Function Model Construction and Application - AP PreCalculus Study Guide

The Core Idea: Function Model Construction and Application

Function model construction is the process of translating a real-world scenario into the language of mathematics. The fundamental goal is to create a function that accurately represents the relationship between two quantities. This process allows us to analyze, predict, and understand the behavior of these quantities using the tools of algebra and calculus.

The information used to build these models can come from various sources. A model can be constructed from a verbal description, where a relationship is explained in words (e.g., cost as a function of items produced). It can also be derived from a geometric context, where formulas for area, volume, or distance are used to relate dimensions. Finally, a function model can be determined from a set of collected data points, where we seek the best mathematical curve to fit the observed information. The type of function chosen—be it linear, quadratic, exponential, or another type—must be appropriate for both the context of the problem and the visual pattern of the data.

Key Modeling Approaches

This topic does not introduce new, specific formulas but rather focuses on the methods used to construct functions from given information. The primary approaches are based on the source of the information.

1. Modeling from a Verbal Description

This involves translating a written description of a relationship into a mathematical equation.

• Process:

  1. Identify the independent and dependent quantities in the description.

  2. Assign variables to represent these quantities (e.g., $t$ for time, $C$ for cost).

  3. Translate the words describing the relationship (e.g., "initial fee," "per item," "doubles every hour") into mathematical operations ($+$, $\ast$, ^).

  4. Combine these elements to write an equation that expresses the dependent variable as a function of the independent variable.

2. Modeling from a Geometric Description

This involves using established geometric formulas to create a function based on shapes and their properties.

• Process:

  1. Identify the geometric shapes involved and the quantities of interest (e.g., area, volume, length).

  2. Write down the standard geometric formulas relevant to the problem (e.g., Area of a rectangle $A=l\cdot w$, Volume of a cylinder $V=\pi r^{2}h$).

  3. Use a given constraint (e.g., a fixed perimeter, a total amount of material) to write an equation that relates the variables in the geometric formula.

  4. Solve the constraint equation for one of the variables.

  5. Substitute this expression into the primary geometric formula to create a function of a single variable.

3. Modeling from a Data Set

This involves finding a function that best fits a collection of data points.

• Process:

  1. Examine the data and the context of the problem.

  2. Create a scatterplot of the data to visualize the relationship between the variables.

  3. Analyze the shape of the scatterplot and consider the problem's context to determine the most appropriate type of function model. The possible models include:

     • Linear: Data points form a straight line.

     • Quadratic/Cubic/Quartic: Data points form a parabola or a curve with one or more turns.

     • Exponential: Data points show growth or decay that increases or decreases at an ever-faster rate.

     • Logarithmic: Data points show growth that slows down over time.

     • Sinusoidal: Data points show a repeating, wave-like pattern.

     • Logistic: Data points show initial exponential-like growth that levels off at a carrying capacity.

  4. Use a graphing calculator or statistical software to perform a regression analysis and find the specific equation for the chosen model type.

Understanding Model Selection

The most critical nuance in this topic, particularly when working with data, is the selection of the most appropriate function model. This decision is not arbitrary; it is guided by two key factors as specified in the Essential Knowledge.

First, the context of the problem provides crucial clues. For example, if a problem describes a population with unlimited resources, an exponential model is a logical starting point. If it describes the height of a thrown object under gravity, a quadratic model is physically appropriate. If it describes a population in a limited environment, a logistic model is more suitable than an exponential one.

Second, the scatterplot of the data provides visual evidence for the relationship's nature. A linear model is only appropriate if the points lie roughly on a straight line. If the points form a U-shape, a quadratic model is a better choice. If they rise slowly at first and then more and more steeply, an exponential model should be considered. The visual pattern must align with the chosen function type. Often, you must weigh both the context and the visual evidence to make the best choice. An AP Precalculus question may provide a scatterplot and ask you to justify why a certain model (e.g., linear) is or is not appropriate.

Core Concepts & Rules

• A function model can be constructed by translating a verbal description of a relationship between two quantities into a mathematical equation.

• A function model can be constructed by using geometric formulas and given constraints to relate two quantities.

• A function model can be determined by analyzing a set of data that relates two quantities.

• The selection of the best function model for a data set depends on both the real-world context of the problem and the visual shape of the data's scatterplot.

• The types of function models that can be used include linear, quadratic, cubic, quartic, exponential, logarithmic, sinusoidal, and logistic functions.

Step-by-Step Example 1: Modeling from a Geometric Description

Problem: A farmer has 2400 feet of fencing to enclose a rectangular field that borders a straight river. No fencing is needed along the river. Construct a function $A(w)$ that represents the area of the field as a function of its width, $w$.

Step 1: Define variables and identify formulas.

Let $w$ be the width of the field (the side perpendicular to the river) and $l$ be the length of the field (the side parallel to the river).

The area of a rectangle is given by the formula $A=l\cdot w$.

The amount of fencing represents the perimeter of the three sides that need fencing. The formula for the fencing used is $P=l+2w$.

Step 2: Use the given constraint.

The problem states that the farmer has 2400 feet of fencing. This is our constraint.

$$2400=l+2w$$

Step 3: Express one variable in terms of the other.

Our goal is to create a function for Area in terms of width, $A(w)$. The current area formula, $A=l\cdot w$, has two variables, $l$ and $w$. We must use the constraint equation to eliminate $l$.

Solve the constraint equation for $l$:

$$l=2400-2w$$

Step 4: Substitute to create the final function model.

Now, substitute the expression for $l$ into the area formula:

$$A=(2400-2w)\cdot w$$

$$A(w)=2400w-2w^2$$

This is the final function model. It is a quadratic function that gives the area of the field for any given width $w$.

Step-by-Step Example 2: Modeling from a Data Set

Problem: The temperature of a cup of coffee cooling in a room is recorded at various times. The data is shown in the table below, where $t$ is the time in minutes and $T$ is the temperature in degrees Celsius.

A scatterplot of the data is provided. Based on the context and the scatterplot, determine the most appropriate type of function model and use a calculator to find the function.