Glossary
Assumptions
Conditions or premises taken as true when constructing a function model, which influence the type of model chosen.
Example:
When using a linear model for sales growth, an assumption might be that the sales increase at a constant rate each month.
Domain and Range
The domain is the set of all possible input values (x-values) for which a function is defined, while the range is the set of all possible output values (y-values).
Example:
For a function modeling the cost of producing items, the domain would be the number of items (non-negative integers), and the range would be the total cost (non-negative currency).
Function Models
Mathematical representations used to describe and predict real-world phenomena, built upon restrictions, transformations, and regressions.
Example:
A scientist might use a function model to predict the growth of a bacterial population over time.
Inversely Proportional
A relationship between two quantities where an increase in one quantity results in a proportional decrease in the other, often modeled by rational functions.
Example:
The volume of a gas is inversely proportional to its pressure at a constant temperature; as pressure increases, volume decreases.
Parent Functions
The simplest form of a family of functions, serving as a base from which more complex functions can be derived through transformations.
Example:
The parent function for all quadratic equations is , from which parabolas can be shifted, stretched, or reflected.
Piecewise-Defined Functions
Functions composed of multiple sub-functions, each defined over a specific interval of the domain.
Example:
A cell phone plan's cost, which charges one rate for the first 100 minutes and a different rate for minutes over 100, can be modeled by a piecewise-defined function.
Predictions
Estimates of future values or behaviors derived from a constructed function model.
Example:
Based on a model of population growth, we can make predictions about the city's population in the next decade.
Rates of Change
Measures of how one quantity changes in relation to another, which can be constant (linear) or changing (non-linear) within a function model.
Example:
The rate of change of a car's position over time is its speed, which can be constant on a highway or changing in city traffic.
Rational Functions
Functions expressed as the ratio of two polynomials, often used to model relationships where quantities are inversely proportional.
Example:
The time it takes to complete a task is often inversely proportional to the number of workers, which can be modeled using a rational function.
Regressions
Statistical methods used with technology to find the 'best-fit' function model for a given set of data points.
Example:
Using a calculator to perform a linear regression on a scatter plot of temperature versus ice cream sales helps predict future sales based on temperature.
Restrictions
Limitations on the domain and range of a function model, ensuring it makes sense within the real-world context it represents.
Example:
When modeling the height of a ball thrown upwards, the restrictions on the domain would mean time cannot be negative, and the range would mean height cannot be below ground.
Transformations
Operations (shifts, stretches, compressions, reflections) applied to a parent function to alter its graph and fit specific data or scenarios.
Example:
Applying a vertical stretch and a horizontal shift are examples of transformations that can make a basic sine wave model ocean tides more accurately.
Units
Standard measures (e.g., meters, seconds, dollars) that must be consistently applied and included in the interpretation of function models and their results.
Example:
If a model predicts the distance traveled, the answer must include appropriate units like 'kilometers' or 'miles' to be meaningful.