Function Model Selection | undefined Unit 1 Study Guide

The Core Idea: Function Model Selection and Assumption Articulation

In mathematics, we use functions to model real-world phenomena. The central task of this topic is to determine which type of function—linear, quadratic, or exponential—best represents a given situation or set of data. The choice of model is not arbitrary; it is dictated by the underlying patterns in the data, specifically how the output variable changes in relation to the input variable. A key indicator is the rate of change: is it constant, is it changing at a constant rate, or is it changing multiplicatively?

Furthermore, creating a mathematical model is an act of simplification. No model perfectly captures all the complexities of reality. Therefore, a critical part of the modeling process is to clearly articulate the assumptions made when choosing the model and to recognize its inherent limitations. This involves understanding what conditions must be true for the model to be valid and acknowledging the boundaries beyond which the model may no longer provide accurate predictions.

Key Model Characteristics

The selection of an appropriate function model is based on analyzing the rate of change in the provided data. Each function family has a unique rate of change characteristic.

Linear Models: Constant Rate of Change

A linear model, represented by the form $f (x) = m x + b$ , is appropriate when the rate of change between the output and input variables is constant.

Test: For a set of data where the input values ( $x$ ) increase by a constant interval, a linear model is suitable if the difference between consecutive output values ( $y$ ) is constant. This is known as having constant first differences.
Formula: $Δ y = y_{2} - y_{1} = constant$ for equally spaced $x$ values.

Exponential Models: Constant Multiplicative Change

An exponential model, represented by the form $f (x) = a \cdot b^{x}$ , is appropriate when the output variable changes by a constant multiplicative factor for each unit increase in the input variable.

Test: For a set of data where the input values ( $x$ ) increase by a constant interval, an exponential model is suitable if the ratio of consecutive output values ( $y$ ) is constant.
Formula: $\frac{y _{2}}{y _{1}} = constant$ for equally spaced $x$ values.

Quadratic Models: Linearly Changing Rate of Change

A quadratic model, represented by the form $f (x) = a x^{2} + b x + c$ , is appropriate when the rate of change itself is changing at a constant rate.

Test: For a set of data where the input values ( $x$ ) increase by a constant interval, a quadratic model is suitable if the second differences of the output values ( $y$ ) are constant. The second differences are the differences between consecutive first differences.
Formula: $Δ (Δ y) = (y_{3} - y_{2}) - (y_{2} - y_{1}) = constant$ for equally spaced $x$ values.

Understanding Model Assumptions

Every function model operates under a set of assumptions and is subject to limitations. Articulating these is a fundamental part of the modeling process.

Assumptions are conditions we accept as true for the model to be valid. For example, when modeling population growth with an exponential function, we might assume unlimited resources and no predators. When modeling a projectile's motion with a quadratic function, we typically assume that air resistance is negligible.
Limitations define the boundaries of a model's applicability. A model might be accurate only for a specific domain of input values. For example, a linear model for a company's profit might be valid for the first five years but may not accurately predict profits in the 20th year due to changing market conditions. Acknowledging that a model is an approximation of reality, not a perfect representation, is crucial.

Core Concepts & Rules

Model selection is data-driven: The choice between a linear, exponential, or quadratic model is determined by analyzing the patterns within the context and data, not by random guessing.
Rate of change is the key determinant: The specific type of function is identified by how the output variable changes with respect to the input variable.
Constant first differences imply a linear model: If the output values increase or decrease by the same constant amount for each uniform step in the input values, the relationship is linear.
Constant ratios imply an exponential model: If the output values are multiplied by the same constant factor for each uniform step in the in input values, the relationship is exponential.
Constant second differences imply a quadratic model: If the rate of change itself changes by a constant amount (i.e., the first differences form a linear pattern), the relationship is quadratic.
All models have assumptions and limitations: It is essential to identify and state the conditions under which a model is considered valid and to recognize the circumstances where it might fail.

Step-by-Step Example 1: Selecting a Model from a Data Table

Problem: A set of data is collected in a lab experiment. Determine whether a linear, quadratic, or exponential function would be the best model for the data. Justify your answer.

$x$ (time in seconds)	$f (x)$ (distance in meters)
0	5
1	6
2	11
3	20
4	33

Step 1: Analyze the First Differences

Calculate the difference between consecutive $f (x)$ values to check for a constant rate of change (linear model).

From $x = 0$ to $x = 1$ : $6 - 5 = 1$
From $x = 1$ to $x = 2$ : $11 - 6 = 5$
From $x = 2$ to $x = 3$ : $20 - 11 = 9$
From $x = 3$ to $x = 4$ : $33 - 20 = 13$

The first differences are $1, 5, 9, 13$ . Since they are not constant, the model is not linear.

Step 2: Analyze the Ratios

Calculate the ratio of consecutive $f (x)$ values to check for a constant multiplicative factor (exponential model).

From $x = 0$ to $x = 1$ : $\frac{6}{5} = 1.2$
From $x = 1$ to $x = 2$ : $\frac{11}{6} \approx 1.83$
From $x = 2$ to $x = 3$ : $\frac{20}{11} \approx 1.82$

The ratios are not constant, so the model is not exponential.

Step 3: Analyze the Second Differences

Calculate the difference between the consecutive first differences found in Step 1.

