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 Applications of Linear Functions 

 

Lesson Summary

In this lesson, students will explore how linear functions model real-world situations. They will learn to identify scenarios that can be represented by linear equations, formulate these equations, and interpret their components within context. The lesson emphasizes understanding the relationship between variables and applying linear models to solve practical problems.

Lesson Objectives

  • Identify real-life situations modeled by linear functions
  • Write linear equations to represent situations
  • Solve problems involving linear functions

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.BF.1 - Write a function that describes a relationship between two quantities.

Prerequisite Skills

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

Key Vocabulary

  • Linear Model: A mathematical representation of a situation where there is a constant rate of change between two variables, depicted by a straight line on a graph.
  • Constant Rate of Change: The unvarying ratio of the change in the dependent variable to the change in the independent variable in a linear relationship.
  • Independent Variable: The variable that represents the input or cause and is plotted along the x-axis; its variation does not depend on other variables in the context of the function.
  • Dependent Variable: The variable that represents the output or effect and is plotted along the y-axis; its value depends on the independent variable.
  • Slope-Intercept Form: A way to express linear equations in the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
  • Point-Slope Form: A form of linear equations given by \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \( (x_1, y_1) \) is a specific point on the line.
  • Standard Form: A representation of linear equations as \( Ax + By = C \), where \( A \), \( B \), and \( C \) are integers, and \( A \) is non-negative.

Multimedia Resources

 

 

Warm Up Activities

Activity 1: Real-World Applications of Rates

Before diving into linear functions, students will explore real-world examples where rates play a crucial role. This activity will help students recognize linear relationships in everyday life.

Instructions:

  1. Present the following real-world scenarios and have students identify the rate of change:
    • A taxi company charges a base fee of \$5 plus \$2 per mile traveled.
    • A gym offers a membership plan for \$30 per month.
    • A water tank is filled at a rate of 10 gallons per minute.
    • A factory produces 50 units of a product per hour.
  2. Ask students to:
    • Determine the independent and dependent variables.
    • Identify the constant rate of change.
    • Write an equation in slope-intercept form for each scenario.
  3. Have a class discussion on how rates represent linear functions and why they are useful in modeling real-world situations.

Activity 2: Using Desmos to Graph Linear Functions

Students will use the Desmos Graphing Calculator to explore the visual representation of linear functions.

Instructions:

  1. Open the Desmos Graphing Calculator.
  2. Have students enter the following linear equations and observe their graphs:
    • \( y = 3x + 2 \) (positive slope, positive y-intercept)
    • \( y = -2x + 5 \) (negative slope, positive y-intercept)
    • \( y = 0.5x - 4 \) (positive slope, negative y-intercept)
    • \( y = -x + 7 \) (negative slope, positive y-intercept)
  3. Ask students to:
    • Identify the slope and y-intercept for each equation.
    • Describe how changing the slope affects the steepness of the line.
    • Discuss how changing the y-intercept shifts the graph up or down.
  4. Encourage students to input their own linear equations to explore different slopes and y-intercepts.

Activity 3: Linear Equation Models

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

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?

Standard Form

Linear equations are often written in standard form:

\[ Ax + By = C \]

where:

  • \( A \), \( B \), and \( C \) are integers.
  • \( A \) is non-negative.
  • The equation represents a straight-line relationship between \( x \) and \( y \).

To graph an equation in standard form, students can:

  1. Find the x-intercept by setting \( y = 0 \) and solving for \( x \).
  2. Find the y-intercept by setting \( x = 0 \) and solving for \( y \).
  3. Plot both intercepts and draw a straight line through them.

Alternatively, you can write the equation in slope-intercept form.

Example: Budgeting for a School Event

Problem: A school is planning an event with a \$500 budget for food and decorations. The cost of food is \$10 per person, and decorations cost \$200. Write an equation in standard form and graph it.

  1. Define variables:
    • \( x \) = number of people.
    • \( y \) = money spent on decorations.
  2. Write the equation: \[ 10x + y = 500 \]
  3. Find intercepts:
    • Set \( y = 0 \):
    • Set \( x = 0 \):
  4. Graph the line using the intercepts \( (50,0) \) and \( (0,500) \).
  5. Interpretation: If no money is spent on decorations, up to 50 people can attend. If all money is used on decorations, no attendees can be accommodated.

