1.5 Identifying assumptions and limitations of function models

• What is a function model?

  Mathematical equation for real-world situations
• Match the aspect with its description:

  Accuracy ↔️ Precise math

  Complexity ↔️ Simplified formulas

  Scope ↔️ Specific variables
• Exponential function models are suitable for discrete data.

  False
• Trigonometric function models are appropriate for data without cyclical patterns.

  False
• The usefulness of a function model depends on how closely it matches the real-world conditions
• What type of shape does a quadratic function assume?

  Parabolic
• What type of relationship does a rational function assume?

  Inverse or reciprocal
• Steps to use function models effectively:

  1️⃣ Understand the real-world situation

  2️⃣ Choose an appropriate function model

  3️⃣ Identify assumptions and limitations

  4️⃣ Evaluate the model's predictions
• What is one aspect where function models differ from real-world situations?

  Complexity
• Function models simplify dynamic interactions into simplified formulas.


  True
• Order the steps involved in using a function model to represent a real-world situation:

  1️⃣ Identify the real-world situation

  2️⃣ Choose an appropriate function model

  3️⃣ Collect relevant data

  4️⃣ Validate the model with data

  5️⃣ Adjust the model if needed
• Match the function model with its key characteristic:

  Linear ↔️ $y =$$mx +$$b$

  Exponential ↔️ $y =$$a \cdot b^{x}$

  Quadratic ↔️ $y =$$ax^{2} +$$bx +$$c$

  Logarithmic ↔️ $y =$$a \ln(x) +$$b$
• A quadratic function model assumes a parabolic shape.


  True
• What is a key limitation of using function models to represent real-world situations?

  Simplified representation
• Steps to effectively use function models in real-world applications

  1️⃣ Understand limitations

  2️⃣ Simplify variables

  3️⃣ Create manageable equations

  4️⃣ Evaluate model critically
• A linear function model assumes a constant rate of change
• Each function model has limitations in representing the complexities of real-world phenomena
• Function models can perfectly capture the complexity of real-world situations.

  False
• Match the aspect with its representation in function models and real-world situations:

  Accuracy ↔️ Precise math ||| Complex, uncertain

  Complexity ↔️ Simplified formulas ||| Dynamic interactions

  Scope ↔️ Specific variables ||| All influencing factors
• What is a key limitation of function models regarding complexity?

  Simplified equations
• A function model simplifies complex relationships into a manageable form for analysis
• What is the formula for a linear function model?

  $y =$$mx +$$b$
• Why are assumptions necessary in function models?

  To simplify complexity
• The accuracy of a function model depends on its underlying assumptions.


  True
• What is a key difference between the accuracy of a function model and the complexity of a real-world situation?

  Precise math vs. uncertain
• What type of relationship does a trigonometric function model assume?

  Periodic, sinusoidal
• Function models always simplify real-world complexity


  True
• A linear function model is defined by the equation $y =$$mx +$$b$, which assumes a constant rate of change
• A logarithmic function model is represented by the equation $y =$$a \ln(x) +$$b$, which assumes a logarithmic relationship
• Function models simplify complex relationships into a manageable form for analysis and prediction
• Function models always involve assumptions and simplifications that limit their precision
• Function models use precise math, while real-world situations involve uncertainty
• Function models always involve assumptions and simplifications that limit their precision.


  True
• Linear function models assume a constant rate of change
• Function models focus on primary variables, while real-world situations consider multiple variables.
• Function models often focus on primary variables
• Function models always involve assumptions and simplifications.

  True
• Match the function model with its key assumption:

  Linear ↔️ Constant rate of change

  Quadratic ↔️ Parabolic shape

  Exponential ↔️ Continuous growth or decay

  Logarithmic ↔️ Logarithmic relationship