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Module 5 Piecewise Functions

prepared by Nell Sloan, Pender County Schools Helen Whitley, New Hanover County Schools

Chapter

5.1. North Carolina Standard Course of Study........................................................................144

5.2. vocabulary ......................................................144

5.3. Introductory material ..................145

5.4. background discussion of mathematics................................................ 146

5.5. graphing piecewise-defined functions..........................................................149

5.6. ties to textbooks ..............................150

5.7. worked examples...................................151

5.8. piecing together piecewise functions..........................................................154

5.9. Activity..................................................................155

5.10. applications of piecewise functions (teacher notes) ....157

5.11. applications of piecewise functions.......................................................... 161

5.12. piecewise defined functions as models............................................................ 162

5.13. Student handout(olymics).......166

5.16. student handout(New jersey turnpike) .............................................................. 167

5.17 practice problems.............................. 174

5

143 Sponsored by NC Math and Science Education Network

Advanced Functions and Modeling Workshop Summer 2004

5.1. North Carolina Standard Course of Study

GOAL 2: The learner will use functions to solve problems.

2.02 Use piecewise-defined functions to model and solve problems; justify results.

a) Solve using tables, graphs, and algebraic properties. b) Interpret the constants, coefficients, and bases in the context of the problem. c) Create and use simulations for probability models. d) Find expected values and determine fairness. e) Identify and use discrete random variables to solve problems. f) Apply the Binomial Theorem.

5.2. Vocabulary

Graph

Independent

Dependent

Domain

Range

Minimum

Maximum

Increasing

Decreasing

Global vs. Local Behavior

Continuous

Discrete

System of Equations

Solve Equations, Justifying Steps Used

Module 5: Piecewise Defined Functions 144


5.3. Introductory material

You can have the students start thinking about “real world” examples of piecewise functions. You can have them select several topics that interest them from the list below or you can have your students generate their own list. After working some real world problems in class, I would have each group of students select several ideas and create several “real world” problems. The students should define the interval of values for the domain, solve their problem, and graph their solution. This could be a mini group project.

This is a topic were the textbooks usually do not list as many real world examples. This project would give the teacher several problems that interest the students for future quizzes, homework, or test.

List of possible ideas • An ideal weight chart for men(women) based on height • Average blood alcohol concentration of a certain weight and gender based on the number of

drinks the individual consumed • The periodic table of elements • Blood alcohol level for DUI and DWI • High school scales for Honor Rolls • A dart board with the outer circle worth 25 pts and the inner worth 50pts • Shipping and handling charge from a catalog • Astrological signs based on date of ones birthday • Dog food • Cell phone charges • Parking meters • Wrestling weight classes • Price of a hockey(basketball) ticket based on where the seat is located • Price of concert tickets based on where the seat is located • Sizes of T-shirt • Suggested snowboard lengths based on weight • Airfare prices for first class • Number of points on your license based on mph over the speed limit • Sales tax • Income tax • Cost of tanning salon based on time • Time Schedule for high school classes (1st period: 8:00-9:30) • Shades for hair dying • Number of ticket awarded on skee ball based on score



5.4. Background discussion of mathematics

Piecewise Function Definitions3

We can define functions with more than one formula applied along different parts of the function's domain.

EXAMPLE 1: Define the function yx

=1 so that there is a value of 0 at x=0:

f x xx

xb g = ≠

=

RS|T|

1 0

0 0

,

,

The domain and range for this function are:

( ) ( ): , : ,D R−∞ ∞ −∞ ∞

EXAMPLE 2: Define a 3-part function:

f xx xx x

x xb g=

>

≤ <≥

RS|T|

,,,

00 2

4 2

3

The domain and range for this function are: ( ) ( ): , : ,D R−∞ ∞ −∞ ∞

3 Taken from http://www.npac.syr.edu/REU/reu94/williams/ch1/subsection3_3_6.html#SECTION0003600000000000000



EXAMPLE 3: Define a “step-wise'' function (a function that jumps between sections) with [ 3,3]D= −

1, 3 1( ) 0, 1 1

1, 1 3

xf x x

x

⎧− − ≤ ≤−⎪⎪⎪⎪= − < <⎨⎪⎪ ≤ <⎪⎪⎩

The range for this function is three separate y-values: R y y= = −{ ,1 0 1, }



The following is a site to help students graph piecewise functions when the following restraints apply D: − ≤ ≤40 40x and R : − ≤ ≤40 40x . It uses the idea of vertical strips. This site would need to be utilized very early in the unit before you teach them how to graph piecewise functions using a graphing calculator.

http://kerbaugh.uncfsu.edu/piecewise/piecewise.html

The following site give the students and teachers computational practice with the answers explained as well as one “real world” problem.

http://projects.hprtec.org/~newberry/cqdemo/FieldGuide/piecewise/quiz.html


http://kerbaugh.uncfsu.edu/piecewise/piecewise.html

http://projects.hprtec.org/~newberry/cqdemo/FieldGuide/piecewise/quiz.html


The following material gives students and teachers instructions for graphing piecewise functions using graphing calculators.