Point-Slope Form

Another useful form of a linear equation is the point-slope form:

\[ y - y_1 = m(x - x_1) \]

where:

  • \( (x_1, y_1) \) is a known point on the line.
  • \( m \) is the slope.

This form is particularly useful when given a point and the rate of change but not the y-intercept.

Example: Car Rental Costs

Problem: A car rental company charges a \$50 base fee plus an additional \$15 per day. Write an equation in point-slope form.

  1. Identify a known point:
    • After 2 days, the cost is \$80 → \( (2, 80) \).
  2. Identify the slope (rate of change): \[ m = 15 \]
  3. Write the equation: \[ y - 80 = 15(x - 2) \]
  4. Interpretation: The equation models the total cost of renting a car based on the number of days rented.

Real-World Applications of Linear Functions

Example 1: Slope-Intercept Form (Positive Slope) – Weekly Savings

Problem: Jake saves \$20 per week in his bank account. He starts with \$100. Write a linear equation and determine how much he will have after 8 weeks.

  1. Identify key information:
    • Initial amount: \$100 (y-intercept, \( b \))
    • Rate of savings: \$20 per week (slope, \( m \))
  2. Write the equation in slope-intercept form:\[ y = 20x + 100 \]
  3. Find \( y \) after 8 weeks:\[ y = 20(8) + 100 = 160 + 100 = 260 \]
  4. Interpretation: After 8 weeks, Jake will have \$260 in his account.

Example 2: Slope-Intercept Form (Negative Slope) – Cooling Water

Problem: A substance cools at a rate of 5°F per minute. It starts at 200°F. Write a linear equation and determine the temperature after 10 minutes.

  1. Identify key information:
    • Initial temperature: 200°F (y-intercept, \( b \))
    • Cooling rate: -5°F per minute (slope, \( m \))
  2. Write the equation:\[ y = -5x + 200 \]
  3. Find \( y \) after 10 minutes:\[ y = -5(10) + 200 = -50 + 200 = 150 \]
  4. Interpretation: After 10 minutes, the water will be 150°F.

Example 3: Standard Form to Slope-Intercept Form (Positive Slope) – Ticket Sales

Problem: A school fundraiser sells adult tickets for \$10 and student tickets for \$5. The total revenue is \$500. Write the equation in standard form and convert it to slope-intercept form.

  1. Write the equation in standard form:\[ 10x + 5y = 500 \]
  2. Solve for \( y \) (convert to slope-intercept form):\[ 5y = -10x + 500 \] \[ y = -2x + 100 \]
  3. Interpretation: The equation shows the relationship between adult and student ticket sales. The slope of -2 means that for every 2 adult tickets sold, 1 fewer student ticket is needed to reach \$500.

Example 4: Standard Form to Slope-Intercept Form (Negative Slope) – Budgeting

Problem: A family is planning a vacation with a \$2,000 budget. They can spend money on flights and hotels. Each flight costs \$400, and each night in a hotel costs \$200. Write the equation and convert to slope-intercept form.

  1. Write the equation in standard form:\[ 400x + 200y = 2000 \]
  2. Solve for \( y \):\[ 200y = -400x + 2000 \]\[ y = -2x + 10 \]
  3. Interpretation: The equation shows that for every extra flight purchased, two fewer hotel nights can be afforded.

Example 5: Point-Slope Form (Positive Slope) – Gym Membership

Problem: A gym charges a monthly fee of \$30. After 3 months, a member has paid \$140. Write an equation in point-slope form to model the cost.

  1. Identify a known point and slope:
    • After 3 months, cost is \$140 → (3,140) .
    • Rate of change: \$30 per month ( m = 30 ).
  2. Write the equation in point-slope form:\[ y - 140 = 30(x - 3) \]
  3. Convert to slope-intercept form (optional for further exploration):\[ y = 30x + 50 \]
  4. Interpretation: The equation shows that the gym charges a \$50 initial fee (y-intercept) and \$30 per month (slope).

