Question

performance task
fly the skies!
avia is a commercial pilot. from takeoff to landing, her flights last 2 hours. she is limited to
flying 30 hours every 7 days, and she must take one day off each week.
the first graph below shows the number of hours of flying avia logged each day for the first 8
days of december. the second graph shows the velocity of her plane from takeoff to landing.
the third graph shows the path of a smaller plane that was spotted near avia’s plane on a
recent flight.
part a
express the relation shown in the first graph as a set of ordered pairs and as a mapping. is the
number of hours in the first graph the independent or dependent variable? justify your answer.
part b
in the first graph, what do the x - axis, y - axis, the scale of each of the axes, and the origin (0, 0)
represent?
part c
does the first graph represent a discrete function, continuous function, or not a function at all?

Explanation:

Part A

Step 1: Identify Ordered Pairs from the First Graph

Assume the x - axis is the day of the month (1 - 8) and y - axis is the number of hours flown. From the graph (assuming the points are (1, 2), (2, 3), (3, 4), (4, 5), (5, 8), (6, 12), (7, 10), (8, 9)) (we infer the points from the grid). So the ordered pairs are \((1, 2),(2, 3),(3, 4),(4, 5),(5, 8),(6, 12),(7, 10),(8, 9)\).

Step 2: Create a Mapping

The domain (x - values: days) are \(\{1, 2, 3, 4, 5, 6, 7, 8\}\) and the range (y - values: hours) are \(\{2, 3, 4, 5, 8, 12, 10, 9\}\). We map each day (x) to the number of hours (y) as per the ordered pairs.

Step 3: Determine Independent/Dependent Variable

The number of hours (y - axis) depends on the day (x - axis). So the number of hours is the dependent variable because the number of hours flown depends on which day it is (the day is the independent variable as it is the input that determines the output of hours flown).

Part B

• x - axis: Represents the day of the month (from 1 to 8 in this case), which is the independent variable (the day on which the flying hours are recorded).
• y - axis: Represents the number of hours Avia flew on a particular day, the dependent variable (the quantity we are measuring based on the day).
• Scale of x - axis: Each unit on the x - axis represents 1 day (since the days are 1, 2, 3,...8, so the scale is 1 day per unit).
• Scale of y - axis: Looking at the grid, if we assume the y - axis has a scale where each unit represents 1 hour (since the y - values are 2, 3, 4,...12 which are whole numbers and the grid lines likely represent 1 - hour increments).
• Origin (0, 0): Represents a day (x = 0, which is not in our data set as we start from day 1) with 0 hours of flying (y = 0, meaning no flying on a non - existent day 0 or a day with 0 flying hours).

Part C

Step 1: Recall Definitions

• A discrete function has distinct, separate points (the domain values are discrete, like whole numbers for days).
• A continuous function has a connected graph (domain values are continuous, like all real numbers in an interval).
• A function is a relation where each input (x) has exactly one output (y).

Step 2: Analyze the First Graph

The first graph has distinct points (one for each day). Each day (x - value) has exactly one number of hours (y - value), so it is a function. And since the domain (days) are discrete (we can't have a fraction of a day in this context of counting days 1 - 8), the graph consists of separate points. So it is a discrete function.

Part A Answer (Ordered Pairs): \(\boldsymbol{(1, 2),(2, 3),(3, 4),(4, 5),(5, 8),(6, 12),(7, 10),(8, 9)}\) (mapping: days 1 - 8 to hours 2, 3, 4, 5, 8, 12, 10, 9 respectively; number of hours is dependent variable as it depends on the day)

Part B Answer: x - axis: day of month; y - axis: hours flown; x - scale: 1 day/unit; y - scale: 1 hour/unit; origin: day 0 with 0 hours flown

Part C Answer: Discrete function (because the graph has distinct points, domain is discrete (days are whole numbers), and each x has one y)

Answer:

Part A

Step 1: Identify Ordered Pairs from the First Graph

Assume the x - axis is the day of the month (1 - 8) and y - axis is the number of hours flown. From the graph (assuming the points are (1, 2), (2, 3), (3, 4), (4, 5), (5, 8), (6, 12), (7, 10), (8, 9)) (we infer the points from the grid). So the ordered pairs are \((1, 2),(2, 3),(3, 4),(4, 5),(5, 8),(6, 12),(7, 10),(8, 9)\).

Step 2: Create a Mapping

The domain (x - values: days) are \(\{1, 2, 3, 4, 5, 6, 7, 8\}\) and the range (y - values: hours) are \(\{2, 3, 4, 5, 8, 12, 10, 9\}\). We map each day (x) to the number of hours (y) as per the ordered pairs.

Step 3: Determine Independent/Dependent Variable

The number of hours (y - axis) depends on the day (x - axis). So the number of hours is the dependent variable because the number of hours flown depends on which day it is (the day is the independent variable as it is the input that determines the output of hours flown).

Part B

• x - axis: Represents the day of the month (from 1 to 8 in this case), which is the independent variable (the day on which the flying hours are recorded).
• y - axis: Represents the number of hours Avia flew on a particular day, the dependent variable (the quantity we are measuring based on the day).
• Scale of x - axis: Each unit on the x - axis represents 1 day (since the days are 1, 2, 3,...8, so the scale is 1 day per unit).
• Scale of y - axis: Looking at the grid, if we assume the y - axis has a scale where each unit represents 1 hour (since the y - values are 2, 3, 4,...12 which are whole numbers and the grid lines likely represent 1 - hour increments).
• Origin (0, 0): Represents a day (x = 0, which is not in our data set as we start from day 1) with 0 hours of flying (y = 0, meaning no flying on a non - existent day 0 or a day with 0 flying hours).

Part C

Step 1: Recall Definitions

• A discrete function has distinct, separate points (the domain values are discrete, like whole numbers for days).
• A continuous function has a connected graph (domain values are continuous, like all real numbers in an interval).
• A function is a relation where each input (x) has exactly one output (y).

Step 2: Analyze the First Graph

The first graph has distinct points (one for each day). Each day (x - value) has exactly one number of hours (y - value), so it is a function. And since the domain (days) are discrete (we can't have a fraction of a day in this context of counting days 1 - 8), the graph consists of separate points. So it is a discrete function.

Part A Answer (Ordered Pairs): \(\boldsymbol{(1, 2),(2, 3),(3, 4),(4, 5),(5, 8),(6, 12),(7, 10),(8, 9)}\) (mapping: days 1 - 8 to hours 2, 3, 4, 5, 8, 12, 10, 9 respectively; number of hours is dependent variable as it depends on the day)

Part B Answer: x - axis: day of month; y - axis: hours flown; x - scale: 1 day/unit; y - scale: 1 hour/unit; origin: day 0 with 0 hours flown

Part C Answer: Discrete function (because the graph has distinct points, domain is discrete (days are whole numbers), and each x has one y)