Regression Modeling — SL 2.7 Regression modelling (linear, quadratic, exponential) using technology (intro-to-standard) - IB Mathematics Applications and Interpretation

SECTION 1

Historical Context & Motivation

Humans have always looked for patterns in numbers. Farmers in ancient civilisations tracked seasonal rainfall and crop yields, hoping to predict the future from past observations. The mathematical leap — turning scattered data points into a smooth, predictive curve — took centuries of development. Regression modelling is the modern toolset that lets us do exactly that: find the equation that best fits a set of data so we can interpolate (estimate values within the range) or extrapolate (predict values outside the range).

1805 • Legendre & Least Squares

French mathematician Adrien-Marie Legendre published the method of least squares, the core algorithm still used in regression today. It minimises the sum of squared differences between observed and predicted values.

1886 • Galton Coins 'Regression'

Sir Francis Galton studied how the heights of parents and children relate, noticing that extreme values tended to move back toward the average — he called this regression toward the mean, giving the technique its name.

1960s • Computers Transform Statistics

With electronic computers, fitting curves to thousands of data points became practical. Quadratic, exponential, and many other models could be compared quickly.

2000s–Today • GDC & Software Regression

Graphing calculators (GDC) and apps like Desmos, GeoGebra, and spreadsheets let any student run a regression in seconds. The IB curriculum embraces this technology-driven approach.

The central question regression addresses is straightforward: Given a set of data points, what type of function — linear, quadratic, or exponential — best describes the underlying relationship, and how well does that function actually fit? In SL 2.7 you will learn to answer this question using technology rather than manual calculation.

SECTION 2

Core Principles & Definitions

Before you press any buttons on your calculator, it helps to understand the ideas that make regression work. Every regression analysis rests on a handful of foundational concepts that guide your choice of model and your interpretation of results.

Scatter Plot

A graph that plots every (x, y) data pair as a dot. The visual shape of the cloud of dots is your first clue about which model to try.

Regression Line / Curve

The function (line or curve) that the technology calculates to pass as close as possible to all data points, minimising the overall error.

Coefficient of Determination (r² or R²)

A value between 0 and 1 that tells you the proportion of variation in y explained by the model. An r² close to 1 means an excellent fit.

Correlation Coefficient (r)

For linear regression only, r ranges from −1 to 1. Positive means y increases with x; negative means y decreases with x; zero means no linear trend.

Residuals

The vertical distance between each data point and the regression curve. A good model keeps residuals small and randomly scattered rather than forming a pattern.

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KEY TAKEAWAY

Think of regression like choosing the right-shaped cookie cutter for a batch of dough. A straight edge (linear) works if your data is arranged in a strip. A curved edge (quadratic or exponential) works when the data bends. The r² value tells you how cleanly the cutter matches — the closer to 1, the less dough (error) is left over.

SECTION 3

Visual Explanation — Choosing the Right Model

The diagram below shows the same set of data fitted with three different regression models — linear, quadratic, and exponential. Notice how the shape of each curve changes and how closely each one follows the data points. This visual comparison is exactly what you do on a GDC when deciding which model to report.

The scatter plot shows eight data points (grey dots). The dashed blue line is the linear regression, the solid violet curve is the quadratic regression, and the dotted green curve is the exponential regression. All three fit reasonably well, but the R² values in the legend help you decide which is best.

When you look at a scatter plot, ask yourself three questions. First, does the overall trend go steadily upward or downward in a roughly straight band? If so, start with a linear model. Second, does the data curve like a hill or valley, rising then falling (or vice versa)? That suggests a quadratic model. Third, does the data grow slowly at first and then shoot upward (or decay rapidly toward zero)? That pattern points to an exponential model. After your visual guess, you confirm with technology by comparing R² values.

SECTION 4

Mathematical Framework

Each of the three regression types produces an equation in a specific form. You do not need to derive these equations by hand — your GDC or software handles the heavy algebra. However, you do need to recognise the general form, understand what each parameter means, and use the equation to make predictions.

LINEAR REGRESSION

y = ax + b

a is the gradient (slope) — the change in y for each unit increase in x. b is the y-intercept — the value of y when x = 0. The correlation coefficient r (and its square r²) measure fit quality.

QUADRATIC REGRESSION

y = ax² + bx + c

a controls whether the parabola opens upward (a > 0) or downward (a < 0). b shifts the vertex horizontally. c is the y-intercept. R² measures how well the parabola fits the data.

EXPONENTIAL REGRESSION

y = a × bˣ (or equivalently y = a · e^(kx))

a is the initial value (y when x = 0). b is the growth factor per unit of x. If b > 1 the function grows; if 0 < b < 1 it decays. R² again measures goodness of fit.

COEFFICIENT OF DETERMINATION

r² = 1 − (SS_res / SS_tot)

SS_res is the sum of squared residuals (errors). SS_tot is the total variation. An r² of 0.95 means 95 % of the variation in y is explained by your model. You will read this value from your GDC output.

