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2. Linear functions
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United States of America
Grades 9-12

2.07 Modeling linear relationships

Lesson
Practice

Introduction

Modeling makes the connection between the mathematics we learn in the classroom and the mathematics that drives the world we live in. Real-world problems are often messy and complex. As we learn to model these real-world problems we will work through the modeling cycle:

A modeling cycle. Starting with the phrase Identify the problem inside a circle. An arrow pointing to the right where the phrase Create a model is inside a rectangle. Next is an arrow pointing downward where the phrase Apply and analyze is inside a rectangle. Then, an arrow pointing to the right where the phrase Interpret results is inside a rectangle. Next is an arrow pointing upward where the phrase Verify the model is inside a triangle. Then, an arrow pointing to the right where the phrase Report findings is inside a circle. There is an arrow pointing to the left from the phrase Verify the model to Create a model.

Each time we model a real-world situation we should:

  1. Identify the essential features of the problem

  2. Create a model using a diagram, graph, table, equation or expression, or statistical representation

  3. Analyze and use the model to find solutions

  4. Interpret the results in the context of the problem

  5. Verify that the model works as intended and improve the model as needed

  6. Report on our findings and the reasoning behind them

Identifying the problem

Modeling problems are generally open-ended and require creative or unique approaches. When identifying the problem, our goal is to restate the open-ended question by clarifying what, specifically, our model will be designed to output. Depending on the question at hand, we may need to do some research to narrow the focus of the question and our approach to solving it. Finally, we will need to list any assumptions or simplifications required to create our model.

Assumption

A statement which is introduced into an argument and temporarily accepted as true.

We introduce assumptions to make the problem more manageable and so that the model fits the context, but we should limit how many assumptions we make because they decrease the accuracy of the model. When we verify the model, we should address the impact of the assumptions.

Exploration

Government assistance programs like the Supplemental Nutrition Assistance Program (SNAP) rely on calculations of the minimum required cost of feeding Americans to help determine how much aid to provide per family. To help determine the support needed for SNAP the government must regularly analyze and answer the question, "what does it cost to feed a family?".

  1. How would you go about answering this question?
  2. What questions would you ask?
  3. What research might you complete?

It is important to consider as many factors as possible that might affect the result in a situation. Starting with a list of questions and then researching them is always a good place to start.

Examples

Example 1

What does it cost to feed a family in the United States?

a

Brainstorm a list of questions that can be answered to help solve this problem.

Worked Solution
Apply the idea

A possible list of brainstormed ideas might be:

  • What are the different food requirements based on size or age?

  • How many people are in a family?

  • What are the current food costs in my area?

Reflect and check

The brainstorming will help identify assumptions that need to be made in order to create a model.

b

Research and determine how a model can be used to solve the problem.

Worked Solution
Create a strategy

In order to answer the questions in our brainstorm, there are a few things we need to decide:

  • How the dietary needs of people differ.
  • How our model can apply to different types of families.
  • How food costs are calculated.
  • What span of time we are using: daily cost, weekly cost, monthly cost, or other.
Apply the idea

One possible approach to this problem could be:

We will create a model that outputs the daily cost to feed a single person based on their age. The cost to feed a family can then be determined by combining the costs of each individual in the family.

The caloric needs will come from dietaryguidelines.gov and we will find the cost of a meal plan for a single day that meets the main nutrition requirements (calories, proteins, carbohydrates, and fats) for the different age groups. Instead of creating different models for different genders, we will use the average nutritional needs between males and females of the same age group. We will research the average cost of each food on the meal plan, paying attention to cost-effective choices, since we are trying to calculate the minimum costs.

Reflect and check

Some of the assumptions made in this problem statement include:

  • Age is the main influence in caloric need.
  • One meal plan can be used to calculate the costs for every day.
  • The average nutritional needs of males and females will make a good representation of the nutritional needs of any person.

As we build and test our model, we can revisit these assumptions to see if they should be changed or updated to better reflect the problem.

