# Applications of Linear Functions

## 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
• 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:

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 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:

## 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?

1. y = 60x + 75; \$255 2. Fixed cost; \$312
3. \$170 4. 40 feet per second 5. y = 3x + 200; \$275