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IB Exam Tip

On the IB exam you are expected to use a GDC (e.g., TI-84, TI-Nspire, or Casio fx-CG50) to find regression equations and r² values. You must state which model you chose and justify it using the R² value. Always round coefficients to three significant figures unless told otherwise.

SECTION 5

Detailed Breakdown — Choosing & Interpreting Models

Selecting the right model is a two-stage process. First you make a visual judgement from the scatter plot, then you confirm quantitatively by comparing R² values. The diagram below walks through the decision-making flowchart that experienced IB students use.

Start at the top: plot the data, judge the shape, run candidate regressions on your calculator, then pick the model with the highest R² value.

After you have selected a model, you should also consider whether the equation makes sense in context. For example, an exponential growth model that predicts a population of ten billion people in a small town within two years is probably being extrapolated beyond its useful range. Always check whether your predictions are reasonable given the real-world situation.

Feature	Linear	Quadratic	Exponential
General form	y = ax + b	y = ax² + bx + c	y = a × bˣ
Number of parameters	2 (a, b)	3 (a, b, c)	2 (a, b)
Shape of graph	Straight line	Parabola (U or ∩)	J-curve or decay curve
Typical real-world use	Constant rate of change, e.g. distance vs. time at constant speed	Projectile height, profit vs. price	Population growth, radioactive decay, compound interest
Fit statistic given by GDC	r and r²	R² only	R² (or r for log-transformed data)

Comparison of the three regression types in SL 2.7

SECTION 6

Worked Example — Modelling Plant Growth

A biology student measures the height (cm) of a sunflower seedling every week for six weeks. The data are shown in the table below. We will determine the best regression model, write the equation, and use it to predict the height at week 8.

Week (x)	Height in cm (y)
1	3.2
2	5.1
3	8.7
4	14.0
5	22.8
6	36.5

Sunflower seedling height data

Finding the Best Regression Model

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Step 1 — Enter Data into GDC

Enter the week numbers into List 1 (L₁) and the heights into List 2 (L₂) on your calculator. On a TI-84, press STAT → Edit and type each value.
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Step 2 — Create a Scatter Plot

Turn on a STAT PLOT with L₁ on the x-axis and L₂ on the y-axis. Press ZOOM → 9:ZoomStat to see the points. The data rises slowly at first then accelerates — this suggests exponential growth rather than a straight line.
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Step 3 — Run Three Regressions

Under STAT → CALC select LinReg, QuadReg, and ExpReg in turn. Record each equation and R² value.

Linear: y = 6.31x − 5.20, r² = 0.949. Quadratic: y = 1.14x² − 1.86x + 3.49, R² = 0.994. Exponential: y = 1.96 × 1.62ˣ, R² = 0.999.
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Step 4 — Compare R² Values and Select

The exponential model has the highest R² value of 0.999, confirming our visual impression. We therefore choose the exponential model: y = 1.96 × 1.62ˣ.
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Step 5 — Predict Height at Week 8

Substitute x = 8 into the equation: y = 1.96 × 1.62⁸. Calculate 1.62⁸ ≈ 48.2, so y ≈ 1.96 × 48.2 ≈ 94.5 cm.

Predicted height at week 8 ≈ 94.5 cm. Note: this extrapolation assumes the exponential trend continues, which may not hold if the plant's growth slows down.

SECTION 7

Strengths, Limitations & Comparisons

Every regression model is a simplification of reality. Understanding when each model works well — and when it breaks down — helps you make responsible predictions and earn full marks on the IB exam.

Model	Strengths	Limitations
Linear	Simple to interpret; gradient gives constant rate of change; reliable for interpolation within the data range.	Cannot capture curvature; may over-simplify trends that accelerate or decelerate.
Quadratic	Captures a single turning point (max or min); good for projectile motion and optimisation scenarios.	Extrapolation is unreliable — parabolas eventually go to ±∞. Not suitable if data has no turning point.
Exponential	Excellent for growth/decay processes; naturally models percentage change per unit of x.	Grows without bound — unrealistic over long time frames. Cannot model data that levels off (use logistic instead).

Strengths and limitations of each regression model

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KEY TAKEAWAY

A high R² value does not automatically mean a model is appropriate. Imagine fitting a quadratic to temperature data over 24 hours — the R² might be great, but the model would predict the temperature shooting to infinity the next day. Always check that the model's behaviour outside the data range makes physical or contextual sense before using it to extrapolate.

SECTION 8

Connection to Advanced Topics

The regression skills you develop in SL 2.7 form a springboard to more sophisticated modelling in IB Higher Level, university statistics, and data science. Here is a glimpse of how the ideas extend.

SL 2.7 Concept	Advanced Extension
Linear regression (y = ax + b)	Multiple linear regression with several independent variables: y = a₁x₁ + a₂x₂ + … + b
R² as a measure of fit	Adjusted R², AIC, BIC — metrics that penalise over-fitting when many parameters are used
Choosing between three model types	Logistic, sinusoidal, power, and piecewise regression — plus machine-learning algorithms like random forests and neural networks
Visual residual checks	Formal residual analysis: normality tests, heteroscedasticity detection, and diagnostic plots

How SL 2.7 topics evolve at higher levels

Even if you never take another statistics course, the core idea — fitting a mathematical model to data and measuring how well it fits — is a skill you will use in economics, biology, engineering, and everyday decision-making. The SL 2.7 toolkit gives you a solid, technology-supported foundation for all of these applications.

