Applications of Linear Functions 

Lesson Objectives

  • Apply linear functions to real-world situations
  • Model real-world problems with linear functions
  • Interpret slope and y-intercept in context
  • Solve problems using linear equations and graphs

Florida BEST Standards

  • MA.912.AR.2.5: Solve and graph problems modeled with linear functions.
  • MA.912.F.1.8: Determine the best function model for a real-world situation.
  • MA.912.AR.2.1: Write and solve one-variable linear equations in context.
  • MA.912.AR.2.3: Write linear two-variable equations from various representations.
  • MA.912.F.1.2: Determine key features of function graphs including extrema, end behavior, intercepts, domain, and range.
  • MA.912.F.1.3: Calculate and interpret average rate of change in various representations.

Prerequisite Skills

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

Key Vocabulary

  • Linear model
  • constant rate of change
  • independent/dependent variable

Warm-up Activity (5 minutes)

Go over three scenarios that can be modeled by linear functions. Use this resource:

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

 For each example highlight the slope and y-interpret and interpret them relative to the situation.

Teach (25 minutes)

Definitions

Review the following definitions:

If necessary provide these visualizations of the independent and dependent variables:

Next, discuss real-life situations modeled by linear functions, emphasizing constant rate of change. For each application, identify the dependent and independent variables.

Linear Function Model: y = mx

Introduce Hooke's Law as a real-world application of linear functions. Use this slide show:

https://www.media4math.com/library/slideshow/application-linear-functions-hookes-law

Explain that Hooke's Law states that the force (F) exerted by a spring is directly proportional to its displacement (x) from its equilibrium position. Displacement refers to either stretching the spring or compressing it.  

  • Write the equation: F = kx, where k is the spring constant.
    • x is the independent variable. This is the amount of stretch or compression in the sprint.
    • F is the dependent variable. The amount of force is dependent on the amount of stretch or compression.
  • Derive the linear function model for Hooke's Law:
    • Step 1: Start with the basic equation F = kx
    • Step 2: Recognize that this equation is in the form of y = mx, where: 
      y = F (force) 
      m = k (spring constant) 
      x = x (displacement)
    • Step 3: To account for initial tension in the spring, add a y-intercept (b): 
      F = kx + b
    • Step 4: Now we have the general form of a linear equation: y = mx + b

Linear Function Model: y = mx + b

You've seen distance-vs.-time graphs. When a car is moving at a constant speed, the distance-vs.-time graph is a line. What happens when a car is accelerating?

When a car accelerates, it changes its speed. Suppose a car starts at a constant speed of 20 mph. It then increases its speed by 2 mph every second. 

Use this Desmos activity to explore this situation. 

https://www.desmos.com/calculator/vuwaig6swc

The data for the first five seconds is shown in a table. A linear function with a slider for m is set up. Have students:

  • Find the value for m that has the line crossing the data points.
  • What are the units for the slope of this line?
  • Why is the slope of this line called the acceleration?

Review (15 minutes)

Have students explore this application of linear functions, which covers distance, speed, and acceleration. It also includes a brief introduction to quadratic models:

https://www.media4math.com/library/slideshow/applications-linear-functions-speed-vs-acceleration 

Assess (10 minutes)

Administer a 9-question quiz to evaluate understanding of applications of linear functions.

Quiz

  1. A plumber charges \$75 for a house call plus \$60 per hour. Write an equation for the total cost (y) in terms of hours worked (x). If the plumber works for 3 hours, what's the total cost?

     
  2. A company produces custom t-shirts. The cost (y) to produce x shirts is given by y = 3x + 12. What does the 12 represent in this context? How much would it cost to produce 100 shirts?

     
  3. A car rental company charges \$40 per day plus \$0.25 per mile driven. How much would it cost to rent for 3 days and drive 200 miles?

     
  4. A baseball is thrown upward from a height of 6 feet. Its height (h) in feet after t seconds is given by h = -16t^2 + 40t + 6. What was the initial velocity of the ball?

     
  5. A moving company charges based on distance. They charge \$200 for moves up to 50 miles, and an additional \$3 per mile beyond that. Write an equation for the cost (y) in terms of miles (x) for distances over 50 miles. How much would a 75-mile move cost?

     
  6. The temperature of a cooling cup of coffee decreases by 2°C every 5 minutes. If it starts at 80°C, write an equation for the temperature (T) after x minutes. How long will it take for the coffee to cool to 60°C?

     
  7. A company's profit (P) in thousands of dollars is given by P = 0.5x - 100, where x is the number of units sold. How many units must be sold to break even? What's the profit when 300 units are sold?

     
  8. A taxi service charges a base fare of \$2.50 plus \$0.50 per mile. Write an equation for the fare (F) in terms of miles traveled (m). What does the slope represent in this context?

      
  9. The population of a town is increasing by 300 people per year. If the current population is 15,000, write an equation for the population (P) after t years. What will the population be in 5 years?

Answer Key

  1. y = 60x + 75; \$255
  2. Fixed cost; \$312
  3. \$170
  4. 40 feet per second
  5. y = 3x + 200; \$275
  6. T = -0.4x + 80; 50 minutes
  7. 200 units; $50,000
  8. F = 0.50m + 2.50; Cost per mile
  9. P = 300t + 15,000; 16,500 people

 

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