Competing Function Model | undefined Unit 2 Study Guide

The Core Idea: Competing Function Model Validation

In mathematics, we often use functions to model real-world phenomena. However, for a given set of data, several different types of functions—such as linear, quadratic, or exponential—might appear to be a reasonable fit. The core task of competing function model validation is to determine which function type provides the best model for the data. This process is not about finding a function that is merely "good," but about systematically analyzing the data and using quantitative measures to identify and justify the single best choice among competing options.

This validation process follows two main approaches. The first involves analyzing the structure of the data itself. By examining the rates of change between data points, we can identify characteristic patterns. A constant additive rate of change points to a linear model, while a constant multiplicative rate of change suggests an exponential model. If the rate of change of the rate of change (the second differences) is constant, a quadratic model is the most appropriate. The second approach is used when no model is a perfect fit. In this more common scenario, we use statistical measures like the sum of the squares of the residuals ( $S$ ) and the coefficient of determination ( $R^{2}$ ) to quantify how well each model's predictions match the actual data. The model that minimizes the error (a smaller $S$ ) and explains the most variance in the data (an $R^{2}$ closer to 1) is validated as the best fit.

Key Rules for Model Identification

This topic relies on a set of rules for identifying the ideal function model from a data set, as well as quantitative measures for comparing imperfect models.

Rules Based on Rates of Change

For a data set with consistently spaced input values:

Linear Model Test (Constant Additive Rate of Change): A data set is best modeled by a linear function if the differences between consecutive output values (the "first differences") are constant.
- If for a set of points $(x_{1}, y_{1}), (x_{2}, y_{2}), \dots, (x_{n}, y_{n})$ with equally spaced $x$ values, the value of $y_{i + 1} - y_{i}$ is constant for all $i ‘, t h e d a t ai s p er f ec tl y l in e a r .2. * * E x p o n e n t ia lM o d e lT es t (C o n s t an tM u lt i pl i c a t i v e R a t eo f C han g e) : * * A d a t a se t i s b es t m o d e l e d b y an e x p o n e n t ia l f u n c t i o ni f t h er a t i oso f co n sec u t i v eo u tp u t v a l u es a reco n s t an t . * I ff or t h es am ese t o f p o in t s, t h e v a l u eo f$ \frac{y_{i+1}}{y_i} $is constant for all `i`, the data is perfectly exponential. 3. **Quadratic Model Test (Constant Second Differences):** A data set is best modeled by a quadratic function if the differences of the first differences (the "second differences") are constant. * First, calculate the first differences: $d_i = y_{i+1} - y_i$ .
- Then, calculate the second differences: $d_{i + 1} - d_{i}$ . If this value is constant, the data is perfectly quadratic.

Measures of Fit for Comparing Models

When data is not perfectly linear, exponential, or quadratic, we use quantitative measures to determine the "best" fit.

Sum of the Squares of the Residuals (S): This value measures the total squared error between the data's actual output values and the output values predicted by the function model. A residual is the vertical distance between a data point and the model's curve, calculated as $(y_{a c t u a l} - y_{p re d i c t e d})$ .
- Formula: $S = \sum (y_{a c t u a l} - y_{p re d i c t e d})^{2}$
- Interpretation: The model with the smallest value of $S$ is considered the best fit because it has the least overall error.
Coefficient of Determination ( $R^{2}$ ): This value, ranging from 0 to 1, represents the proportion of the variance in the output variable that is predictable from the input variable.
- Range: $0 \leq R^{2} \leq 1$
- Interpretation: The model with the $R^{2}$ value closest to 1 is considered the best fit. An $R^{2}$ of 1 indicates that the model perfectly explains all the variability of the response data around its mean. An $R^{2}$ of 0 indicates that the model explains none of the variability.

Understanding Model Justification

The primary goal of this topic is not just to identify the best model, but to justify that choice using clear evidence from the provided data. The Essential Knowledge statements provide the two pillars of this justification.

The first pillar is pattern recognition within the data. This method is definitive when a perfect pattern exists. If you calculate the first differences of a data set and find they are all exactly 2.5, you can state with certainty that the data is best modeled by a linear function. Your justification is the clear statement: "The data is best modeled by a linear function because the first differences are constant." This type of analysis is powerful but is often only applicable to idealized, textbook-style data sets.

The second, more broadly applicable pillar is quantitative comparison of imperfect models. In most real-world scenarios, data contains "noise" and does not perfectly fit any simple function type. You might be given a data set and three different regression models (e.g., linear, quadratic, exponential) that have been generated by a calculator. Here, you cannot rely on calculating differences or ratios, as they will not be perfectly constant. Instead, your justification must be based on the provided measures of fit. The best model is the one whose function outputs are, on average, closest to the actual data outputs. $S$ and $R^{2}$ are the tools to measure this "closeness."