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

A student runs a linear regression and finds r = −0.92. Explain what this value tells you about the relationship between x and y, and comment on whether the linear model is a good fit.

Show answer

The value r = −0.92 indicates a strong negative linear correlation: as x increases, y tends to decrease. Since |r| is close to 1, the linear model fits the data well. The corresponding r² = (−0.92)² = 0.8464, meaning about 84.6 % of the variation in y is explained by the linear model. This is a good, though not perfect, fit — some scatter around the line remains.

Question 2: BASIC CALCULATION

A GDC gives the exponential regression equation y = 5.00 × 1.20ˣ with R² = 0.987. Calculate the predicted value of y when x = 10, and state the percentage of variation explained by the model.

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Substitute x = 10: y = 5.00 × 1.20¹⁰. First compute 1.20¹⁰ ≈ 6.192. Then y ≈ 5.00 × 6.192 = 30.96. So the predicted value is approximately 31.0 (3 s.f.). The R² of 0.987 means 98.7 % of the variation in y is explained by the exponential model, indicating an excellent fit.

Question 3: INTERMEDIATE

Temperature data collected from 8 AM to 3 PM is modelled by a quadratic regression y = −2.5x² + 25x + 10, where x is hours after 8 AM and y is temperature in °C. (a) Find the predicted maximum temperature and the time at which it occurs. (b) Comment on whether this model should be used to predict the temperature at 10 PM.

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(a) The vertex of y = −2.5x² + 25x + 10 occurs at x = −b/(2a) = −25/(2 × (−2.5)) = 5. So the maximum is predicted 5 hours after 8 AM, i.e. at 1 PM. The maximum temperature is y = −2.5(25) + 25(5) + 10 = −62.5 + 125 + 10 = 72.5 °C. This is physically unreasonable, suggesting the coefficients might need checking or the data applies on a different scale; however, mathematically the vertex is at x = 5. (b) At 10 PM, x = 14. The model gives y = −2.5(196) + 25(14) + 10 = −490 + 350 + 10 = −130 °C, which is absurd. This illustrates the danger of extrapolating a quadratic model far beyond the data range.

Question 4: APPLIED

A small business tracks monthly revenue (in thousands of dollars) over its first 5 months: (1, 2.4), (2, 3.8), (3, 6.0), (4, 9.5), (5, 15.1). Using your GDC, find the linear and exponential regression equations and their R² values. Which model would you recommend, and what revenue does it predict for month 7? Comment on the reliability of this prediction.

Show answer

Running LinReg on these data yields approximately y = 3.07x − 1.48 with r² ≈ 0.966. Running ExpReg yields approximately y = 1.50 × 1.59ˣ with R² ≈ 0.999. The exponential model has a higher R², so it is recommended. For month 7: y = 1.50 × 1.59⁷ ≈ 1.50 × 25.1 ≈ 37.7 thousand dollars. This prediction extrapolates two months beyond the data, which is moderately risky. In reality, market saturation or competition may slow growth, so the prediction should be treated with caution and labelled as an estimate.

Question 5: CRITICAL THINKING

A student claims: 'I found R² = 0.998 for my quadratic regression, so it is definitely the correct model.' Identify at least two flaws in this reasoning, and describe what additional checks the student should perform before concluding that a quadratic is the best model.

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Flaw 1: A high R² does not prove that the model type is correct — it only measures how well that particular curve fits the data. An exponential model might yield an even higher R², or a linear model might yield R² = 0.997, in which case the simpler model should be preferred (principle of parsimony). Flaw 2: With only a few data points, any model with enough parameters can achieve a high R² due to over-fitting. The student should compare R² across all three candidate models (linear, quadratic, exponential), plot the residuals to check for systematic patterns, and consider whether a quadratic makes contextual sense — for example, does the real-world situation actually have a maximum or minimum? Without these checks, a high R² alone is insufficient justification.

SUMMARY

Lesson Summary

Summary

Regression modelling is the process of fitting a mathematical function to a set of data points using technology. In SL 2.7 you work with three model types: linear (y = ax + b) for constant-rate relationships, quadratic (y = ax² + bx + c) for data with a turning point, and exponential (y = a × bˣ) for growth or decay patterns. You select the best model by first inspecting the scatter plot shape, then comparing R² values from your GDC — the model with the highest R² generally fits best.

Always remember that a high R² does not guarantee the model is appropriate for extrapolation. Check that your predictions make sense in context, and state your regression equation with coefficients rounded to three significant figures. The correlation coefficient r (for linear models) tells you the direction and strength of the linear association. Mastering these skills prepares you for real-world data analysis and for the IB examination, where you are expected to justify your model choice with technology-generated evidence.