# Competing Function Model Validation

> AP Precalculus · Exponential and Logarithmic Functions
> Source: https://www.owlsprep.com/study/ap-precalculus-u2-competing-function-model-validation/

This module covers methods to compare competing linear, quadratic, and exponential function models for bivariate data using residual analysis and residual plots, including reading the sign of a residual to identify over- and underestimates, aligned to AP Precalculus CED requirements.

**Prerequisites:** Fitting linear, quadratic, and exponential models to bivariate data; Calculating residuals for a fitted model

## Learning objectives

- Compare competing linear, quadratic, and exponential function models for bivariate data
- Use residual analysis and residual plots to assess model goodness of fit
- Identify whether a model overestimates or underestimates using the sign of residuals
- Justify model selection per AP Precalculus exam requirements

## What is Competing Function Model Validation?

Competing function model validation is the process of testing two or more candidate function models (most commonly linear, quadratic, and exponential) against real-world bivariate data to select the model that best describes the underlying relationship. This topic accounts for 2-3% of the total AP Precalculus exam score, and appears in both multiple-choice and free-response sections.

On the exam, you will typically be given a scatterplot, data table, or pre-fit candidate models, then asked to justify which model is most appropriate using quantitative or graphical evidence. Unlike fitting a single model, validation focuses on comparing competing options, a critical skill for applied data analysis that is heavily weighted for justification points on FRQs.

## Graphical Residual Analysis

**Residual** — The difference between the observed response value $y_i$ and the model's predicted response value $\hat{y}_i$ for a given data point $(x_i, y_i)$: $e_i = y_i - \hat{y}_i$

*Notation:* $e_i$

*Example:* For model $\hat{y} = 2x + 1$ and point $(3, 8)$, $e_i = 8 - 7 = 1$

Residual analysis is the most intuitive and widely tested method for comparing model fit on the AP exam. If a model fits well, residuals will be randomly scattered around the horizontal axis $e=0$, with no clear systematic pattern (like a curve, trend, or funnel shape). If residuals show a clear pattern, the model is missing the underlying trend, so another competing model is a better choice.

**Worked example:** An ecologist measures the population of deer in a protected forest over 12 years. Residual plots for two candidate models: Model 1 (linear growth) has residuals that start positive, turn negative in the middle, then positive again, forming a clear upward-opening parabolic pattern. Model 2 (exponential growth) has residuals randomly scattered between $-18$ and $15$ deer, centered evenly around $e=0$. Which model is a better fit? Justify your choice.

1. Recall that a well-fitting model produces residuals with no systematic pattern, randomly distributed around the zero line.
2. Evaluate Model 1: The clear parabolic pattern means the linear model fails to capture the non-linear trend in deer population growth, so it is a poor fit.
3. Evaluate Model 2: Residuals have no systematic pattern and are evenly scattered around zero, which indicates the model correctly captures the underlying trend.
4. Conclude that Model 2 (exponential growth) is the better fitting model.

> **Exam tip:** On AP FRQ, you must explicitly reference the presence/absence of a pattern in residuals to earn the justification point; just saying "the residuals are better" will not get you full credit.

## AP-Style Concept Check

**Check your understanding**

Test your understanding with this multiple-choice question:

1. A researcher compares three models for the height of a growing tomato plant over time, all fit to the original height data (in cm), and examines each model's residual plot:
- Linear model: residuals form a clear upward-curved (U-shaped) pattern
- Exponential model: residuals are randomly scattered around 0 with no pattern
- Quadratic model: residuals form a clear downward-curved pattern
Based on the residual plots, which model is the most appropriate?

   - Linear model, because its residuals form a clear U-shaped pattern
   - Exponential model, because its residuals are randomly scattered around 0 with no pattern
   - Quadratic model, because its residuals form a downward curve
   - All three models are equally appropriate

   *Answer:* Exponential model, because its residuals are randomly scattered around 0 with no pattern

   *Why:* Correct. A model fits well when its residuals are randomly scattered around 0 with no systematic pattern. The linear and quadratic models both show clear curved patterns, so they fail to capture the trend; only the exponential model's residuals show no pattern.

## Common pitfalls

- **Wrong:** Claiming a model is bad just because one residual is much larger than the rest
  - Why it fails: A single outlier is not the same as a systematic pattern across all data points
  - Correct: Judge model fit based on the overall pattern of all residuals, not just one extreme outlier
- **Wrong:** Reading a positive residual as the model overestimating the observed value
  - Why it fails: A residual is observed minus predicted ($e_i = y_i - \hat{y}_i$), so a positive residual means the observed value is above the prediction
  - Correct: Match the sign to the direction: a positive residual means the model underestimates (prediction too low); a negative residual means it overestimates (prediction too high)
- **Wrong:** Judging model fit only by how small the residuals are, ignoring their pattern
  - Why it fails: Even small residuals can form a systematic curve or funnel, which signals the wrong type of model
  - Correct: Always look for a pattern in the residual plot; random scatter around $e=0$ — not just small size — is what indicates an appropriate model

## Cheatsheet

| Category | Formula / Rule | Notes |
| --- | --- | --- |
| Residual Calculation | $e_i = y_i - \hat{y}_i$ | $y_i$ = observed $y$, $\hat{y}_i$ = predicted $y$ from the model |
| Good Residual Pattern | No systematic pattern, random around $e=0$ | Indicates a well-fitting model |
| Bad Residual Pattern | Curve, linear trend, or funnel shape | Indicates a poorly fitting model |
| Over/Underestimate | $e_i>0$: underestimate; $e_i<0$: overestimate | The sign of the residual gives the direction of the error |
| Model Selection Rule | Choose the model whose residual plot shows no systematic pattern | Random scatter around $e=0$ = appropriate model |

## What's next

Competing function model validation is the capstone modeling skill for Unit 2: Exponential and Logarithmic Functions, and it prepares you for all future modeling-focused content on the AP Precalculus exam. The core skills of residual analysis, goodness-of-fit comparison, and model justification you learn here apply to every type of model you will encounter for the rest of the course and on the final exam. Immediately after mastering this topic, you will move into Unit 3, which covers polynomial and rational functions, where you will apply the same core model validation skills to compare polynomial models of different degrees to bivariate data. Without mastering these foundational skills here, justifying model selection for non-linear polynomial models will be much more difficult, as the same principles apply directly.

- [AP Precalculus Unit 2 Overview](https://www.owlsprep.com/study/ap-precalculus-u2-overview/)
- [Composition of Functions](https://www.owlsprep.com/study/ap-precalculus-u2-composition-of-functions/)
- [AP Precalculus Inverse Functions](https://www.owlsprep.com/study/ap-precalculus-u2-inverse-functions/)

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