Name____________________________ Period______ Date_________Linear Functions – Activity 2.1
Linear Functions – Activity 2.1
Walking for Fitness: Suppose you are a member of a health and fitness club. A special diet and exercise
program has been developed for you by your personal trainer. At the beginning of the program, and
once a week thereafter, you are tested on the treadmill. The test consists of how many minutes it takes
you to walk, jog, or run 3 miles on the treadmill. The following data gives your time, 𝑡, over an 8 week
period.
End of Week, w
Time, t (in minutes)
0
45
1
42
2
40
3
39
4
38
5
38
6
37
7
39
8
36
Note that 𝑤 = 0 corresponds to the first time on the treadmill, 𝑤 = 1 is the end of the first week, 𝑤 =
2 is the end of the second week, and so on.
1. a. Is time, 𝑡, a function of weeks, 𝑤? If so, what are the input and output variables?
b. Plot the data points using ordered pairs of the form (𝑤, 𝑡)
Definition - Scatterplot
a set of points in the plane whose coordinate pairs represent input/output pairs of a data set.
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Now use the Nspire to provide a more accurate representation of your data:
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1. scroll to top row and enter week in A
2. scroll to right and enter time in B
3. scroll to the left and down to the row called 1 and enter the data for week by hitting
enter after each number
4. after entering 8, scroll to the right and up to row 1 under time.
5. Enter the time data by hitting enter after each number
6. Enter /I 5 and you will see a scatter plot
7. Scroll to the bottom of the screen and click on it. Scroll down to week and hit enter. You
will see the data move.
8. Scroll to the left and up until you see a box. Hit enter and scroll to time.
9. Sketch this graph below and label the x and y axis:
2. a. What was your treadmill time at the beginning of the program?
b. What was your treadmill time at the end of the first week?
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An important question that can be asked about this situation is, how did your time change from one
week to the next?
3. a. During which week(s) did your time increase?
b. During which week(s) did your time decrease?
c. During which week(s) did your time remain unchanged?
Procedure:
Determining Total Change
The change in time, t, is represented by the symbol ∆𝑡. The symbol ∆ (delta) is used to represent
“change in.” You generally calculate the change in time, 𝑡, from a first (initial) value to a second (final)
value of 𝑡. The first time is represented by 𝑡1 (“𝑡 sub 1”), and the second time is represented by 𝑡2 (“𝑡
sub 2”). The change in 𝑡 is then calculated by subtracting the first (initial value) from the second (final)
value. This is mathematically represented by
∆𝑡 = 𝑡2 − 𝑡1 or ∆𝑡 =final time – initial time.
Because 𝑡 is the output variable, ∆𝑡 is the change in output.
Similarly ∆𝑤 = 𝑤2 − 𝑤1 or ∆𝑤 =final week – initial week.
Because 𝑤 is the input variable, ∆𝑤 is the change in input.
4. Your time decreased during each week of the first 4 weeks of the program.
a. Determine the total change in time, 𝑡, during the first 4 weeks of the program (i.e., from 𝑡 =
45 to 𝑡 = 38). Why should your answer contain a negative sign? Explain.
b. Determine the change in weeks, w, during this period (i.e., from 𝑤 = 0 to 𝑤 = 4).
5. Use the ∆ notation to express your results in Problems 4a and 4b.
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Average Rate of Change
Neither the change in treadmill time nor the change in the number of weeks completely describes your
∆𝑡
progress during the first 4 weeks. The ratio of the change in 𝑡, ∆𝑡, to the change in 𝑤, ∆𝑤, written as ∆𝑤,
provides more relevant information about the effect of the exercise program over time. This ratio,
∆𝑡
∆𝑤
,
shows how the time changed on average over the 4-week period.
∆𝑡
6. Use your results from Problem 5 to determine the ratio ∆𝑤 during the first 4-week period.
Interpret your answer.
7. a. What are the units of measurement of the ratio determined in Problem 6?
b. On your graph from Problem 1, connect points (0, 45) and (4, 38) with a line segment. Does the
output increase, decrease, or remain unchanged over the interval?
The ratio
∆𝑡
∆𝑤
is called the average rate of change of time, 𝑡, with respect to weeks, 𝑤.
Definition - Average Rate of Change
If 𝑥 represents the input and 𝑦 represents the output, then the quotient
change in output
change in input
∆𝑦
𝑦 −𝑦
= ∆𝑥 = 𝑥2 −𝑥1
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Is called the average rate of change of 𝑦 with respect to 𝑥 over the 𝑥-interval from 𝑥1 to 𝑥2 . The units
∆𝑦
of measurement of the quantity ∆𝑥 are 𝑦-units per 𝑥 units.
8. a. Determine the average rate of change of t with respect to w during the sixth and seventh weeks
(from the point where 𝒘 = 𝟓 to the points where 𝒘 = 𝟕).
b. What is the significance of the positive sign of the average rate of change to this situation?
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c. Connect the data points (5, 38) and (7, 39) on your graph from Problem 1, using a line segment.
Is the output increasing, decreasing, or constant on the interval?
