1.5 Identifying assumptions and limitations of function models

• A quadratic model assumes that the relationship between variables follows a parabolic shape.
• A function model is a mathematical representation of a real-world relationship
• Assumptions in function models are the conditions that must be true for the model to accurately represent the phenomenon.
• Limitations of function models arise when their underlying assumptions are violated
• What is the primary purpose of function models?

  Mathematical representation of relationships
• Evaluating the validity of function models involves assessing their accuracy
• Improving function model accuracy involves refining its assumptions and parameters.
• Match the function model type with its description:

  Linear ↔️ Constant rate of change

  Quadratic ↔️ Changing rate of change

  Exponential ↔️ Continuously changing rate of change

  Rational ↔️ Relationship with numerator and denominator
• A rational function model represents a relationship with a numerator and denominator
• Assumptions are the conditions that must be true for a function model to accurately represent a real-world phenomenon.
• What is a typical assumption of a linear function model?

  Constant rate of change
• An exponential function model assumes that variables change at a continuously changing rate
• A linear model for population growth may accurately represent growth under resource constraints.

  False
• One limitation of function models is oversimplification, where they fail to capture all the complexities of a real-world situation
• How can data limitations affect the reliability of a function model?

  Impact its predictive power
• Steps for using function models effectively:

  1️⃣ Identify underlying assumptions

  2️⃣ Evaluate limitations

  3️⃣ Account for external factors
• Function models may not capture all the complexities of a real-world situation
• What is one consequence of violating the assumptions of a function model?

  Inaccurate representation
• The quality of data used in a function model does not affect its reliability.

  False
• Steps to using function models effectively

  1️⃣ Identify the underlying assumptions

  2️⃣ Evaluate the assumptions

  3️⃣ Understand how violations affect accuracy
• What is oversimplification in function models?

  Failure to capture complexities
• Match the function model with its purpose:

  Linear ↔️ Constant rate of change

  Quadratic ↔️ Parabolic relationship

  Exponential ↔️ Continuously changing rate

  Rational ↔️ Inverse relationship
• Assumptions in function models are conditions or constraints
• What is a typical assumption of a linear model?

  Variables change at a constant rate
• Quadratic models assume that the relationship has a parabolic shape.
• What type of relationship does an exponential model represent?

  Continuously changing rate
• Match the model type with its typical assumption:

  Linear ↔️ Constant growth rate

  Quadratic ↔️ Parabolic relationship

  Exponential ↔️ Continuous decay

  Rational ↔️ Inverse proportionality
• Why might a linear model for population growth be inaccurate?

  Constant growth rate
• Recognizing limitations of function models allows for more informed decisions.
• Function models may not capture all real-world complexities due to oversimplification
• How can data limitations affect a function model?

  Reduced reliability
• What is one common source of error in function models that occurs when models may not capture all the complexities of a real-world situation?

  Oversimplification
• A function model may become inaccurate if its underlying assumptions are violated.
• The reliability of a function model is unaffected by the quality of the data used to fit it.

  False
• Match the function model type with its typical assumption:

  Linear ↔️ Relationship is linear

  Quadratic ↔️ Relationship has a parabolic shape
• What type of relationship is assumed in an exponential function model?

  Exponential growth or decay
• A linear model for population growth may be invalid if the growth follows an exponential pattern.
• A function model can only be used to describe phenomena, not to predict or analyze them.

  False
• Order the types of function models based on their typical characteristics:

  1️⃣ Linear: Represents a constant rate of change

  2️⃣ Quadratic: Represents a changing rate of change

  3️⃣ Exponential: Represents a continuously changing rate of change

  4️⃣ Rational: Represents a relationship with a numerator and denominator
• What type of relationship does a linear function model represent?

  Constant rate of change