A justification using $S$ would sound like: "The quadratic model is the best fit because its sum of squared residuals ( $S = 0.15$ ) is smaller than the $S$ values for the linear ( $S = 2.4$ ) and exponential ( $S = 1.8$ ) models."
A justification using $R^{2}$ would sound like: "The exponential model is the best fit because its coefficient of determination ( $R^{2} = 0.998$ ) is closer to 1 than the $R^{2}$ values for the linear ( $R^{2} = 0.912$ ) and quadratic ( $R^{2} = 0.975$ ) models."

It is critical to understand that both $S$ and $R^{2}$ measure the same underlying concept—the goodness of fit—but from different perspectives. $S$ is an absolute measure of total error (in squared units of the output variable), while $R^{2}$ is a relative measure of explanatory power (a unitless proportion).

Core Concepts & Rules

Linear Model: A linear model is indicated if the rate of change is a constant addition for each uniform step in the input. This is verified by calculating the first differences ( $y_{n + 1} - y_{n}$ ) and finding them to be constant.
Exponential Model: An exponential model is indicated if the rate of change is a constant multiplier for each uniform step in the input. This is verified by calculating the ratios of consecutive outputs ( $y_{n + 1} / y_{n}$ ) and finding them to be constant.
Quadratic Model: A quadratic model is indicated if the first differences are not constant but change by a constant amount. This is verified by calculating the second differences (the differences of the differences) and finding them to be constant.
Principle of Best Fit: When comparing multiple function models for a single data set, the "best" model is the one whose predicted outputs are closest to the actual outputs of the data.
Validating with $S$ : The sum of the squares of the residuals, $S$ , quantifies the total error of a model. A smaller $S$ value signifies less error and therefore a better fit.
Validating with $R^{2}$ : The coefficient of determination, $R^{2}$ , quantifies the proportion of the output's variation that the model can explain. An $R^{2}$ value closer to 1 signifies a better fit.

Step-by-Step Example 1: Validating a Model Using Differences

Problem: A ball is rolled up a ramp. Its distance from the ground, $h$ , in centimeters, is measured at different times, $t$ , in seconds. The data is recorded in the table below. Determine and justify which function type (linear, quadratic, or exponential) best models this data.

$t$ (seconds)	0	1	2	3	4
$h (t)$ (cm)	0	15	24	27	24

Step 1: Test for a Linear Model (Calculate First Differences)

We calculate the difference in $h (t)$ for each 1-second interval.

From $t = 0$ to $t = 1$ : $15 - 0 = 15$
From $t = 1$ to $t=2`: `24 - 15 = 9` * From $t=2$ to $t = 3$ : $27 - 24 = 3$
From $t = 3$ to $t=4`: `24 - 27 = -3` The first differences are $15, 9, 3, -3$ . Since these values are not constant, the data is not best modeled by a linear function.

Step 2: Test for an Exponential Model (Calculate Ratios)

We calculate the ratio of consecutive $h (t)$ values. Note that we cannot start with the first point because division by zero is undefined. We will start from the second point.

From $t = 1$ to $t = 2$ : $\frac{24}{15} = 1.6$
From $t = 2$ to $t = 3‘ :$ \frac{27}{24} = 1.125$
From $t = 3$ to $t = 4$ : $\frac{24}{27} \approx 0.889$

The ratios are $1.6, 1.125, 0.889`. Since these values are not constant, the data is not best modeled by an exponential function. **Step 3: Test for a Quadratic Model (Calculate Second Differences)** Using the first differences we found in Step 1 ($15, 9, 3, -3$ ), we now calculate the differences between these values.

$9 - 15 = - 6$
$3 - 9 = - 6$
$- 3 - 3 = - 6$

The second differences are all $- 6$ .

Step 4: Justify the Conclusion

Since the second differences are constant ( $- 6$ ), the data is best modeled by a quadratic function.

Justification: The function type that best models this data is quadratic. This is because the second differences of the output values are constant for uniformly spaced input values.

Step-by-Step Example 2: Comparing Imperfect Models Using `R^2`

Problem: A biologist is studying the population of a certain bacteria in a petri dish. The population is recorded every hour for 5 hours. A statistics software package is used to generate three regression models for the data. The models and their corresponding coefficients of determination (R^2) are shown below. Identify and justify which function type is the best model for the data.

Linear Model: $P (t) = 180.4 t + 88.8$ , with $R^{2} = 0.923$
Quadratic Model: $P (t) = 18.1 t^{2} + 90.1 t + 121.2$ , with $R^{2} = 0.971$
Exponential Model: $P (t) = 135.2 (1.55)^{t}$ , with $R^{2} = 0.998$

Step 1: Understand the Goal

The goal is to find the best model among the three options. Since the models are imperfect (as shown by the fact that three different models were generated), we cannot rely on calculating differences or ratios. We must use the provided measure of fit, which is the coefficient of determination, R^2.

Step 2: Analyze the $R^{2}$ Values

The $R^{2}$ value measures the proportion of the variation in the output (population) that is explained by the input (time). A value closer to 1 indicates a better fit.