Example 6: Point-Slope Form (Negative Slope) – Fuel Consumption

Problem: A truck starts a trip with an unknown amount of fuel. After 4 hours, it has 60 gallons left. After 8 hours, it has 40 gallons left. Write an equation in point-slope form.

  1. Find the slope using the two points \( (4,60) \) and \( (8,40) \):\[ m = \frac{40 - 60}{8 - 4} = \frac{-20}{4} = -5 \]
  2. Choose one point, e.g., \( (4,60) \), and write the equation in point-slope form:\[ y - 60 = -5(x - 4) \]
  3. Convert to slope-intercept form (optional for further exploration):\[ y = -5x + 80 \]
  4. Interpretation: The equation models fuel consumption, showing that the truck started with 80 gallons and uses 5 gallons per hour.

 

 

Review

Summary of Concepts Covered

In this lesson, students explored different forms of linear equations—slope-intercept form, point-slope form, and standard form—while applying them to real-world scenarios. Key takeaways include:

  • Slope-Intercept Form: \( y = mx + b \) is useful when given the slope and y-intercept.
  • Point-Slope Form: \( y - y_1 = m(x - x_1) \) is useful when given one point and the slope.
  • Standard Form: \( Ax + By = C \) is useful when working with constraints and can be converted to slope-intercept form for graphing.
  • How to interpret slope as a rate of change in real-world contexts.
  • How to determine the intercepts of a linear function.

Key Vocabulary

  • Slope: The measure of the steepness of a line, calculated as rise over run.
  • Rate of Change: A ratio describing how one quantity changes in relation to another.
  • Y-Intercept: The point where a line crosses the y-axis, representing the output when \( x = 0 \).
  • Slope-Intercept Form: The equation \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
  • Point-Slope Form: The equation \( y - y_1 = m(x - x_1) \), used when given a point and a slope.
  • Standard Form: The equation \( Ax + By = C \), where \( A \), \( B \), and \( C \) are integers.
  • Linear Model: A mathematical equation representing a situation with a constant rate of change.

Multimedia Resources

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 

Additional Examples

Example 1: Slope-Intercept Form (Car Rental Cost)

Problem: A car rental company charges a \$25 base fee plus \$15 per hour. Write the equation and determine the cost after 6 hours.

  1. Identify slope and y-intercept:
    • Base fee (y-intercept): \$25
    • Rate of change (slope): \$15 per hour
  2. Write the equation:\[ y = 15x + 25 \]
  3. Find \( y \) for \( x = 6 \):\[ y = 15(6) + 25 = 90 + 25 = 115 \]
  4. Interpretation: After 6 hours, the total rental cost is \$115.

Example 2: Point-Slope Form (Home Value Appreciation)

Problem: A house was worth \$250,000 five years ago and has been increasing in value by \$12,000 per year. Write an equation using point-slope form to model its price over time.

  1. Identify a known point and slope:
    • Five years ago, the house was worth \$250,000 → (5,250000).
    • Rate of appreciation: \$12,000 per year ( m = 12000).
  2. Write the equation in point-slope form:\[ y - 250000 = 12000(x - 5) \]
  3. Convert to slope-intercept form (optional):\[ y = 12000x + 190000 \]
  4. Interpretation: The equation predicts the home's value based on the number of years passed.

Example 3: Standard Form to Slope-Intercept Form (Concert Ticket Sales)

Problem: A concert venue sells general admission tickets for \$50 and VIP tickets for \$100. The total revenue is \$5,000. Write the equation in standard form and convert it to slope-intercept form.

  1. Write the equation in standard form:\[ 50x + 100y = 5000 \]
  2. Convert to slope-intercept form:\[ 100y = -50x + 5000 \]\[ y = -0.5x + 50 \]
  3. Interpretation: The equation shows the relationship between general admission and VIP ticket sales. The slope of -0.5 means that for every additional two general admission tickets sold, one fewer VIP ticket is needed to reach \$5,000.

 

 

Quiz

Answer the following questions.

  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