Princeton Review

 Modeling and Analyzing Linear Functions 

 Lesson Objectives

  • Model and analyze linear functions in real-world contexts
  • Interpret key features of linear functions
  • Represent linear relationships in various forms

Florida BEST Standards

  • MA.912.AR.2.5: Solve and graph problems modeled with linear functions.
  • MA.912.F.1.2: Determine key features of function graphs including extrema, end behavior, intercepts, domain, and range.
  • MA.912.AR.2.3: Write linear two-variable equations from various representations.
  • MA.912.AR.2.4: Graph linear functions and determine key features.
  • MA.912.F.1.3: Calculate and interpret average rate of change in various representations.
  • MA.912.F.1.8: Determine the best function model for a real-world situation.

Prerequisite Skills

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

Key Vocabulary

  • Linear model
  • Extrapolation
  • Interpolation
  • Limitations
  • Rate of change
  • Dependent variable
  • Independent variable

Warm-up Activity (5 minutes)

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

Teach (20 minutes)

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.

Review (15 minutes)

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

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

Complete the following table:

Linear Modely = 220 - x
y-intercept220
slope-1
Domain0 ≤ x ≤ 220
Range0 ≤ y ≤ 220

 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?

Assess (10 minutes)

Administer a 10-question quiz to evaluate student understanding.

Quiz

  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?

Answers

  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

 

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