# Modeling and Analyzing Linear Functions

## 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
- 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* = 10*x* + 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 = 10
*x*+ 2000;*x*= 300 t-shirts

- If 500 t-shirts are produced, what are the total costs?

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 Model | y = 220 - x |

y-intercept | 220 |

slope | -1 |

Domain | 0 ≤ x ≤ 220 |

Range | 0 ≤ 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

- In the sunflower growth model
*h*= 2*d*+ 10, what does the 2 represent?

- How tall will the sunflower be after 12 days?

- On which day will the sunflower reach a height of 50 cm?

- What is the initial height of the sunflower in this model?

- If we measure a sunflower's height as 35 cm, how many days has it been growing according to this model?

- Why might this linear sunflower growth model be inaccurate for very long time periods?

- If the growth rate slows to 1.5 cm per day after the first week, write the new growth function starting from day 7.

- Could you use a linear function for population growth>

- 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.

- Using the function from question 6, what would be the air pressure at an altitude of 1500 meters?

## Answers

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

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