1.14 Function Model Construction and Application

Choosing the right model

When constructing a function model from a real-world scenario, first identify the pattern of change. Linear models have constant rate of change, quadratic/polynomial models fit data with changing rates and multiple turning points, and rational models arise from ratios or quantities with asymptotic behavior. Match the context's behavior (symmetry, end behavior, extrema) to the appropriate function family.

Building the model from data or context

Use given information — points, rates, intercepts, or constraints — to determine parameters (coefficients, degree, zeros) of the function. Regression on a data set can suggest degree; contextual clues like symmetry or maximum/minimum values help verify the model. Always check that the model's domain and behavior make sense for the real-world situation, such as excluding negative time or ensuring outputs stay non-negative when required.

Applying and interpreting the model

Once built, use the function to answer questions: evaluate at specific inputs, solve for inputs given outputs, or analyze extrema, zeros, and asymptotes in context. Interpret results in terms of the original units and scenario, and consider whether the model reasonably extrapolates or if it should be restricted to the domain of the given data.

Key terms

Steps to construct a function model

Identify the pattern of change, choose an appropriate function family, use given data/constraints to find parameters, and verify domain/behavior fits the context.

When to use a quadratic model

When data shows one extremum (max or min) and symmetric behavior, such as projectile height or area with fixed perimeter.

When to use a rational model

When the scenario involves ratios, rates, or values approaching a limit (asymptotic behavior), such as average cost per unit.

Why restrict a model's domain

To ensure only realistic, meaningful inputs (e.g., non-negative time) are used, even if the function is mathematically defined elsewhere.

Interpreting model output

Translate numerical results back into the original context and units, checking that answers make sense for the real-world situation.

Try an AP-style question

A ball's height is h(t) = −16t² + 64t + 5 feet. What is its maximum height?

1. 64 feet
2. 69 feet
3. 133 feet
4. 5 feet