5.5. Graphing Piecewise-Defined Functions4

Example: Graph 2

2 3( )

2x if x

f xx if x

+ <⎧= ⎨ > −⎩

2−

STEP 1: Go to o. Let Y1= 2X + 3 and Y2 = x^2.

STEP 2: Under Y1 enter

2X + 3/(X < -2)

and in Y2 enter

X^2/(X > -2).

To find the inequality signs, go to y under the : menu.

STEP 3: Before you graph the function, go to z and use the blue down arrow key to go to the line that reads

Connected Dot. Use the blue right arrow key to highlight Dot and hit Í.

4 Taken from http://www.mtsu.edu/~math141/piece.html



4. Now press s.

Piecewise functions can also be entered as one equation.

STEP 1: Go to o. Under Y1 enter (2x + 3)(x < -2) + (x^2)(x > -2)

2. Follow steps 3 and 4 above to graph.

5.6. Ties to sections in textbooks

Algebra and Trigonometry: Study guide p. 76: Examples 4 and 5 Textbook: p. 213 example 3; p. 221-223

Functions Modeling Change: Instructors Manual: Suggested Lesson Plan p. 25 and 26 and Worksheet on p. 202.

Textbook: p. 71-74; problem 16 to be used as an introduction in addition to the example of the carpeting found in the manual.



5.7. Worked examples

Piecewise Functions Piecewise function — a function where different equations are used for different intervals of the

domain. The definition of the function on the intervals determines if the function pieces connect.

Example 1: The absolute value function is a piecewise function. Consider ( )f x = x . The absolute value of a number is defined to be the distance from that number on the number line to 0. The absolute value of a number is always nonnegative. The table provided lists a specific domain and the corresponding range value. Notice that the domain of the graph consist of all real numbers; however, the range only includes only the nonnegative real numbers. The absolute value function is an example of a continuous function since you can trace the graph without lifting your pencil off the paper.

x f(x)

–5 5

–3 3

0 0

3 3

6 6

( )

0( )

0

f x x

x if xf x

x if x

=⎧ ≥⎪⎪=⎨⎪− <⎪⎩

Example 2: Todd traveled at a rate of 40 mph for 6 minutes. Then he stopped at a stoplight for 2 minutes. Then for 5 minutes he traveled 20 mph through a school zone. He sat at the school for 4 minutes while his sister got out of the car. Then he traveled home at 45 mph.

This is another example of a continuous piecewise function. You do not pick since you can trace the graph without lifting your pencil off the paper. Let’s graph Todd’s travel to school.

In pairs I want each of you to create a piecewise graph for your travel to school. Estimate your times and speed.

Example 3: A piecewise function in which the graph appears to be a set of stairs is called a step function. The step function is an example of a piecewise function in not continuous since you can not trace the function without lifting your pencil. One type of step function is called the greatest integer function. The symbol x means the greatest integer not

greater than x. For example, because 9 would be greater than 8.3. Another

example is because –3 would be greater than –3.4.

8.3 8=

3.4 4− =−



x f(x)

− ≤ ≤ −4 3x –4 − ≤ ≤ −3 2x –3 − ≤ ≤ −2 1x –2 − ≤ ≤1 0x –1 0 1≤ ≤x 0 1 2≤ ≤x 1 2 3≤ ≤x 2 3 4≤ ≤x 3

( )f x x=

Example 4: Together let’s utilize the chart below that list the shipping charges that apply to order from a catalog to find the following:

Value of Order Shipping, Packing and Handling Charge

$0.00-$25.00 $3.00 $25.01-75.00 $5.75 $75.01-125.00 $7.75 $125.01 and up $9.75

This is an example of a “step” piecewise function. a) Write the shipping charges as a function of the value of the order.

v is the value of the order s(v) is the shipping

s v

if vif vif vif v

( )

.

. . .. .. .

=

≤ ≤≤ ≤≤ ≤

≤

RS||

T||

300 0 25575 2501 75007 75 7501 125009 75 12501

.

b) Graph the function.

c) Select a catalogue and write down the name of your catalogue at the top of your group’s transparency. On the transparency, write the shipping and handling charges located in the catalogue you selected. Now your group is to define a piecewise function that will enable someone to calculate the shipping and handling charges they would owe if they ordered from your catalogue. On the transparency, write the piecewise function your group defined and graph your function.



Describe each of the following functions as either an absolute value function or as a step function. Write a rule for the function and graph each function.

i. The stated weight of a box of cereal is 15.5 ounces. The company randomly chooses boxes to test to see whether their equipment is dispensing the right amount of cereal. If the discrepancy is more than .4 ounce, the production line is stopped for adjustments.

ii. On a certain telephone rate plan, the price of a cellular telephone call is .25 per minute or fraction of a minute.

iii. The labor cost for repairing a computer is $40 per hour or fraction of a hour.

iv. The table provided shows the charge for renting a bicycle from a rental shop on Bald Head Island for different amounts of time.

