Linear Situations

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The graph above compares the position for two runners in the women's 100-meter dash in two
different Olympics. Both women won the gold medal in their respective Olympics. The position at
time, t, for Betty Robinson is represented by the solid line, and the position of Florence GriffithJoyner at time, t, is represented by the dotted line.
1.
2.
a)
State the independent variable.
b)
State the dependent variable.
a)
Both ladies won the gold medaL Who do you think won the gold medal (1 st place) in 1928
and who won the gold medal in 1988?
b)
Explain the reasoning behind your choice of winners in part (a).
c)
d)
3.
·af
b)
4.
a)
If the runners had run the race together, at what time would one runner pass the other?
Round the answer to the nearest second.
b)
Approximately how far along the course were Griffith-Joyner and Robinson when this
happened?
How does Robinson's speed over the interval from 0 to 2 seconds compare with
Griffith-Joyner's speed over the same interval? Explain your answer using the graph.
Find Griffith-Joyner' average speed over the interval from 5 to 9 seconds.
7.
Suppose the ladies are running on the same track at the same time and that you are the
commentator for the race. Write a paragraph describing the race that both ladies ran. Indicate
in your narrative when or if they increased or decreased their speed. Indicate their positions in
comparison to each other. In addition, be sure to speculate what Robinson was planning at
about 5 seconds.
Connecting a Verbal Description to Table and Graph
The Math and Science Team at Smedley Middle School must decide on several fund-raisers to pay entry
fees for contests and other expenses. In August, the students decide to have a Problem-A- Thon. Each
student will work as many problems as possible in 20 minutes. They will get sponsors to give donations
based on how many problems they work correctly. Ginny gets pledges from her friends and parents
totaling $2 per problem. Genaro gets pledges of $3 per problem.
1.
In the previous lesson, you determined the independent and dependent variables in the verbal
description above. Write a sentence using "is a function of" that describes the relationship
between those two variables.
2.
Complete the following table to show how much money Ginny and Genaro raise for each number
of problems that they work correctly. Be sure you understand the labels for each column. The
first column, number of problems, refers to the number of problems worked correctly and is used
for both Ginny and Genaro's columns. In the process column you will show how you figured
Ginny's and Genaro's total money raised which you will put in the money raised column. If
everyone in your group understands these 5 columns, finish this page. If your group has
questions, raise your hands and I will come to you.
Number of
Problems
Process Column
Ginny
Money
Raised by
Ginny
Process
Column
Genaro
Money
Raised by
Genaro
5
10
15
20
30
50
70
90
100
3
6.
How many problems will Ginny have to work correctly to earn $120? Remember to
explain how you got your answer:
Number of
Problems
12.
Process Column
Ginny
Process Column
Genaro
Money
Raised by
Genaro
Ginny is very frustrated that Genaro got more in pledges per problem than she did. What
variables determine how much she makes? How could she make the same amount of money that
Genaro makes? How could she make more?
13.
On a coordinate plane, the independent variable is always graphed on the x-axis (the horizontal
axis) while the dependent variable is always graphed on the y-axis (the vertical axis). Carefully
considering your independent variable (x), your dependent variable (y), and the data in your table,
determine an appropriate viewing window to use to graph both Ginny's and Genaro's functions.
In the space provided below, give the minimum and maximum values that you will use and your
scale. Justify your choices.
xmin
xmax
xscale
ymm
ymax
yscale
14.
Sketch each function below. Be sure to label your both your axes and your graph.
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Use Tables and Graphs to Determine the Better Deal
In September, the Smedley team determined that they had enough money for the first 4 contests
- NOT ENOUGH FOR THE YEAR. The planning commitieedecided to have members sell
programs for the high school football games. The team must choose between two offers. The first
offer is a flat fee of $40 plus $1 for each program sold. The second offer gives $10 plus $2 for each
program sold. Now the team must decide which contract to sign.
As you answer the questions in this lesson, be sure to justify each answer.
1.
In this situation, what variables might affect the success of the project?
2.
Complete the following table to show the money made for offer 1 and for offer 2 based on the
number of programs sold.
Programs
Sold
Process Column
Offer 1
Income
Offer 1
Process
Column
Offer 2
Income
Offer 2
5
10
15
20
30
40
50
70
100
5.
Considering the patterns that you noticed extend the table to fill in the nth row below.
Programs
Sold
Process Column
Offer 1
Income Offer Process Column Income Offer
1
Offer 2
2
6.
In this scenario, what is the independent variable? What is the dependent variable? Write a
sentence using "is a function of' that shows the relationship between these two variables.
7.
ChooserneaniIlgfulletters to represent your independent variable and dependent variable.
Using these letters, write a function rule for the income from offer 1.
8.
Using the same letters that you used in number 7, write a function rule for the income from
offer 2.
10.
How many programs must the team sell to make $90 with offer I? Remember, justify your
. answer.
12.
You are on the committee to decide which offer to choose. Carefully consider which offer to use.
Once you decide and sign a contract, you cannot change your mind. Be sure to justify your
answer.
13. l~then.:_liJlillnbeIQfpmgramsthatcanbesiLld>-wh~Lejh_ejncQmefrolUbQth_0ffets_isjh~same? Justify your ans-weL--
15.
Determine an appropriate viewing window for the graphs of both functions, and justify your
choice of those windows.
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17.
What effect does the $1 income for each program have on the graph of offer I? What effect does
the $40 have on the graph?
