Linear and Exponential Models: Functions, Graphs, and Applications

Linear and Exponential Models

Introduction

This study guide explores the foundational concepts of linear and exponential models, focusing on their definitions, representations, and real-world applications. Both types of models are essential for understanding patterns of change in mathematics, science, and economics. We will examine how to identify, construct, and interpret these models using analytical, tabular, verbal, and graphical methods.

Linear Models

Definition and General Form

• Linear Model: A function that describes a constant rate of change between two variables. The general form is $y = mx + b$ or $f(x) = mx + b$, where m is the slope and b is the y-intercept.

• Key Properties:

  • The graph is always a straight line.

  • The slope m represents the rate of change (how much y increases for each unit increase in x).

  • The y-intercept b is the value of y when x = 0.

Example 1: Uber Ride Cost

• Scenario: An Uber driver charges a flat rate of $5 plus $2 for every additional mile driven.

• Table of Values:

  Miles driven, x	Total cost, y
  0	5
  1	7
  2	9
  3	11
  4	13
  5	15

• Equation: $y = 2x + 5$

• Interpretation: For each mile driven, the cost increases by $2. The independent variable is miles driven (x), and the dependent variable is total cost (y).

• Example Calculation: For 20 miles: $y = 2(20) + 5 = 45$

• Graph: The graph is a straight line with slope 2 and y-intercept 5.

• Function Definition: A function assigns exactly one output for each input. Here, each number of miles yields one unique cost.

Example 2: Business Revenue Prediction

• Scenario: Predicting total revenue based on the number of employees.

• Table of Values:

  Employees, x	Revenue, y
  0	1,000,000
  10	1,500,000
  20	2,000,000
  30	2,500,000
  40	3,000,000

• Equation: $y = 50,000x + 1,000,000$

• Interpretation: For every 10 additional employees, revenue increases by $500,000, or $50,000 per employee.

• Example Calculation: For x = 70 (2024): $y = 50,000(70) + 1,000,000 = 4,500,000$

Example 3: Seeds per Row

• Scenario: Number of seeds that can be planted in a row as a function of row length.

• Table of Values:

  Row length, x (ft)	Seeds per row, y
  16	4
  24	6
  52	13
  64	16
  72	18

• Equation: $y = 0.25x$

• Interpretation: For each additional foot, 0.25 more seeds can be planted. The model is linear.

• Example Calculation: For x = 200: $y = 0.25(200) = 50$ seeds

Exponential Models

Definition and General Form

• Exponential Model: A function where the rate of change is proportional to the current value, leading to growth or decay by a constant percentage. The general form is $Q(t) = Q_0(1 + r)^t$, where Q0 is the initial value and r is the percent change (as a decimal).

• Key Properties:

  • Exponential growth: output increases by a fixed percentage each period.

  • Exponential decay: output decreases by a fixed percentage each period.

  • Graph is a curve that increases (growth) or decreases (decay) rapidly after a certain point.

Example 1: Doubling Paper Layers

• Scenario: Each time a sheet of paper is torn, the number of pieces doubles.

• Table of Values:

  Number of tears, x	Total papers, y
  0	1
  1	2
  2	4
  3	8
  4	16
  5	32
  6	64

• Equation: $y = 2^x$

• Interpretation: Each tear doubles the number of pieces (100% increase per tear). This is exponential growth.

• Example Calculation: For x = 50: $y = 2^{50}$ (a very large number, far exceeding the number of papers you could physically stack!)

Example 2: COVID Case Growth

• Scenario: COVID cases increase by 60% every 10 days.

• Table of Values:

  Date	Total COVID Counts
  11/20/2021	100,000
  11/30/2021	160,000
  12/10/2021	256,000
  12/20/2021	409,600
  12/30/2021	655,360
  01/09/2022	1,048,576

• Equation: $Q(t) = 100,000(1.60)^t$ (where t is the number of 10-day periods since 11/20/2021)

• Interpretation: Each period, the number of cases increases by 60%. This is exponential growth.

• Example Calculation: For t = 6 (60 days): $Q(6) = 100,000(1.60)^6 \approx 1,677,722$

Example 3: U.S. Population Growth

• Scenario: U.S. population grows at 0.9% per year from a base of 309 million in 2010.

• Equation: $Q(t) = 309,000,000(1.009)^t$ (where t is years since 2010)

• Example Calculation: For t = 50 (year 2060): $Q(50) = 309,000,000(1.009)^{50} \approx 483,634,000$

Example 4: Population Decay (China)

• Scenario: China's population declines at 0.5% per year from 1.3 billion in 2009.

• Equation: $Q(t) = 1,300,000,000(0.995)^t$ (where t is years since 2009)

• Interpretation: This is exponential decay, as the population decreases by 0.5% each year.

• Application: Use the equation to predict future population and assess policy goals.

Identifying Growth Types

Linear vs. Exponential vs. Quadratic

• Linear: Constant difference between outputs (e.g., +50 students per year).

• Exponential: Constant percentage change (e.g., doubling, halving, or increasing by a fixed percent).

• Quadratic: Output changes by a variable amount, often forming a parabolic graph (e.g., death rate vs. hours of sleep).

Additional Examples

• Student Population Growth: $P(t) = 50t + 1700$ (linear, +50 students/year)

• Computer Storage Doubling: $Q(t) = 16(2)^t$ (exponential, doubling every 2 years)

• TV Price Decay: $Q(t) = 1000(0.75)^t$ (exponential decay, -25% per year)

• Death Rate vs. Sleep: Quadratic relationship, minimum death rate at about 7.1 hours of sleep

Applications of Exponential Models

• Population growth and decline

• Bacteria and virus spread

• Inflation and compound interest

• Natural resource consumption

• Loudness of sound, radioactive decay

• Social phenomena (e.g., viral trends)

Other Key Takeaways

• Exponential functions can model both rapid growth and rapid decay.

• Simple interest grows linearly, while compound interest grows exponentially.

• Functions must assign exactly one output for each input.

• Exponential and factorial functions grow faster than linear or quadratic functions.