Time Price ½ hour $7.00 1 hour $12.00 2 hours $17.00 All day $25.00

v. The cost of mailing a letter is $0.37 for the first ounce and $0.22 for each additional ounce or portion of an ounce.

vi. A can of coffee is supposed to contain one pound of coffee; however the can including the coffee weighs one pound. How would the actual weight of the coffee compared to one pound be represented?

vii. A company charges $200 a month to organize a company’s payroll for up to 20 employees and an additional $100 a month for each 20 employees over 20.



5.8. Piecing Together Piecewise Functions

DESCRIPTORS: Graphing equations, cooperative learning, constructive learning

EXPLANATION: Different groups are asked to graph several different functions. Then they are asked to transfer these graphs onto trace paper, but only over certain parts of the domain. After this is done, students pair up with classmates form different groups and piece their new functions together creating piecewise-defined functions. Graphing several piecewise functions on a graphing calculator enhances their understanding further.

SOURCE: King, Sybrina L. "Piecing Together Piecewise Functions." The Mathematics Teacher 90 (October 1997): 550-552.

http://web.centre.edu/edu/Student_pages/MATH/ACT5~1.HTM


http://web.centre.edu/edu/Student_pages/MATH/ACT5~1.HTM


5.9. Activity: Piecewise Functions

Overview: Given a description of the cost per square yard of carpet, students are asked to find the cost of specific amounts of carpet, to graph the data points found, and to find a function that gives the cost of carpet as a function of yardage.

Estimated Time Required: The activity should take approximated 30 minutes.

Technology: none

Prerequisite Concepts: Piecewise function

Discussion: Point out that a function may or may not have a formula giving its rule. In the same manner,

there is no reason that one function cannot have a rule that requires more than one formula. Students may have trouble deriving the equations for the different amounts of carpet.

Encourage them to calculate the prices for different purchases of carpet and then use those points to calculate each of the equations of the linear functions involved. If you have students do this, be aware that there are many ways that students can go wrong here. For example, if they calculated the line between a point where x < 100 and a point where x > 100. If you can afford the class time, these failures are highly illustrative of the basic principle of formulas for piecewise function that the given formulas only apply over a limited range of input values, not over the entire domain of the function.

After students complete part a and b of the activity, you can introduce the standard notation for representing piecewise function and their domains. Be sure to emphasize that this is one function that just happens to use two different formulas in its rule.



Activity: Piecewise-defined Functions Student Handout

Suppose a carpet store sells carpet for $10 per square yard for the first 100 square yards purchased, and then lowers the price to $7 per square yard after the first 100 yards have been purchased.

a) Find the cost of buying the yardages indicated in the table:

Square Yards

(x)

Cost (C)

0

25

50

75

100

125

150

175

200

b) Graph the points you found above. Label your scale clearly.

c) Find a function C=f(x) that gives the cost of purchasing any number of square yards of carpet between 0 and 200.

http://www2.austincc.edu/lochbaum/03-4%20Piecewise%20Functions.pdf


http://www2.austincc.edu/lochbaum/03-4 Piecewise Functions.pdf


5.10. Applications of Piecewise Functions Teacher’s Notes

Physics Topics: Free fall with Variable Air Resistance Math Topics: Linear and Quadratic Equations, Domain, Range, Dependent and Independent Variables, Profit, Cost, Revenue

I. Overview This activity is designed to provide proficiency in modeling with piecewise functions in various applications. Two of the applications have data sets provided and should be performed with a spreadsheet, while the other two are analytical and can be performed manually. As an option, a graphing calculator may be used for the latter two applications. Sample spreadsheet solutions are attached.

II. Goals and Objectives The student will be able to:

o connect physical phenomena to characteristics of a scatterplot; determine from this whether a piecewise defined function should be used to model the data.

o determine the realistic domain and range for a piecewise function. o use interval inequality notation to define piecewise functions over appropriate intervals. o recognize the need for a linear piecewise function from the wording of a problem, and

formulate the model using the slope - intercept or point - slope formula. o use the appropriate interval to make predictions with the piecewise model. o explain why piecewise functions are functions vis - a - vis the definition of a function. o explain the physical significance of the vertical intercept and slope of the models. o explain the meaning of cost, profit, revenue, and the break - even point in the cost vs.

profit problem.

III. Concepts This activity is appropriate for students at a level of Algebra I and beyond. Concepts covered include graphical analysis, linear and quadratic equations, inequality notation, domain, range, dependent vs. independent variable, free fall with variable air resistance, and cost vs. profit functions.

IV. Time Frame This activity should take two class periods, or one class period with a homework assignment.

V. Facilities / Equipment Options A spreadsheet is needed for the first two scenarios. A graphics calculator may be used for the last two scenarios. Graph paper is needed for the manual graphs produced for the last two scenarios.

VI. Procedure See the student handout.



VII. Assessment The paragraphs in the student report section should provide the rationale for using a piecewise model vs. a single model from a particular family for the entire domain. Some explanation should be given stating the conditions in the problem which dictate that the model take different forms over different intervals in the domain. Pay particular attention to the inequality notation defining intervals, regarding the use of, e.g., "less than" vs. "less than or equal," vis-à-vis the definition of a function (single output for each input). A statement as to how the dependent and independent variables were chosen should be included.