Idea summary

A modeling problem is often open-ended with many unknowns. Our job is to research and brainstorm the problem, determine how a model can solve the problem, and make necessary assumptions to support our model.

Creating and analyzing a model

A variable is used to represent a quantity in our problem that can take on many values. A parameter is a known value that may remain constant in the model.

When creating a model, we need to define the independent variable, or input of the model, and the dependent variable, or the output of the model that is influenced by the value of the input.

The model itself may take on different forms. Models can be presented as equations, graphs, tables, diagrams, and maps. What's important is that the model conveys the information in a way that is accessible to a viewer.

Examples

Example 2

Consider the model from the previous section:

We will create a model that outputs the daily cost to feed a single person based on their age. The cost to feed a family can then be determined by combining the costs of each individual in the family.

a

Identify the independent and dependent variables of the model.

Worked Solution
Apply the idea

The independent variable is the age of the person and the dependent variable is the daily cost to feed that person.

b

Identify the parameters of the model.

Worked Solution
Apply the idea

The calories for each age category are a parameter since they are known, fixed, values. For example, for the age group 9-13 we will have a calorie requirement of 1700 calories.

The amount of daily macronutrients is also a parameter since it is a known fixed value once the number of calories is determined:

A table about the daily nutritional goals of ages 2 and older. Speak to your teacher for more information.
Reflect and check

We've uncovered additional assumptions by defining a few parameters in the problem. For example, the average calories needed in the age group 9-13 years old is 1700, but the chart for macronutrients only lists 1600 and 1800. We will need to revisit our original assumption (averaging the gender needs) or make a new assumption (average the macronutrient needs).

Example 3

Consider the table below:

Age381318305070
Calories1000130017002000220020001800
Daily cost\$ 5.07\$ 6.33\$8.19\$ 8.51\$9.17\$8.20\$7.75
a

Create a model that can be used to determine the cost of feeding an individual based on their age.

Worked Solution
Create a strategy

There are many different models that all satisfy different requirements. Since we are provided data in a table, we can efficiently convert this to a visual model with a graph.

When creating a graph, the independent variable should be represented by the x-axis and the dependent variable by the y-axis. Be sure to choose a scale that shows the domain and range.

Apply the idea

We can start by plotting each point in the table:

5
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70
\text{Age in years}
5
5.5
6
6.5
7
7.5
8
8.5
9
9.5
\text{Daily cost in dollars}

Analyzing the graph, we can see that the daily cost increases until age 30 and then decreases until age 70. This analysis helps guide our choice of model.

If we assume that the daily cost changes at an approximately linear rate, then we can use a linear piecewise model:

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\text{Age in years}
5
5.5
6
6.5
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7.5
8
8.5
9
9.5
\text{Daily cost in dollars}
Reflect and check

If we don't make the assumption of linear rate of change, we could use a step function to model the daily costs instead:

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\text{Age in years}
5
5.5
6
6.5
7
7.5
8
8.5
9
9.5
\text{Daily cost in dollars}

Since the points on the graph are not very symmetric, an absolute value function is not a good model for the data in the table:

5
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\text{Age in years}
5
5.5
6
6.5
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7.5
8
8.5
9
9.5
\text{Daily cost in dollars}
b

Create a different model that can be used to determine the cost of feeding an individual based on their age.

Worked Solution
Create a strategy

Algebraic models like equations and functions, are useful since they can be quickly evaluated for any value in the domain. For a secondary model, we'll create an equation.

Since the cost is increasing from age 3 to 30 and then decreasing from 30 to 70, we'll need two separate equations to model this data.

Apply the idea

Create a piecewise-defined function for each age range:f(x) = \begin{cases} ⬚, & 3 \lt x \leq 30 \\ ⬚, & 30 \lt x \leq 70 \end{cases}

To create a linear function for ages 3 to 30, we can use the points \left(3,5.07\right) and \left(30,9.17\right).