First Differences: $1, 5, 9, 13$
Second Difference 1: $5 - 1 = 4$
Second Difference 2: $9 - 5 = 4$
Second Difference 3: $13 - 9 = 4$

The second differences are constant ( $4$ ).

Step 4: Justify the Model Selection

Conclusion: A quadratic function is the most appropriate model for this data.
Justification: The model is quadratic because the second differences of the output values are constant for equally spaced input values.

Step 5: Articulate an Assumption

Assumption: We assume that the observed pattern of a constant second difference of 4 continues for all values of $x$ beyond the measured data points.

Step-by-Step Example 2: Modeling a Real-World Scenario

Problem: The value of a rare collectible is recorded over several years as shown in the table below.

Year ( $t$ )	Value ( $V (t)$ in dollars)
0	1200
1	1500
2	1875
3	2343.75

(a) Identify the most appropriate function type (linear, quadratic, or exponential) to model the value of the collectible. Provide a clear justification for your choice.

(b) Articulate one key assumption made when using this model to predict future value.

Part (a): Model Selection and Justification

Step 1: Check for a Linear Model (First Differences)

$1500 - 1200 = 300$
$1875 - 1500 = 375$
$2343.75 - 1875 = 468.75$

The first differences are not constant. The model is not linear.

Step 2: Check for an Exponential Model (Ratios)

$\frac{1500}{1200} = 1.25$
$\frac{1875}{1500} = 1.25$
$\frac{2343.75}{1875} = 1.25$

The ratios of consecutive values are constant ( $1.25$ ).

Step 3: Conclusion and Justification

Conclusion: An exponential function is the most appropriate model.
Justification: The data exhibits a constant multiplicative change. The ratio of the collectible's value in any year to its value in the preceding year is constant at $1.25$ . This indicates the value is growing by 25% each year.

Part (b): Assumption Articulation

Assumption: The model assumes that the collectible's value will continue to grow at a constant rate of 25% per year indefinitely.
Limitation (Implied): This model may become unrealistic over the very long term, as market conditions, collector demand, or the item's condition could change, affecting its rate of appreciation.

Using Your Calculator

While the primary method for model selection is the analytical process of checking differences and ratios, a graphing calculator can be used to confirm your findings through regression analysis.

Problem: Confirm the model choice for the data in Example 1 using a TI-84 style calculator.

Data:

$x$ = {0, 1, 2, 3, 4}

$f (x)$ = {5, 6, 11, 20, 33}

Step 1: Enter the Data into Lists

Press STAT, then select 1:Edit....
In list L1, enter the $x$ values: $0, 1, 2, 3, 4$ .
In list $L 2$ , enter the $f (x)$ values: $5, 6, 11, 20, 33$ .

Step 2: Perform Regression Analyses

Press STAT, move the cursor to the CALC menu.
Select 4:LinReg(ax+b) for a linear model. Press ENTER until it calculates. Note the $R^{2}$ value.
Press STAT, $C A L C$ , then select 5:QuadReg for a quadratic model. Press ENTER. Note the $R^{2}$ value.
Press STAT, CALC, then select 0:ExpReg for an exponential model. Press ENTER. Note the $R^{2}$ value.

Step 3: Interpret the Results

The calculator will provide an $R^{2}$ value (the coefficient of determination) for each regression. This value ranges from 0 to 1. The closer $R^{2}$ is to 1, the better the function model fits the data.

For this data, QuadReg will yield $R^{2} = 1$ .
LinReg and ExpReg will yield $R^{2} < 1$ .

The $R^{2}$ value of 1 for the quadratic regression confirms that the data perfectly fits a quadratic model, which aligns with our analytical conclusion that the second differences were constant.

AP Exam Quick Hit

Common Question Types

Model Justification from a Table: You will be given a table of values and asked to determine and justify which function type (linear, quadratic, or exponential) is the best fit.
- Example: "The table below shows the height $h (t)$ of a plant at time $t$ . Is a linear, quadratic, or exponential model most appropriate for this data? Justify your answer."
Model Selection from a Scenario: You will be given a verbal description of a real-world situation and asked to identify the correct model type and articulate an assumption.
- Example: "A town's population grows by approximately 500 people each year. What type of function best models the population? State one assumption required for your model to be accurate."
Interpreting Rates of Change: You will be asked to connect a description of a rate of change to a function family.
- Example: "The velocity of an object is increasing at a constant rate. Since velocity is the rate of change of position, what type of function models the object's position?" (Answer: Quadratic, because the rate of change is linear).

Common Mistakes

Confusing Additive vs. Multiplicative Change: Incorrectly identifying a situation with a constant percentage increase (e.g., "grows by 10%") as linear, when it is exponential.
Stopping at First Differences: Calculating the first differences, seeing they are not constant, and immediately concluding the model must be quadratic without checking the ratios for an exponential model first.
Calculation Errors: Simple arithmetic mistakes when calculating first differences, second differences, or ratios, leading to an incorrect conclusion.
Providing a Weak or No Justification: Stating that a model is "quadratic" without explaining why (i.e., "because the second differences are constant"). The justification is as important as the answer.
Stating an Irrelevant Assumption: When asked for an assumption, providing one that is not relevant to the mathematical model. For example, for a model of a company's revenue, stating "we assume the CEO does not change" is less relevant than "we assume the market demand for the product remains constant."

Function Model Selection and Assumption Articulation - AP PreCalculus Study Guide