The following questions are some that the instructor may ask of the students to aid in the assessment of their understanding, as well as to provoke further investigation.

• What would you change in the free fall problem to eliminate the need for a piecewise model?

• Label the free fall height vs. time curve with physical causes for each distinctive part of the graph (e.g., why did the graph go up like this? etc.)

• What would the free fall graph look like if the parachute were larger? • An earlier activity studied free fall without considering air resistance. In that activity, the

student was asked to determine the relationship between the leading coefficient of and the acceleration of gravity. That question could be revisited for

the portion of the graph in the free fall scenario before the parachute was opened but before terminal velocity was reached. Equilibrium conditions are discussed later in Unit 4; as a preliminary discussion, ask the students what the forces look like at terminal velocity.

200.5y gt v t= + + 0y

VIII. Comments Emphasize the need for providing a title, appropriate scale, and labeled axes for all graphs, including the spreadsheet graphs.



Applications of Piecewise Functions

Introduction The following tasks involve piecewise functions. In formulating the math model for these problems, consideration must be given to prevailing constraints in different parts of the problem.

Tools Provided A spreadsheet is needed to perform several tasks.

Task 1::A study was commissioned to predict the number of high school graduates in Wake County, NC over time. Since Wake County is experiencing rapid growth, this information is vital for planning purposes for local high schools, colleges, and universities. Data Set 1.

• Use a spreadsheet to scatterplot and find the best trendline for the data. • Are any of the trendline models adequate? What’s wrong? (Hint - separate the data into

two or more intervals and see if the results are better). • How many graduates do you predict for the year 2007?

Task 2: A sky diver jumps from an aircraft flying at a height of 3000 m. After 35 he deploys his parachute for the remainder of the fall to earth. Data Set 2.

• Use a spreadsheet to scatterplot the data and find the best trendline (considering breaking the data into parts as above).

• What happened to the graph when the parachute opened? • What other characteristics of the graph can be associated with physical phenomena?

Task 3: A dental insurance policy states its family coverage as: • $250 deductible • 80% payment for charges that exceed $250 but does not exceed $1500 • 50% payment for charges that exceed $1500

Find a model that relates payments and charges. Which should be the dependent variable? State the realistic domain and the realistic range.

• A second policy states that it pays 65% of all charges with no deductible. For what amount of charges is the first policy better? Solve analytically and graphically by using a spreadsheet to plot both plans on the same axes.

Task 4: In manufacturing a certain item, Smoof Deluxe Widgets, Inc. must pay a set-up cost of ( ) $2,650,000S x x=

1 25,000x< < ( ) $12,500 / monthF x, where x is the number of items it makes in a production run, with

. The fixed cost is = and the variable cost is $8/item. • Write the cost function for setting up and manufacturing x items a month. • If the item will sell for $22.25/item, write a function for the revenue from selling x items. • Write a function for the profit for manufacturing and selling x items. Assume the price

gets discounted for over 2,200 items. Use this function to determine the point at which Smoof, Inc. breaks even.



Report • Describe the method(s) used to analyze each of the scenarios. • Provide all spreadsheet outputs including scatterplots with trendlines. Write a paragraph

about each scenario giving the solutions to each problem. State your recommendations based on your results, and your rationale for your recommendations.

• For the last two scenarios, graph and annotate the models on graph paper. • What is different about the graphs produced in this activity? Why are they called

piecewise functions?

Data Set 1

Year High School Graduates

1992 3660 1993 3668 1994 3660 1995 3747 1996 3739 1997 3937 1998 4272 1999 4361 2000 4648 2001 4855 2002 5021 2003 5452 2004 5740

Task 1 data set (from the Division of Statistical Services,

North Carolina Department of Public Instruction, February, 1995)



Data Set 2

Time (s) Height (m) Time (s) Height (m) Time (s) Height (m) 0 3000 5.5 2873.37 46 1130.78 0.2 2999.95 6.0 2851.91 48 1112.93 0.3 2999.80 7 2806.07 50 1095.08 0.6 2998.78 8 2757.15 55 1050.45 0.8 2997.60 9 2705.93 60 1005.81 1.0 2996.04 10 2653.04 65.1 960.29 1.2 2994.10 15 2368.00 70.1 915.66 1.4 2991.78 20 2083.08 75.1 871.03 1.6 2989.08 25 1798.44 80.1 826.40 1.8 2986.01 30 1513.69 90.1 737.14 2.0 2982.59 34.5 1263.18 100.1 647.88 2.2 2978.80 35.0 1234.60 110.1 558.62 2.4 2974.67 35.5 1225.37 120.1 469.36 2.6 2970.21 36.0 1220.28 130.1 380.10 2.8 2965.41 36.5 1215.65 140.1 290.84 3.0 2960.29 37 1211.13 150.1 201.58 3.4 2949.13 38 1202.19 160.1 112.31 3.8 2936.80 39 1193.26 170.1 23.04 4.2 2923.40 40 1184.34 180.1 – 66.00 4.6 2909.00 42 1166.48 5.0 2983.68 44 1148.63