18.
What effect does the $2 income for each program have on the graph of offer 2? What effect does
the $10 have on the graph?
19.
At one point, the two graphs intersect. What are the coordinates of this point of intersection?
What is the significance of this point?
20.
In question 12, you had to decide which contract you would sign. Now that you have graphed this
data, would you reconsider? Why or why not? Justify your answer.
Connecting Table, Graph, and Function Notation
Lakeshia is a science team member who was not able to participate in the Problem-A- Thon or to
sell programs. She is, however a whiz at computers. Ms. Alvarez, the language arts teacher, will
pay her a starting wage of $10 and an additional $6 an hour to keep her computer records
current. Ms. Alvarez will pay Lakeshia for whole hours only.
As you answer these questions, remember that you are expected to justifY each answer.
1.
In this situation, what variables might affect the amount that Lakeshia earns?
2.
Complete the following table to show how much money Lakeshia can earn by working for Ms.
Alvarez. (I know, I know! By now you are an expert at this!)
Number of
Hours
1
2
3
4
Process Column
Money earned
5
10
15
20
3.
Look for patterns in the table you just completed and describe in words a pattern you see in the
amount of money earned.
4.
Using your pattern, fill in the process column for n hours.
Number of Hours
Process Column
Money earned
n
6.
Lakeshia is primarily concerned with how many hours she will work and, of course, how much
money she makes. Make a sentence with these variables using "depends" or "is a function of'.
7.
Choose and define meaningful variables to represent your independent variable and dependent
variable. Using these letters, write a function rule for the amount of money Lakeshia earns.
8.
Based on your pattern or function rule, how much would Lakeshia earn if she works 14 hours for
Ms. Alvarez? (Remember to explain your answer in words or symbols.)
10.
Determine an appropriate viewing window for the graphs of both functions and justify your choice
of viewing window.
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12.
What effect does the $6 pay for each hour have on the graph of the function? What effect does the
$10 have on the graph?
13.
Let's revisit question 7. For the sake of consistency, let n be number of the hours Lakeshia
worked, and let E represent the money she earned. Write an algebraic equation for the amount
earned E in terms of n.
14.
The algebraic equation for E in terms of n describes E as a function of n. There is a specific
notation that is often used for functions. The expression E(n) is read the function E ofn. This
means that E(n) would give the amount of money Lakeshia earned for n hours. Rewrite your
algebraic equation from question 13 using this notation.
15.
In the function that you wrote in question 14, EO) means to replace n with 7 and find the value of
the function. What is the value of E(7)? Of E(ll)?
16.
The information from the table in question 2 told us that E was 34 when n was 4. Write this fact
using function notation.
17.
An important term that is often used in algebra is substitution. To find the amount of money
earned for a specific number of hours, we replaced n with a number, that is we substituted a
number for n. In problem 16, we substituted 4 for n and got $34 for the amount of money earned.
Which of the following values ofn can be meaningful substitutions? Why are they meaningful?
n=6
n
=
11
n= 3.7
n=O
1
n=2-
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Slope ..Investigation
Consider the following situation: At her annual physical, Mrs. Smith, the health teacher at Smedley
Middle School, found that she was 153 centimeters tall and that she weighed 120 kilograms. After
being diagnosed with hypertension (high blood pressure) and high cholesterol, she decided that she
would follow a diet planned and supervised by her doctor. On this diet, Mrs. Smith planned to lose 0.5
kg per week.
1.
Fill in the table at right to show Mrs. Smith's
weight during the first 10 weeks of her diet, if
she continues this pattern. Also give her weight
for the nth week in the process column.
Week
o
Weight
120kg
1
2
3
4
5
6
7
8
9
10
2.
U sing the variables n for week number and w
for weight, write a function rule to describe Mrs.
Smith's weight after n weeks. Be sure to use
function notation.
3.
In a previous lesson, you leamed how to find rate of change. Using the ordered pairs (n, wen)),
write the ordered pairs that represent her weight for weeks 3 and 4. What is Mrs. Smith's rate of
change in kilograms per week between weeks 3 and 4?
6.
Stop now and compare your answers with those of your group. What conclusion can you draw
about the rate of change in this function?
7.
Determine an appropriate viewing
window for the graph of this function, and
justify your choice of windows.
8.
Graph the function on the grid provided
below. Be sure to draw and label your
axes, label the tick marks, and label the
graph.
xmin
xmax
xscale
ymin
ymax
yscale
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9.
Discuss with your group the appearance of the points that you graphed above. Write any
conclusions that you consider important about that appearance in the space below.
In every situation in which there is a constant rate of change between points, the points will always lie
on a line. This constant rate of change is called the slope of the line.
In mathematics the Greek symbol ~ means "change". In problems 3,4, and 5, you found that the rate
of change was -0.5 kg per week. You found this by finding the change in kilograms and dividing it by
the change in weeks. This is exactly what slope is. If you have two ordered pairs (Xl, YI) and (X2, Y2),
then the slope is:
10.
Pick any two points on the graph that you made in problem 8. List those two points as ordered
pairs and show how to determine the slope between them.
11.
Pick two points, different from the ones you chose for problem 10. List those two points as
ordered pairs and determine the slope between them.
12.
Consider the function that you wrote in problem 2, the rate of change that you found in problems
3 - 5, and the graph that you made in problem 8. Explain how to find the slope from the function,
from the rate of change, and visually from the graph.