First, find the slope:

\displaystyle \frac{9.17-5.07}{30-3}\displaystyle =\displaystyle 0.15Evaluate slope formula

Then, use the point-slope formula to find the equation:

\displaystyle y-y_1\displaystyle =\displaystyle m(x-x_1)Point-slope formula
\displaystyle y-5.07\displaystyle =\displaystyle 0.15(x-3)Substitute \left(3,5.07\right)
\displaystyle y-5.07\displaystyle =\displaystyle 0.15x-0.45Distributive property
\displaystyle y\displaystyle =\displaystyle 0.15x+4.62Addition property of equality

Then we need to repeat this process for age 30 to 70:

Use \left(30,9.17\right) and \left(70,7.75\right) to find the slope and equation:

\displaystyle \frac{7.75-9.17}{70-30}\displaystyle =\displaystyle -0.04Evaluate slope formula
\displaystyle y-y_1\displaystyle =\displaystyle m(x-x_1)Point-slope formula
\displaystyle y-9.17\displaystyle =\displaystyle -0.04(x-30)Substitute \left(30,9.17\right)
\displaystyle y-9.17\displaystyle =\displaystyle -0.04x-1.2Distributive property
\displaystyle y\displaystyle =\displaystyle -0.04x+7.97Addition property of equality

So, with both equations our piecewise function is:f(x) = \begin{cases} 0.15x+4.62, & 3 \lt x \leq 30 \\-0.04x+7.97, & 30 \lt x \leq 70 \end{cases}

c

Determine how the assumptions used in parts (a) and (b) could be revised to create a better model.

Worked Solution
Create a strategy

First, we need to acknowledge where our model is less accurate. Then, we can brainstorm ways to make improvements.

Apply the idea

Consider our models from parts (a) and (b).

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\text{Age in years}
5
5.5
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7.5
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8.5
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9.5
\text{Daily cost in dollars}

f(x) = \begin{cases} 0.15x+4.62, & 3 \lt x \leq 30 \\-0.04x+7.97, & 30 \lt x \leq 70 \end{cases}

We can see on the graph, that our linear model underestimates the daily cost for 8, 13, and 18 year-olds. If we use our piecewise-defined function we can see that the estimated daily cost for a 13 year old is \$ 6.57.

\displaystyle f\left(x\right)\displaystyle =\displaystyle \begin{cases} 0.15x+4.62, & 3 \lt x \leq 30 \\-0.04x+7.97, & 30 \lt x \leq 70 \end{cases}Piecewise-defined function
\displaystyle f\left(13\right)\displaystyle =\displaystyle 0.15(13)+4.62Substitute into the first equation of the function since 3<13\leq 30
\displaystyle f\left(13\right)\displaystyle =\displaystyle 6.57Evaluate the multiplication and addition

However, according to the table, the daily cost for a 13 year old should be \$8.19 which means our model underestimates the true cost by \$1.62 which would add up to \$ 591.30 each year.

One way to improve the model is to define a linear function for ages 3-13 where the rate is higher, and then a different linear function for ages 13-30 where the rate is less.

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\text{Age in years}
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8.5
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9.5
\text{Daily cost in dollars}

f(x) = \begin{cases} 0.31x+4.14, & 3 \lt x \leq 13 \\ 0.06x+7.41, & 13 \lt x \leq 30 \\-0.04x+7.97, & 30 \lt x \leq 70 \end{cases}

We can see from the graph that this model is a better estimation for ages 8, 13, and 18 than the previous model.

Reflect and check

Other improvements to the model could be to extend the model to include the daily cost to feed children under 3 years of age and the daily cost to feed adults over the age of 70.

Idea summary

A model may take on a variety of different forms including equations, graphs, tables, and diagrams. Modeling is a cycle that can and should be repeated. The initial assumptions used to create a model can be revised to make improvements or adjustments for the model.

Outcomes

A.CED.A.2

Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

A.CED.A.3

Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context.

F.IF.B.4

For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.

F.IF.B.5

Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.

F.LE.B.5

Interpret the parameters in a linear or exponential function in terms of a context.

S.ID.C.7

Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.

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