Task 2 Data Set (generated by students using a differential equation model for a project)

http://www.waketech.edu/math/Chimp/Unit3/PiecewiseFunctions.htm


http://www.waketech.edu/math/Chimp/Unit3/PiecewiseFunctions.htm


Piecewise Defined Function as Models Advanced Functions and Modeling

Goals: • Describe graphically, algebraically and verbally real-world phenomena as functions;

identify the independent and the dependent variable (3.01) • Translate among • graphic, algebraic, and verbal representations of relations (3.02) • Graph relations and functions and find the zero of functions (3.03) • Write and interpret an equation of a curve (linear) which models a set of data (4.01) • Find the equation of best fit (linear) for set of data. Interpret the constants, coefficients,

and based in the context of the data. Check the equation for goodness of fit and use equation for prediction (4.02)

Materials Needed: 1. Copy of Student Handout 2. Graphing Calculator 3. Paper and Pencil for note taking

Activity One: Since the 1924 Olympics, men and women have competed separately in the 400 Meter Free Style swimming events. The winning times for each Olympics are in seconds and are given the table below:

Year Men Women Year Men Women 1924 304.2 362.2 1968 249.0 271.8 1928 301.6 342.8 1972 240.27 259.04 1932 288.4 328.5 1976 231.93 249.89 1936 284.5 326.4 1980 231.31 248.76 1948 281.0 317.8 1984 231.23 247.1 1952 270.7 312.1 1988 226.95 243.85 1956 267.3 294.6 1992 225.00 247.18 1960 258.3 290.6 1996 227.95 247.25 1964 252.2 283.3 2000 (This problem is taken from the Algebra II indicators prepared by DPI.

It is associated with goal 4.01)

1. Create a scatter plot of the (year, men’s time). In order to make the y-intercept meaningful, move the data to the left to make (1924, 304.2) become (0, 304.2). This will change the meaning of the ordered pairs to be ( years since 1924, men’s time)



Piecewise Defined Functions as Models 1 NCSSM Distance Learning

2. The trend in the data from the earlier Olympics does not seem to be the same as the trend in more recent Olympics. Where would you say the change in the trend occurs? Support your answer. Also, discuss what may be happening in the sport for the trends to change.

This is a question that has several possible answers. For the sake of documentation, we will use 1976 or as the point at which the trend seems to change. The slope seems quite steep before 1976 and is much flatter from 1976 forward. This kind of change could occur because some physical situation like design of the pool or measurement of times changes. However, this is more likely to represent the point at which maximum effort is going into training and physical situations of swimming. From 1976 forward, swimmers could only affect subtle parts of the sport like training or change in technique.

52x =

3. Break the data into two parts, before the change in the trend and after the change in the trend. Find linear equations that are the best fit for each part. Include the set of x-values for which each equation is true.

For years 1924 through 1976 or 0 52x≤ ≤

For years 1976 through 1996 or 52x ≥




Together:

4. Write a short paragraph to contrast the information in the two trends-earlier and most recently. Use the ideas of slope and y-intercepts of the prediction equations in this discussion.

From 1924 to 1976 according to the slope of the line, the winning times were decreasing by 1.301 seconds every year or more than 5 seconds between Olympics. The y-intercept of the line for 1924 to 1976 shows the winning time in 1924 (x = 0) to be 304.7, which is a very accurate description of that point. After 1976, the slope is much flatter which means the winning times were not changing much. The slope of the line after 1976 time show winning times decreased by 0.31 seconds per year or 1.24 seconds between Olympics. This is 25% of the decline that had occurred before. Swimmers were not able to greatly affect their times. Also the after 1976 linear equation predicts the winning time of 1924 (its y-intercept) as 248.2 seconds, while the data show the winning time to be 304.2 seconds for men in 1924.

5. Use each model to predict the winning time in the 400 Meter Free Style for men in the 2000 Olympics and the 2004 Olympics.

Equation Prediction for 2000 Olympics

Prediction for 2004 Olympics

1.301 304.665y x= − + 205.82 seconds 200.62 seconds 0.308 248.153y x= − + 224.75 seconds 223.52 seconds

6. Ian Thorpe, an Australian nicknamed the “Thorpedo,” won gold in the 2000 Olympics for 400 meter free style with a world record: 220.59 seconds. Which of the lines best predicts the time for his swim?

Ian Thorpe’s time is best predicted by the second equation 0.308 248.153y x= − + . Since this is for the 2000 Olympics, it seems appropriate to use the model for years since 1976.

7. Answer the same question for the women’s winning times. Brooke Bennett, a US swimmer, won gold in Sydney for 400-meter free style with a time of 245.80 seconds. Which model best predicts this result?




