Modeling and Analyzing Linear Functions 

Lesson Summary

In this lesson, students will delve into the modeling and analysis of linear functions within real-world contexts, particularly focusing on scientific and business applications. They will learn to construct linear models, interpret their components, and utilize these models to make informed predictions. Additionally, students will explore the limitations inherent in linear models when applied to complex, real-life scenarios.

Lesson Objectives

• Analyze and interpret linear functions in real-life scientific and business contexts
• Use linear models to make predictions and solve problems related to natural phenomena and business scenarios
• Understand limitations of linear models in scientific and business applications

Common Core Standards

• F.LE.1: Distinguish between situations that can be modeled with linear functions and with exponential functions.
• F.LE.5: Interpret the parameters in a linear or exponential function in terms of a context.
• F.IF.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.

Prerequisite Skills

• Understanding linear functions and equations
• Interpreting slope and y-intercept
• Graphing linear functions

Key Vocabulary

• Linear Model: A mathematical representation of a situation where there is a constant rate of change between two variables, depicted as a straight line when graphed.
• Extrapolation: The process of estimating values beyond the known data range by extending the linear model.
• Interpolation: The method of estimating values within the range of known data points using the linear model.
• Limitations: The constraints or restrictions that affect the accuracy and applicability of a linear model in representing real-world situations.
• Rate of Change: The ratio that describes how one quantity changes in relation to another; in linear functions, it is represented by the slope.
• Dependent Variable: The variable in a function whose value depends on the value of the independent variable; typically represented as $y$.
• Independent Variable: The variable in a function that is manipulated or changed to observe its effect on the dependent variable; typically represented as $x$.

Warm Up Activities

Choose from one or more of these activities.

Activity 1: Real-World Application

Display 2 real-world scientific scenarios involving linear relationships. Have students identify the independent and dependent variables and describe the rate of change in each situation.

• As altitude increases, air pressure decreases at a constant rate of about 1 kPa per 100 meters.
• In a study of plant growth, the height of a sunflower increases by about 2 cm per day during its early growth phase.

Use this slide show to illustrate the linear models shown above:

https://www.media4math.com/library/slideshow/building-linear-function-models

Activity 2: Graphing Linear Functions in Desmos

Students will explore how different forms of linear equations (slope-intercept, standard, and point-slope) appear graphically using the Desmos Graphing Calculator.

Step-by-Step Instructions:

1. Open the Desmos Graphing Calculator.
2. Have students enter the following equations and observe their graphs:
   • Slope-Intercept Form: $y = 2x + 3$ (positive slope), $y = -0.5x - 4$ (negative slope)
   • Standard Form: $3x + 4y = 12$, $2x - 5y = 10$
   • Point-Slope Form: $y - 3 = 2(x - 1)$, $y + 5 = -3(x - 4)$
3. Ask students to compare the graphs and answer:
   • What do you notice about the slope and y-intercept in each form?
   • How does rewriting standard form into slope-intercept form affect the graph?
   • Why might different forms of linear equations be useful in different scenarios?

Activity 3: Discussion – What is a Mathematical Model?

Before introducing linear models, engage students in a discussion about what mathematical models are and why they are useful.

Guiding Questions:

• What is a model? (Students may say things like blueprints, prototypes, or diagrams.)
• How do models help us make predictions?
• How do scientists, economists, and businesses use mathematical models?
• What is the difference between an accurate and an oversimplified model?

Example of a Mathematical Model:

Provide an example that is not a linear model, such as exponential population growth.

Scenario: A biologist is studying bacteria growth. Every hour, the number of bacteria doubles. The function representing this growth is:

$$y = 2^x$$

Ask students:

• How does this model help predict future bacteria populations?
• Why wouldn't a linear model work for this situation?

Teach

Building a Linear Model

Introduce a real-life business situation involving a linear model of the form y = mx + b:

"Let's consider a small business that produces and sells custom t-shirts. The business has fixed monthly costs (rent, utilities, etc.) of \$2,000. The cost to produce each t-shirt, including materials and labor, is \$10. We want to model the total monthly costs based on the number of t-shirts produced."

Use this slide show to go over the information below:

https://www.media4math.com/library/slideshow/applications-linear-functions-building-business-model 

Here is a Desmos activity that models this linear functions:

https://www.desmos.com/calculator/ngc5ixgnvk 

Write the linear model on the board: C = 10x + 2000

• C = total monthly costs (dependent variable)
• x = number of t-shirts produced (independent variable)
• 10 = slope (rate: \$10 per t-shirt)
• 2000 = y-intercept (fixed monthly costs)

Analyze and interpret this linear function:

• Explain the meaning of the slope: For each additional t-shirt produced, the total cost increases by \$10.
• Interpret the y-intercept: Even if no t-shirts are produced, the business still has \$2,000 in fixed costs.
• You can usse the model to make predictions:
  • If 500 t-shirts are produced, what are the total costs? C = 10(500) + 2000 = \$7,000
  • If the total costs are \$5,000, how many t-shirts were produced? 5000 = 10x + 2000; x = 300 t-shirts

Discuss limitations of this linear model:

• There are limits to the number of t-shirts that can be produced every month.
• When producing a large number of t-shirts, the cost per t-shirt might go down.

