Understanding Linear Relationships and Applications

Nov 4, 2024

Lecture on Modeling Linear Relationships

Frequent Buyer Card Function

  • Scenario: A department store offers a frequent buyer card.
    • Rewards: 12 points per transaction, 25 points upon sign-up.
  • Objective: Write a function for card value based on transactions.
  • Variables:
    • Card value in points
    • Number of transactions (T)

Function Model

  • Equation: Card value = Initial points + (Points per transaction * Number of transactions)
  • Mathematical Representation: ( C(T) = 25 + 12T )
    • 25: Initial points (y-intercept)
    • 12T: Points per transaction (slope)

Solving for Transactions

  • Goal: Earn 100 points for a gift card.
  • Calculation:
    • Set equation to 100: ( 25 + 12T = 100 )
    • Solve for T: 6.25 transactions
    • Result: 7 transactions needed (round up to next whole number).

Key Components

  • Y-Intercept: 25 (initial signup points)
  • Slope: 12 (rate of points increase per transaction)

Fundraising Linear Model

Objective

  • Scenario: Band boosters club raising $1,000 by selling t-shirts and blanket wraps.
    • T-shirts: $10 each
    • Blanket wraps: $25 each
  • Equation: ( 10T + 25B = 1000 )

Graphing the Equation

  • Axes:
    • X-axis: Number of t-shirts (T)
    • Y-axis: Number of blanket wraps (B)
  • Intercepts:
    • X-intercept (no blankets): 100 t-shirts
    • Y-intercept (no t-shirts): 40 blanket wraps

Interpretation

  • Use graph to find possible combinations to reach $1,000.
  • Discrete Nature: Only whole numbers can be sold (t-shirts and blankets).

Sandwich Shop Problem

Objective

  • Scenario: Sell sandwiches and water to earn $100.
    • Sandwiches: $5 each
    • Water: $1 each
  • Equation: ( 5S + W = 100 )

Graphing the Equation

  • Axes:
    • X-axis: Number of sandwiches (S)
    • Y-axis: Number of water bottles (W)
  • Intercepts:
    • X-intercept (no water): 20 sandwiches
    • Y-intercept (no sandwiches): 100 water bottles

Application

  • If no water sold, 20 sandwiches must be sold.
  • Graph helps visualize solutions to real-world problems.

Key Takeaways

  • Determine units for linear relationships.
  • Graphing helps identify solutions and interpret situations.
  • Real-world applications involve discrete points but graphs can simplify solutions.

Homework: Determine units and model linear relationships for assigned problems. See you in the next class.