It is best to separate the women’s times at 1976 as well. The resulting equation is 1.888 354.868 0 520.151 256.772 52

x xy

x x− + ≤ ≤⎧

= ⎨− + ≥⎩

To predict from either of the equations produces the following results:

Equation Prediction for 2000 Olympics

Prediction for 2004 Olympics

1.888 354.868y x= − + 211.35 seconds 203.80 seconds 0.151 256.722y x= − + 245.22 seconds 244.61 seconds



Student Handout

Since the 1924 Olympics, men and women have competed separately in the 400 Meter Free Style swimming events. The winning times for each Olympics are in seconds and are given the table below:

Year Men Women Year Men Women 1924 304.2 362.2 1968 249.0 271.8 1928 301.6 342.8 1972 240.27 259.04 1932 288.4 328.5 1976 231.93 249.89 1936 284.5 326.4 1980 231.31 248.76 1948 281.0 317.8 1984 231.23 247.1 1952 270.7 312.1 1988 226.95 243.85 1956 267.3 294.6 1992 225.00 247.18 1960 258.3 290.6 1996 227.95 247.25 1964 252.2 283.3 2000 (This problem is taken from the Algebra II indicators prepared by DPI.

It is associated with goal 4.01)

1. Create a scatter plot of the (year, men’s time). In order to make the y-intercept meaningful, move the data to the left to make (1924, 304.2) become (0, 304.2). This will change the meaning of the ordered pairs to be ( years since 1924, men’s time)

2. The trend in the data from the earlier Olympics does not seem to be the same as the trend in more recent Olympics. Where would you say the change in the trend occurs? Support your answer. Also, discuss what may be happening in the sport for the trends to change.

3. Break the data into two parts, before the change in the trend and after the change in the trend. Find linear equations that are the best fit for each part. Include the set of x-values for which each equation is true.

4. Write a short paragraph to contrast the information in the two trends-earlier and most recently. Use the ideas of slope and y-intercepts of the prediction equations in this discussion.

5. Use each model to predict the winning time in the 400 Meter Free Style for men in the 2000 Olympics and the 2004 Olympics.

6. Ian Thorpe, an Australian nicknamed the “Thorpedo,” won gold in the 2000 Olympics for 400 meter free style with a world record: 220.59 seconds. Which of the lines best predicts the time for his swim?

7. As a follow-up problem, answer the same question for the women’s winning times. Brooke Bennett, a US swimmer, won gold in Sydney for 400-meter free style with a time of 245.80 seconds. Which model best predicts this result?



Student Handout Name

New Jersey Turnpike: A Toll Road

This table shows exits and tolls on the New Jersey Turnpike. When a car enters the Turnpike, the driver is given a ticket specifying the entry exit. When the driver exits the Turnpike, the total toll is collected. 1. Use a map to become familiar with the Turnpike and its exits. 2. Using mileage and cost create a scatter plot of the data. 3. The data point at exit 6 seems out of place on the scatter plot.

If you take exit 6, where will it take you? Remove exit 6 from your data set and re-create the scatter plot.

4. Which function best fits the data: a linear, a quadratic, or an exponential function? Support your answer. Use residuals to help with this judgment.

5. The data seems to be in two parts—both linear. Separate the data and find models for each part. Write a piecewise defined function for the data and discuss goodness of fit.

6. For each linear expression developed to fit the data, explain the meaning of the slope and the y-intercept in the context of the problem.

7. What toll would your model predict for: 50 miles (Exit 6), 95 miles (Exit 12) and 112 miles (Exit 16W)?

8. If a new exit is added between Exits 2 and 3 at 20 miles, what would be an appropriate cost?

9. Go back to a map or description to look at the two parts of the data. Is there a geographical reason for the tolls having two models? If so, what is the reason? Which of the two models is most expensive? Why?

Exit Number Mileage Cost 1 0 $0.00 2 12 $0.55 3 25 $0.85 4 33 $1.15 5 43 $1.45 6 50 $2.20 7 52 $1.75 7A 59 $1.85 8 67 $2.05 8A 73 $2.20 9 82 $2.60 10 87 $2.65 11 90 $2.90 12 95 $3.20 13 99 $3.65 13A 100 $4.25 14 104 $4.25 14A 107 $4.90 14B 109 $5.00 14C 110 $5.30 15E 106 $4.50 15W 108 $4.70 16E 111 $5.00 16W 112 $5.00

10. If we had studied the trip on the toll road beginning at Exit 16W and traveling to Exit 1 (from North to South), how would the models be similar and how would they be different?



Tolls on the New Jersey Turnpike Advanced Functions and Modeling

Students develop a model for the toll structure of the New Jersey turnpike using a piecewise defined linear function developed using data analysis. This problem is based on work done by John A. Goebel for his Algebra 2 class at Durham Academy in 2003.

Goals Addressed in the Lesson: 1. Create and use calculator-generated models of a linear function of bivariate data to solve

problems. (1.01) a. Interpret the constants and coefficients in the context of the data. b. Check models for goodness of fit; use the most appropriate model to draw

conclusions and make predictions. 2. Use piecewise-defined functions to model and solve problems; justify results. (2.02)

a. Solve using tables, graphs, and algebraic properties. b. Interpret the constants and coefficients in the context of the problem.

Materials Needed: 1. Copy of the student handout for each student. 2. Graphing calculator for each student. 3. Graph paper. 4. Access to the internet (if possible). 5. Map of New Jersey or web link to New Jersey map.