Linear $R^2`: $0.923$
Quadratic $R^{2}$ : $0.971$
Exponential $R^2`: $0.998$

Step 3: Compare the $R^{2}$ Values

We compare the three values to see which is closest to 1.

$0.998$ is closer to 1 than $0.971$ .
$0.971$ is closer to 1 than $0.923$ .

The highest value is $0.998$ , which corresponds to the exponential model.

Step 4: Formulate the Justification

The justification must state the chosen model and the reason for the choice, referencing the specific $R^{2}$ values.

Justification: The exponential model is the best fit for the data. The coefficient of determination, $R^{2}$ , measures how well a model explains the variation in the data, with a value closer to 1 indicating a better fit. The exponential model's $R^{2}$ value of $0.998$ is closer to 1 than the $R^{2}$ values for both the quadratic model ( $0.971$ ) and the linear model ( $0.923$ ).

Using Your Calculator

A graphing calculator (like a TI-84) is an essential tool for model validation, especially when comparing imperfect models. It can take a set of data and quickly generate regression models and their corresponding $S$ and $R^{2}$ values.

Problem: Given a data set, use a calculator to find the linear, quadratic, and exponential regression models and determine which is the best fit based on $R^{2}$ .

Data:

$x$	1	2	3	4	5
$y$	3	5	11	20	36

Step 1: Turn on Diagnostic Tools

Before you begin, you must ensure your calculator will display $R^{2}$ .

Press [2nd] $t h e n$ [0] $to access the `CATALOG`. 2. Scroll down to $DiagnosticOn$ and press [ENTER]`.
Press [ENTER] again. The calculator will display "Done."

Step 2: Enter the Data

Press [STAT] and select 1:Edit....
Enter the $x$ values into list L1.
Enter the $y$ values into list L2.

Step 3: Perform the Regressions and Compare $R^{2}$ Values

Linear Regression:
1. Press [STAT], arrow over to CALC.
2. Select 4:LinReg(ax+b).
3. Ensure Xlist:L1 and Ylist:L2$. Scroll down to $C a l c u l a t e$ and press [ENTER]`.
4. Write down the $R^{2}$ value. (For this data, $R^{2} \approx 0.9535$ ).
Quadratic Regression:
1. Press [STAT], arrow over to $C A L C$ .
2. Select 5:QuadReg.
3. Ensure $Xl i s t : L 1$ and $Y l i s t : L 2$ . Scroll down to $C a l c u l a t e$ and press [ENTER].
4. Write down the $R^{2}$ value. (For this data, $R^{2} \approx 0.9989$ ).
Exponential Regression:
1. Press [STAT], arrow over to CALC.
2. Select 0:ExpReg.
3. Ensure Xlist:L1 and Ylist:L2$. Scroll down to $C a l c u l a t e$ and press [ENTER]`.
4. Write down the $R^{2}$ value. (For this data, $R^{2} \approx 0.9718$ ).

Step 4: Conclude and Justify

Compare the R^2 values: $0.9535$ (Linear), $0.9989$ (Quadratic), $0.9718$ (Exponential). The $R^{2}$ value for the quadratic regression is closest to 1. Therefore, the quadratic model is the best fit for this data set.

AP Exam Quick Hit

Common Question Types

Table Analysis: You will be given a data table with equally spaced inputs and asked to determine if a linear, quadratic, or exponential model is most appropriate by calculating first differences, second differences, or ratios. You must show your calculations and provide a written justification.
- Example: "For the data provided in the table, justify whether a linear or quadratic function would be a better model."
Model Comparison: You will be presented with a scenario and a data set, along with two or three pre-calculated regression models (e.g., $f (x) = \dots$ , $g (x) = \dots$ ). You will also be given a measure of fit for each model, typically $R^{2}$ or $S$ . You must select the best model and justify your choice by correctly interpreting the given measure of fit.
- Example: "A linear model for the data has an $R^{2}$ of 0.89 and an exponential model has an $R^{2}$ of 0.98. Which model is a better fit for the data? Justify your answer."

Common Mistakes

Mixing up Difference Rules: Confusing the rule for linear models (constant first differences) with the rule for quadratic models (constant second differences).
Incorrect Ratio Calculation: When testing for an exponential model, calculating the ratio incorrectly (e.g., $\frac{y _{1}}{y _{2}}$ instead of $\frac{y _{2}}{y _{1}}$ ) or not being consistent with the order of division.
Misinterpreting $R^{2}$ : Believing that a lower $R^{2}$ value indicates a better fit, or simply stating that one $R^{2}$ is "higher" without explaining that "closer to 1 is better."
Misinterpreting $S$ : Believing that a larger sum of squared residuals ( $S$ ) indicates a better fit, when the opposite is true (a smaller $S$ means less error and a better fit).
Incomplete Justification: Stating that a quadratic model is best but failing to provide the evidence (e.g., "because the second differences are constant and equal to 4"). A complete justification must link the claim to the evidence.

Competing Function Model Validation - AP PreCalculus Study Guide