
15 3.00
20 4.27
25 5.67
Jay
A scatterplot that shows the relationship between time in minutes (x-axis) and distance in miles (y-axis) plotted points include (0, 0), (5, 0.84), (10, 1..86), (15, 3), (20, 42.7), and (25, 5.67).

What is the average rate of change of the first 10 minutes of the race?

To find the average rate of change of the first 10 minutes of the race, you would look at the first data value on the table and the third data value on the table, since it corresponds to 10 minutes.

Remember, the average rate of change formula is f ( b ) − f ( a ) b − a f ( b ) − f ( a ) b − a , where ( b , f ( b ) ) ( b , f ( b ) ) typically represents your second data point and ( a , f ( a ) ) ( a , f ( a ) ) represents your first data point.

For this situation, it would be ( 0 , 0 ) ( 0 , 0 ) or a = 0 a = 0 and f ( a ) = 0 f ( a ) = 0 and ( 10 , 1.86 ) ( 10 , 1.86 ) or b = 10 b = 10 and f ( b ) = 1.86 f ( b ) = 1.86 .

Substituting the points into the average rate of change formula results in the following:

f ( b ) − f ( a ) b − a f ( b ) − f ( a ) b − a , which is 1.86 10 1.86 10 or 0.186 miles per minute.

Notice that the answer is positive and the points on the graph are higher each time.

Try it

Try It: Finding Rates of Change

Jay was racing bikes with a friend on the way to the park. A table and a graph of the times are shown below.

Time (minutes) Distance (miles)
0 0
5 0.84
10 1.86
15 3.00
20 4.27
25 5.67
Jay
A scatterplot that shows the relationship between time in minutes (x-axis) and distance in miles (y-axis) plotted points include (0, 0), (5, 0.84), (10, 1..86), (15, 3), (20, 42.7), and (25, 5.67).

What was Jay’s average rate of change from the beginning to 25 minutes later?

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