Activity One: The goal of this activity is to develop interest in the issue of toll roads—how do they work, what do they look like, why they exist? We want the students to know something about the New Jersey Turnpike since the data in this problem is from there. Several particular exits will be of interest so students need a way to determine where an exit takes them. Some avenues to help students get acquainted are listed below.

• Look at maps of New Jersey • Go to http://www.state.nj.us/turnpike/nj-vcenter-maps.htm to view maps of the New

Jersey Turnpike. • A list of the interchanges and their locations can be found at

http://www.contrib.andrew.cmu.edu/~mn2n/tollroads.html. • A “road sign” tour is available at http://web.mit.edu/spui/www/freeway/njtp.html • There is a discussion of the issues of toll roads at

http://www.worldbank.org/transport/roads/toll_rds.htm#options. Discuss the purpose and the mechanics of toll roads.

• A web site with a number of cameras on different parts of the New Jersey Turnpike can be seen at http://newyork.metrocommute.com/video/newyork/indexNJTpke.html.


http://www.state.nj.us/turnpike/nj-vcenter-maps.htm

http://www.contrib.andrew.cmu.edu/~mn2n/tollroads.html

http://web.mit.edu/spui/www/freeway/njtp.html

http://newyork.metrocommute.com/video/newyork/indexNJTpke.html


Piecewise Defined Functions as Models NCSSM –A Toll Road Lesson

Activity Two: The following data show the exit number, mileage (from first exit) and toll for the New Jersey Turnpike. These data are for peak hours using an E-Z Pass. The E-Z Pass is an electronic system that deducts tolls from a pre-established account. More information on this pass is at http://www.state.nj.us/turnpike/nj-ezpass.htm.)

Exit Number Mileage Cost 1 0 $ - 2 12 $0.55 3 25 $0.85 4 33 $1.15 5 43 $1.45 6 50 $2.20 7 52 $1.75 7A 59 $1.85 8 67 $2.05 8A 73 $2.20 9 82 $2.60 10 87 $2.65 11 90 $2.90 12 95 $3.20 13 99 $3.65 13A 100 $4.25 14 104 $4.25 14A 107 $4.90 14B 109 $5.00 14C 110 $5.30 15E 106 $4.50 15W 108 $4.70 16E 111 $5.00 16W 112 $5.00

1. Create a scatter plot of the data showing cost versus mileage.

2. Notice the point associated with exit 6 does not seem to fit the trend around it. Delete that point and reconsider the data. Exit 6 takes a driver to the Pennsylvania Turnpike.

3. Fit this data with a linear, quadratic, and exponential model. Discuss goodness of fit for each model. Which model is best?


http://www.state.nj.us/turnpike/nj-ezpass.htm



Note on Goodness of Fit: We will explore goodness of fit through residuals. A residual is defined as the vertical distance between a data point and the model describing the data. The value of a residual is calculated using the following formula For a given datax , the residual . moddata ely y= −The values of the residuals are found on the calculator using the lists. Given the x-values of the data in list L1, the y-values of the data in L2 and the model for the data in Y1, go to the top of L3 and type 3 2 1( 1L L Y L= − )

A residual plot is created by plotting the ordered pair: ( ),datax residual . The residual plot should only show noise and should have no pattern. Otherwise, the model used may not be the best model for that data.

Linear Model: where 0.0473 0.666= −y x =x mileage and =y cost of toll.

On the left, the linear model is shown below superimposed over the data. The residuals verify that this model is not a satisfactory one. The residuals are shown in the middle graph and its window is on the far right.

Quadratic Model: where 20.0049 0.02 1.07= − +y x x =x mileage and =y cost of toll.

On the left, the quadratic model is shown below superimposed over the data. The residuals verify that this model is again not satisfactory (seems a bit better than the line). The residuals have a definite pattern which verifies the concern with the model.




Exponential Model: 0.5243 (1.0205)= ⋅ xy =x mileage and =y cost of toll.

On the left, the exponential model appears to fit the data; however, the residuals (shown in the middle) show a pattern. Thus, the exponential is also not a good model for this data. This fit is very similar to that of the quadratic function.

As we look at these three possible models, the exponential has residuals that are a bit smaller than those of the quadratic function, but a clear pattern is still visible in every residual plot. We need to consider other models.

4. Separate the data into two parts. This will depend on how the student sees the data. For the purpose of producing a solution, we will break the data at exit 12 which is at 95 miles. The point associated with exit 12 will be included in both data sets for the purpose of fitting the line. When we write the model, the point at 95 miles must go with only one part of the piecewise function.

Students may select different breaking points and not choose to include the point in both sets. Hence, there are many answers. This is one example. For exits 1 through 12 or , the regression line (using LinReg from the Stat Calc menu) is where

0 x≤ ≤ 950.0302 0.1249y x= + x = the number of miles from exit 1 and the cost of

the toll when you exit. The line is shown superimposed over the data followed by the residual plot associated with this linear fit.

y =




The residual plot shows that the line fits the beginning points best. The residuals show a fairly random shape as well as the value of the residuals is small which is shown by the y-values of the window.