Example 1: Predicting Future Sales (Positive Slope)

Problem: A small business tracks its monthly revenue growth. The data is recorded as follows:

1. Graph the points and observe that they form a straight line.
2. Find the slope using two points $(1,5000)$ and $(3,7000)$: $$m = \frac{7000 - 5000}{3 - 1} = \frac{2000}{2} = 1000$$
3. Use point-slope form with $(1,5000)$: $$y - 5000 = 1000(x - 1)$$
4. Convert to slope-intercept form: $$y = 1000x + 4000$$
5. Interpretation: The company’s revenue increases by \$1000 per month.

Example 2: Declining Water Levels (Negative Slope)

Problem: A reservoir’s water level is decreasing due to drought. The recorded levels are:

1. Graph the points and observe that they form a straight line.
2. Find the slope using two points $(2,90)$ and $(6,70)$: $$m = \frac{70 - 90}{6 - 2} = \frac{-20}{4} = -5$$
3. Use point-slope form with $(2,90)$: $$y - 90 = -5(x - 2)$$
4. Convert to slope-intercept form: $$y = -5x + 100$$
5. Interpretation: The reservoir loses 5 million gallons of water per week.

Example 3: Budgeting a Business Expense (Standard Form to Slope-Intercept Form)

Problem: A company has \$20,000 to spend on advertising. A TV ad costs \$500 per slot, and a social media ad costs \$250. Write a model and convert it to slope-intercept form.

1. Write the equation in standard form: $$500x + 250y = 20000$$
2. Convert to slope-intercept form: $$250y = -500x + 20000$$ $$y = -2x + 80$$
3. Interpretation: For every two additional TV ads purchased, the company can afford 1 fewer social media ad.

Example 4: Hooke’s Law (Existing Mathematical Model)

Problem: Hooke’s Law states that the force needed to stretch a spring is proportional to the distance stretched: $F = kx$. A particular spring has a spring constant of $k = 200$ N/m. Determine the force needed to stretch it 0.5 meters.

1. Use Hooke’s Law:
2. Substituting $x = 0.5$: $$F = 200x$$
3. Interpretation: A force of 100 N is required to stretch the spring by 0.5 meters.

Example 5: Distance vs. Time – Interpreting Speed

Problem: A car travels at a constant speed. The recorded distances are:

1. Graph the points and see they form a straight line.
2. Find the slope using $(1,50)$ and $(3,150)$: $$m = \frac{150 - 50}{3 - 1} = \frac{100}{2} = 50$$
3. Write the equation in slope-intercept form: $$y = 50x$$
4. Interpretation: The car is traveling at a constant speed of 50 mph.

Example 6: Boyle’s Law (Existing Linear Model)

Problem: Boyle’s Law states that for a fixed amount of gas at a constant temperature, pressure and volume are inversely related: $PV = k$. A gas sample has a volume of 3 liters and a pressure of 80 kPa. Find the pressure if the volume expands to 6 liters.

1. Use Boyle’s Law: $$P_1V_1 = P_2V_2$$
2. Substituting known values: $$80(3) = P_2(6)$$
3. Solve for $P_2$: $$P_2 = \frac{80(3)}{6} = \frac{240}{6} = 40$$
4. Interpretation: When the volume doubles from 3 to 6 liters, the pressure decreases to 40 kPa, demonstrating the inverse relationship.

Review

Show this video and as a class develop a linear model:

https://www.media4math.com/library/21299/asset-preview

Complete the following table:

 Here is a Desmos activity that you can use:

https://www.desmos.com/calculator/diequ4bkqq

Analyze the model and discuss some of its limitations:

• What does the y-intercept represent?
• What does the slope represent?
• What is a more realistic domain for this model?
• What is a more realistic range?

Quiz

Answer the following questions.

1. In the sunflower growth model h = 2d + 10, what does the 2 represent?
   
   
    
2. How tall will the sunflower be after 12 days?
   
   
    
3. On which day will the sunflower reach a height of 50 cm?
   
   
    
4. What is the initial height of the sunflower in this model?
   
   
    
5. If we measure a sunflower's height as 35 cm, how many days has it been growing according to this model?
   
   
    
6. Why might this linear sunflower growth model be inaccurate for very long time periods?
   
   
    
7. If the growth rate slows to 1.5 cm per day after the first week, write the new growth function starting from day 7.
   
   
    
8. Could you use a linear function for population growth>
   
   
    
9. Write a linear function for the air pressure (P) in kPa based on altitude (a) in meters, given that pressure decreases by 1 kPa per 100 m. Assume sea level pressure is 101.3 kPa.
   
   
    
10. Using the function from question 6, what would be the air pressure at an altitude of 1500 meters?

Answer Key

1. The growth rate in cm per day
2. 34 cm
3. Day 20
4. 10 cm
5. 12.5 days
6. Plant growth typically slows down as the plant matures
7. h = 1.5d + 24 (where d is now days since day 7)
8. No. Populations grow in a non-linear manner.
9. P = -0.01a + 101.3
10. 86.3 kPa