Question

Function Representations

In this activity, you will use multiple representations of relationships to identify key features and solve problems.

Ryan conducted a 6-day study observing the effects of an organic plant food on the growth of his sprouting bean plant. He tracked these two pieces of information:

the amount of plant food remaining in the container after each day’s feeding
the height of the plant over time
Part A
Ryan found that the amount of plant food remaining decreased an equal amount each day, and he used the entire 72 milliliters by the end of his study.

Question 1
Question
Which statement is true about the relationship between the amount of plant food remaining and the number of days?

This relationship is not a function because more than one amount of plant food remains each day.
This relationship is a function because only one amount of plant food remains each day.
This relationship is a function because more than one amount of plant food remains each day.
This relationship is not a function because only one amount of plant food remains each day.
Question 2
Question
Write a function rule for the relationship between the amount of plant food remaining, f(x), and the number of days that have passed, x.

Type the correct answer in the box.


Question 3
Question
Determine the domain and the range for this relationship.

Drag the tiles to the correct locations on the image. Not all tiles will be used.

{0, 12, 24, 36, 48, 60, 72}0 ≤ y ≤ 60 ≤ y ≤ 60 ≤ y ≤ 72{0, 1, 2, 3, 4, 5, 6}{0, 1, 2, 3, 4, 5, 6}0 ≤ x ≤ 72{0, 6, 12, 18, 24, 30}0 ≤ x ≤ 6
Question 4
Question
Determine the missing values in the table, and then use the drawing tools to create the graph representing the relationship between the amount of plant food remaining, f(x), and the number of days that have passed, x.

x 0 1 2 3 4 5 6
f(x)

Drawing Tools
Question 5
Question
What does the point
represent?

the length of time it takes for Ryan to use all the plant food
the amount of plant food used each day
the length of time it takes for the height of the plant to reach 72 centimeters
the initial amount of plant food
Part B
Ryan also found that he could model the height of his plant with the equation h = 0.5d + 4, where d is the number of days and h is the height in centimeters.

Question 1
Question
Which statement is true about the relationship between the height of the plant and the number of days?

This relationship is a function because the plant can only be one height at a time.
This relationship is not a function because the plant can only be one height at a time.
This relationship is not a function because the plant can be more than one height at a time.
This relationship is a function because the plant can be more than one height at a time.
Question 2
Question
Determine the domain and the range for this relationship.

Drag the tiles to the correct locations. Not all tiles will be used.

0 ≤ h ≤ 70 ≤ h ≤ 6{4, 5, 6, 7}{0, 1, 2, 3, 4, 5, 6}{0, 1, 2, 3, 4, 5, 6}0 ≤ d ≤ 64 ≤ d ≤ 74 ≤ h ≤ 7{4, 4.5, 5, 5.5, 6, 6.5, 7}{4, 4.5, 5, 5.5, 6, 6.5, 7}
Question 3
Question
The height of the plant is given by the equation h = 0.5d + 4. Rewrite this as a function rule where f(x) is the height, in centimeters, and x is the time, in days. Use the rule to complete the table, and then use the drawing tools to create the graph representing this relationship.

x 0 1 2 3 4 5 6
f(x)

Drawing Tools
Question 4
Question
What does the point
represent?

the height of the plant at the end of the experiment
the amount of plant food Ryan uses each day
the initial height of the plant
the average amount the plant grows each day
Question 5
Question
Use your function rule to complete each statement.

Type the correct answer in each box. Use numerals instead of words.

After 3.5 days, the height of the plant is
5.75
centimeters.

The height of the plant is 6.25 centimeters after
4.5
days.

Part C
Explain the similarities and differences between the two relationships in this situation.

Answer 1

Explanation:

Part A:

Question 1:

The statement that is true about the relationship between the amount of plant food remaining and the number of days is:

This relationship is a function because only one amount of plant food remains each day.

Question 2:

The function rule for the relationship between the amount of plant food remaining, f(x), and the number of days that have passed, x, is: f(x) = 72 - x.

Question 3:

The domain for this relationship is the set of numbers representing the number of days that have passed. In this case, the domain is {0, 1, 2, 3, 4, 5, 6}.

The range for this relationship is the set of numbers representing the amount of plant food remaining. Since Ryan used the entire 72 milliliters of plant food, the range is {0}.

Question 4:

The missing values in the table are as follows:

x 0 1 2 3 4 5 6

f(x) 72 71 70 69 68 67 66

Question 5:

The point (6, 66) represents the amount of plant food remaining after 6 days.

Part B:

Question 1:

The statement that is true about the relationship between the height of the plant and the number of days is:

This relationship is a function because the plant can only be one height at a time.

Question 2:

The domain for this relationship is the set of numbers representing the number of days. In this case, the domain is {0, 1, 2, 3, 4, 5, 6}.

The range for this relationship is the set of numbers representing the height of the plant. The equation h = 0.5d + 4 shows that the height can be any value greater than or equal to 4. Therefore, the range is {h: h ≥ 4} or {h: 4 ≤ h}.

Question 3:

The function rule for the relationship between the height of the plant, f(x), and the number of days that have passed, x, is: f(x) = 0.5x + 4.

The completed table is as follows:

x 0 1 2 3 4 5 6

f(x) 4 4.5 5 5.5 6 6.5 7

Question 4:

The point (0, 4) represents the initial height of the plant.

Question 5:

Using the function rule, after 3.5 days, the height of the plant is f(3.5) = 0.5(3.5) + 4 = 5.75 centimeters.

The height of the plant is 6.25 centimeters after 4.5 days.

Part C:

The similarities between the two relationships are:

- Both relationships involve a variable (number of days) that represents time.

- Both relationships can be represented by a function rule.

- Both relationships can be represented on a graph.

The differences between the two relationships are:

- The first relationship involves the amount of plant food remaining, while the second relationship involves the height of the plant.

- The first relationship is linear and decreases by a constant amount each day, while the second relationship is also linear but increases by a constant rate each day.

- The domain and range of the first relationship are specific sets of numbers, while the domain and range of the second relationship have specific constraints (domain: numbers of days, range: height greater than or equal to 4).