The second part of the data for 95 or exits 12 through 16W is fit by the regression line where

112x≤ ≤0.1117 7.2652y x= − x = the number of miles from exit 1 and the cost of the toll

when you exit. The line is shown superimposed over the data followed by the residual plot associated with this linear fit.

y =

This residual plot shows a very scattered set of points and reflects that the line is a good model for this data.

The middle screen below shows the creation of a piecewise function on the calculator. Find Y1 and Y2 on the VARS menu and the inequalities under TEST. A scatter plot of the full data set with the two lines superimposed over the data is shown.




The slopes and y-intercepts have meaning to the NJ Turnpike. • For the function where 0.0302 0.1249y x= + x = the number of miles from exit 1 and

the cost of the toll when you exit. This function represents exits 1 through 12 or . The slope of 0.0302 measures the cost in dollars per mile on Turnpike for these exits. More generally, the slope represents a cost of 3 cents per mile. The y-intercept 0.1249 measures the initial cost in dollars for just entering the Turnpike.

y =0 x≤ ≤ 95

• For the function which represents 950.1117 7.2652y x= − 112x≤ ≤ or exits 12 through 16W, the slope 0.1117 measures cost in dollars per mile. More simply, it costs 11 cents per mile to travel between exits 12 and 16W. The y-intercept is not really meaningful since x cannot be 0.

Use the function to find values of tolls. 0.0302 0.1249,0 95

( )0.1117 7.2652,95 112

x xt x

x x+ ≤ ≤⎧

= ⎨ − < ≤⎩• At , the model produces a toll of $1.64. At Exit 6, the toll is $2.20. 50x =• At , the model predicts a toll of $2.99. At Exit 12, the toll is $3.20. 95x =• At , the model predicts a toll of $5.24. At Exit 16W, the toll is $5.00. 112x =• If a new exit is added at , the model predicts a toll of $0.72. For ease it would be

changed to $0.75. 20x =

A review of the map shows that once we pass Exit 12, the NJ Turnpike travels through a dense population area near New York City. The cost per mile greatly increases but also the cost of the turnpike does as well. There will have to be more lanes and more repairs and changes in the turnpike since it serves some many more people. Clearly, the most expensive travel is from exits 12 through 16W.



5.17. Suggestions for Practice Problems

A. Algebra and Trigonometry: p. 230 problems 59-72; 75-76 B. Functions Modeling Change:

1. p. 74-75 several problems are available 2. Manual suggests 4, 6, 8, 12, 15, and 18.

Since there are very few homework problems or problems listed in some of the textbooks. I have included a few extra.

1. The degree day is used o measure the demand for heating or cooling. In the United States, 65 o F is considered the desirable temperature for the inside of a building. The number of degree days recorded on a given date is equal to the difference between 65 and the mean temperature for that date. If the mean temperature is above 65 o F, cooling degree days are recorded. Heating degree days are recorded if the mean temperature is below 65 o a) What type of function can be used to model degree days? b) Write a function to model the number of degree days d(f) for a mean temperature of t F? o

c) Graph the function. d) The mean temperature is the mean of the high and low temperatures for a day. How many

degree days are recorded for a day with a high of temperature of 63 o F and a low temperature of 28 F? Are they heating degree days or cooling degree days? o

2. The tax table below list income tax brackets. Income Tax Bracket Up to 10,000 5% greater than $10,000 but less than $20,000 7% more than $20,000 8.5%

a. What type of function is described by the tax rates? b. Write the function if x is income and t(x) is the tax rate. c. Graph the tax brackets for different taxable incomes. d. David Jones lives in the state with this tax bracket. In which tax bracket is Mr. Jones if he

made $38,000 last year?

3. The charge for a taxi ride is $1.50 for the first 1/6 of a mile, and $.20 for each additional 1/6 of a mile (round up to 1/6 mile). a) Make a table showing the cost of a trip as a function of its length. Your table should start

at zero and go up to one mile in 1/6 increments. b) What is the cost for a 5/6 of a mile ride? c) How far can you go for $4.00? d) Graph the cost function in part a)

4. A contractor purchases gravel one cubic yard at a time. a) A gravel driveway L yards long and 7 yards wide is to be poured to a depth of 1 foot.

Find a formula for n(L), the number of cubic yards of gravel the contractor buys, assuming that he buys 10 more cubic yards than are needed to ensure he has enough.



b) Assuming no driveway is less than 6yards long, state the domain and range of n(L). Graph n(L) showing the domain and range.

c) If the function n(L) did not represent an amount of gravel, but was a mathematical relationship defined by the formula in part(a), what is its domain and range?

Suggestions for projects

Once the toll road problem has been explored, students could develop a toll system for I-95 in North Carolina. To complete this task, students would need maps with I-95 exits and distances shown. There have been news articles with information about major goals. For example, follow the link to http://www.newsobserver.com/front/digest/story/3025530p-2771057c.html. Perhaps as students produce their toll system, they can work to have each car that travels the full length of I-95 in North Carolina pay a total of $18, while establishing toll booths as they exit I-95 at certain exits.


http://www.newsobserver.com/front/digest/story/3025530p-2771057c.html

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