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2.1E Homework: Representations of a Linear Context*
Name: Period:
1. The table below represents how much a crew charges to landscape your yard. The crew charges an initial fee of $100 for equipment.
a. Graph the table of values below. Time (hours) Cost (dollars)
0 100
2 500
4 900
6 1300
b. Use the table and graph to find the rate of change.
The crew charges $200 per hour.
c. What is the y-intercept of your graph? What does the y-intercept represent in the context?
(0,100) It represents the crew’s initial fee for equipment.
d. Write an equation for the cost, c, of landscaping for h hours.
e. Is this context linear? Explain why or why not.
Yes, the context is linear. In the context, there is a constant rate of 200 dollars per hour. The graph is a line.
f. How much will it cost if the crew worked for 8 hours? $1,700
g. If you have $5,000 to spend on landscaping, how many hours can the crew work? Is this situation realistic?
They can work for 24.5 hours. This amount of time is realistic, but the job will not be completed in one day. The crew will probably work 8 to 9 hours over three days.
h. How would the context and equation change if the rate of change was 150 and there was no equipment fee? Would the graph of the new line be steeper or less steep?
new equation:
The crew would not charge an initial fee of $100 for equipment; they would only charge an hourly fee of $150. The new line would be less steep and pass through the origin.
2 4 6 8 10 12 14
200 400 600 800 1000 1200 1400
0
Time (hours)
C
os
t
(do
SDUHSD Math B Honors Module #2 – TEACHER EDITION
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2. Serena is always losing her tennis balls. At the beginning of tennis season, she has 20 tennis balls.The table below represents how many tennis balls she has as the season progresses; where x
represents the number of weeks and y represents the number of tennis balls.
Weeks # of tennis balls
2 16
4 12
6 8
8 4
a. Complete the graph below.
a. On average, how many tennis balls does Serena lose each week?
The unit rate is -2 tennis balls per week. Serena loses two tennis balls each week.
b. What is the y-intercept of your graph? What does the y-intercept represent in the context?
(0,20) It represents the amount of tennis balls that Serena has to begin with.
c. Write an equation that represents this relationship.
d. Is this context linear? Explain why or why not.
Yes, the context is linear. In the context, there is a constant rate of -2. Two balls are lost per week. The graph is a line.
e. Explain how to use the equation to determine how many weeks will pass until Serena runs out of tennis balls.
The number of tennis balls is represented by the y-value. If you substitute 0 for y into your equation, then solve for x, you get 10 weeks. Serena will run out of balls in 10 weeks.
f. How would the context and equation change if the rate of change in this problem was -3? Would the graph of the new line be steeper or less steep than the original?
new equation: y = -3x + 20
Serena starts with 20 tennis balls and loses 3 tennis balls each week The new line would be steeper because the rate of change increased.
1 2 3 4 5 6 7 8 9 10
2 4 6 8 10 12 14 16 18 20
0
Weeks
# o
f
T
en
ni
s
B
al
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Directions: In each problem, complete the remaining 3 representations that are not given. When needed, label the columns in the table and the axes on the graph.3. A Community Garden Context
Gavin is buying tomato plants to plant in his local community garden. Tomato plants are $9 per flat (a flat contains 36 plants). No partial flats can be purchased. Consider the relationship between total cost and number of flats purchased.
Table
# of flats Total Cost
0 0
2 18
4 36
6 54
Graph Equation
How many flats can Gavin purchase if he has $150 to spend? Show your work.
flats, so Gavin can purchase 16 flats
Should the points on the graph be connected as a line? Explain your reasoning.
The points on the graph should not be connected because you can not buy part of a flat.
a. What is the rate of change in this problem? What does the rate of change represent in the context?
9; the cost per flat
b. What is the y-intercept of your graph? What does the y-intercept represent in the context?
(0, 0); there is no initial fee for purchasing flats of tomatoes
c. How would the context and equation change if the rate of change in this problem was 10 and there was a $5 entrance fee to the garden ? Would the graph of the new line be steeper or less steep than the original?
new equation:
The garden charges a $5 entrance fee and $10 per flat. The new line would be steeper and have a y-intercept at (0,5)
2 4 6 8 10 12 14 18
36 54 72 90 108 126
0
# of flats
To
ta
l Co
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4.Context
Answers will vary.
Table
Time (hours) # of Fish
0 300
2 270
4 240
6 210
8 180
10 150
Graph Equation
How long will it take for all the fish to be gone?
20 hours
a. What is the rate of change in this problem? What does the rate of change represent in the context?
-15; the number of fish remaining decreases by 15 each hour
b. What is the y-intercept of your graph? What does the y-intercept represent in the context?
(0, 300); start with 300 fish
c. How would you change your context so that the amount of fish remaining, y, after x hours could be represented by the equation If this new equation was graphed, how would the line differ from the original?
There were 200 fish to begin with and the number of fish remaining decreases by 30 each hour. The new line would be steeper and have a y-intercept at (0,200)
2 4 6 8 10 12 14 30
60 90 120 150 180 210 240 270 300
0
Time (hours)
# of
F
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5.Context
Answers will vary.
Table
x y
0 30
1 35
2 40
3 45
4 50
Graph and table labels will depend on context.
Graph Equation
What is the rate of change in this problem? What does the rate of change represent in the context?
+5; answers will vary
What is the y-intercept of your graph? What does the y-intercept represent in the context?
(0, 30); answers will vary
6. Use the graph below to write a table of values that relates a circle’s diameter to its circumference. Round each decimal value to the nearest hundredth.
a. Complete the table of values:
Diameter Circumference
0 0
4 12.5
8 25.12
12 37.68
16 50.24
b. What is the rate of change? 3.14 or
c. Write an equation that represents this relationship.
C = 3.14 d, where C represents circumference and d represents diameter
d. How does a circle’s diameter relate to its circumference? Is this relationship proportional? Explain your reasoning. The circumference of a circle is equal to its diameter multiplied by 3.14 ( ). This situation is proportional and linear. When graphed, the line passes through the origin. There is a constant rate of change.
2 4 6 8 10 12 14 10
20 30 40 50 60 70
0
