Articulating Assumptions (College Board AP® Precalculus): Revision Note

Every mathematical model is built on assumptions

• things that are taken to be true in order for the model to work

Understanding these assumptions is important because it helps you recognize

• when a model is appropriate

• and when it might break down

A model may assume that certain conditions remain consistent throughout the scenario

E.g. a linear model for distance traveled assumes the speed remains constant

• If the speed changes, the model is no longer appropriate

Or a model for population growth might assume a constant growth rate

• In reality, growth rates can change due to environmental factors

A model may assume a specific relationship between how the input and output quantities change

E.g. a linear model assumes that the rate of change of the output quantity is constant

• While a quadratic model assumes that the rate of change itself changes at a constant rate

If the real-world relationship between the quantities changes (e.g. a trend reverses)

• then the model may no longer be valid beyond a certain point

A mathematical function may be defined for all real numbers

• but the context may only make sense for certain input values

The need for domain restrictions can come from

• Contextual clues

  • e.g. time cannot be negative in many scenarios

    • so only time values can be considered

  • or a quantity like total sales can never decrease

    • so domain values must be restricted to values where the model function is not decreasing

• Mathematical clues

  • e.g. a quadratic model will eventually reach a maximum or minimum and then reverse direction

    • which may not make sense in context

• Range of values in the data set

  • The model may only be reliable within the range of the data used to build it

  • Predictions far outside this range (extrapolation) may be unreliable

Ask 'what values of the input make sense in this scenario?'

• Can the input be negative?

  • E.g. time since an event started is usually represented )

  • A geometrical measurement (length, height, etc.) can never be negative

• Is there a natural upper limit?

  • E.g. the model may stop being valid after a certain point)

Ask 'does the mathematical behavior of the model conflict with the context at some point?'

• E.g. if a quadratic model has a maximum, and the quantity being modeled should only increase (like a cumulative total)

  • then the domain must be restricted to the interval before the maximum

• Or if a model predicts negative values for a quantity that cannot be negative (like a population)

  • then the domain must exclude the region where the model gives negative outputs

The output values of a model may need to be restricted based on context

• E.g. if the model predicts the number of items sold, the output should be a whole number

  • the model's output may need to be rounded to the nearest integer

• Or a model may predict values that are physically impossible (like negative distances)

  • those output values should be excluded