9. a. At what average rate did your time change during the fifth week (from 𝑤 = 4 to 𝑤 = 5)?
b. Interpret your answer in this situation.
c. Connect the data points (4, 38) and (5, 38) on your graph from Problem 1, using a line segment.
Is the output increasing, decreasing, or constant on the interval?
10. a. At what average rate is your time changing as w increases from 𝑤 = 3 to 𝑤 = 7?
b. Does your answer mean that your time did not change in this 4-week period? Interpret your
answer in this situation.
11. As part of your special diet and exercise program, you record your weight at the beginning of the
program and each week thereafter. The following data gives your weight, 𝑤, over a 5-week period.
Weeks, w
0
1
2
3
4
5
Weight, y (pounds)
196
183
180
177
174
171
a. Determine the average rate of change of your weight during the first 3 weeks.
b. Determine the average rate of change during the 5-week period.
c. Determine the change in weight during each week of your exercise program.
d. What are the units of measure of the average rate of change?
e. What is the practical meaning of the average rate of change in this situation?
f.
What can you say about the average rate of change of weight during the time interval in this
situation?
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Summary:
1. Let 𝑦1 represent the corresponding output value for the 𝑥1 and 𝑦2 represent the
corresponding output value for the 𝑥2 . As the variable 𝑥 changes in value from 𝑥1 to 𝑥2 ,
a. The change in 𝑥 is represented by ∆𝑥 =_________________
b. The change in 𝑦 is represented by ∆𝑦 =_________________
2. The quotient
∆𝑦
∆𝑥
=
is called the average _________ ___ ____________
of 𝑦 with respect to 𝑥.
3. The line segment connecting the points (𝑥1 , 𝑦1 ) and (𝑥2 , 𝑦2 )
∆𝑦
a. Increases from left to right if ∆𝑥 ____ 0. The slope is __________________.
∆𝑦
b. Decreases from left to right if ∆𝑥 ____ 0. The slope is __________________.
∆𝑦
c. Remains constant from left to right if ∆𝑥 ____ 0. The slope is __________________.
4. The average rate of change indicates how much, and in which direction, the output changes
when the input increases by a single unit. It measures how the output changes on the
aver_______.
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Name_________________________________
Date_______
Lesson 2.1 Practice:
The following table of data from the United States Bureau of Census gives the median age of an
American man at the time of his first marriage:
Year
Median Age
1910
25.1
1920
24.6
1930
24.3
1940
24.3
1950
22.8
1960
22.8
1970
23.2
1980
24.7
1990
26.1
2000 2010
26.8 27.5
Use this data to answer Exercises 1-6.
1. a. Determine the average rate of change in median age per year from 1950 to 2010.
b. Describe what the average rate of change in part a represents in this situation.
2. Determine the average rate of change in median age per year from 1930 to 1960.
3. What is the average rate of change over the 100-year period described in the table?
4. During what 10-year period did the average age increase the most?
5. a. What does it mean in this situation if the average rate of change is negative?
b. Determine at least one 10-year period when the average rate of change is negative.
c. What trend would you observe in the graph of median age if the average rate of change were
negative? That is, would the graph rise, fall, or remain horizontal?
6. a. Is the average rate of change zero over any 10-year period? Is so, when?
b. What does a rate of change of zero mean in this situation?
c. What trend would you observe in the graph during this period? That is, would the graph rise,
fall, or remain horizontal?
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7. The following table gives information about hotel construction in the United States from the fourth
quarter of 2007 through the first quarter of 2009:
Yearly Quarter
Quarters since 2007, Q4, 𝑡
Number of New Hotel Projects, ℎ
2007 Q4
0
439
2008 Q1
1
459
2008 Q2
2
381
2008 Q3
3
403
2008 Q4
4
289
2009 Q1
5
257
a. Plot the data points using ordered pairs of the form (𝑡, ℎ).
b. Determine the average rate of change of new hotel construction projects from 2007 Q4 to 2008 Q1.
c. Determine the average rate of change of new hotel construction projects from 2008 Q3 to 2009 Q1.
d. Compare the average rate of change from 2007 Q4 to 2008 Q1 with the rate of change from
2008 Q3 to 2009 Q1.
e. When the average rate of change is negative, what trend will you observe in the graph? What does
that mean in this situation?
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8. Between 1960 and 2007, the size and shape of automobiles in the United States have changed
almost annually. The fuel consumed by these vehicles has also changed. The following table
describes the average fuel consumed per year per passenger car in gallons of gasoline.
Year, t
Gallons Consumed per
Passenger Car (average), g
1960
668
1970
760
1980
576
1990
520
1995
530
2000
547
2003
550
2007
547
a. Determine the average rate of change, in gallons of fuel used per passenger car, from 1960-1970.
b. Determine the average rate of change, in gallons of gas per year, from 1960 to 1990.
c. Determine the average rate of change, in gallons of gas per year, from 1995 to 2007.
d. Determine the average rate of change, in gallons of gas per year, from 1960 to 2007.
e. What does the result in part d mean in this situation?
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