Linear Functions & Models — SL 2.2 Linear functions and models (rate of change; intercept) - IB Mathematics Applications and Interpretation

SECTION 1

Historical Context & Motivation

Humans have been thinking about straight-line relationships for thousands of years. Ancient surveyors in Egypt noticed that the Nile's flood levels rose at roughly the same rate each day during flood season, making it possible to predict water heights and plan crop planting. This core idea — that a quantity changes at a constant rate — lies at the heart of what we now call a linear function. Over the centuries, mathematicians refined this idea into one of the most powerful and widely used tools in all of mathematics.

~300 BCE • Euclid's Elements

The Greek mathematician Euclid formally defined lines and their properties in his landmark work, laying the geometric foundation for the study of linear relationships.

1637 • Descartes & the Coordinate Plane

René Descartes introduced the Cartesian coordinate system, allowing lines to be described with algebraic equations rather than purely geometric constructions.

1795 • Gauss & Least Squares

Carl Friedrich Gauss developed the method of least squares to fit the best straight line through astronomical data, pioneering the idea of linear modelling.

20th Century • Linear Models Everywhere

Linear functions became indispensable in economics, physics, engineering, and data science — any field where understanding constant rates of change is critical.

The question that drives this topic is deceptively simple: If something changes at a steady pace, how can we describe it, graph it, and use it to make predictions? Answering that question with precision is exactly what SL 2.2 is about.

SECTION 2

Core Principles & Definitions

A linear function connects two variables with a relationship that produces a perfectly straight line when graphed on the Cartesian plane. Before we dive into calculations, let's lock down the four foundational ideas you'll use throughout the course.

Linear Function

A function of the form f(x) = mx + c, where m and c are constants. Its graph is always a straight line.

Gradient (Slope)

The value m measures the rate of change — how much y increases (or decreases) for every 1-unit increase in x.

y-Intercept

The value c is the output when x = 0. It tells you where the line crosses the y-axis, giving you the starting value of the model.

Rate of Change

Calculated as Δy ÷ Δx (change in y divided by change in x). For a linear function this rate is constant — the same between any two points on the line.

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KEY TAKEAWAY

Think of a linear function like a car travelling at a constant speed on a straight highway. The gradient (m) is the speed — it tells you how quickly the distance piles up per hour. The y-intercept (c) is where the car already was when you started timing. Together, speed and starting position let you predict exactly where the car will be at any future moment.

SECTION 3

Visualising Linear Functions

The graph below shows the line y = 2x + 3 plotted on a coordinate plane. Pay close attention to how the gradient triangle (rise over run) and the y-intercept are marked — these two features define every linear function you'll ever meet.

The pink dot marks the y-intercept at (0, 3). The dashed yellow triangle shows the gradient: for every 1 unit you move right (run), the line rises 2 units (rise), giving m = 2.

Every linear function graph shares these features. The line extends infinitely in both directions, the gradient triangle can be drawn between any two points and will always yield the same ratio, and the y-intercept sits exactly where x = 0. When you see a graph in an exam, these are the first two features to identify.

SECTION 4

Mathematical Framework

The IB formula booklet provides several forms for linear functions. Understanding each form — what it tells you and when to use it — is essential for the exam and for real-world modelling.

SLOPE-INTERCEPT FORM

y = mx + c

m = gradient (rate of change), c = y-intercept (value of y when x = 0). This is the form used most often in IB Applications and Interpretation.

GRADIENT FORMULA

m = (y₂ − y₁) / (x₂ − x₁)

Given any two points (x₁, y₁) and (x₂, y₂) on the line, this formula calculates the gradient. Remember: it doesn't matter which point you call 'point 1' as long as you stay consistent.

POINT-SLOPE FORM

y − y₁ = m(x − x₁)

Useful when you know the gradient m and one point (x₁, y₁) on the line but not the y-intercept. You can always rearrange this into slope-intercept form.

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IB Exam Tip

The IB formula booklet uses c for the y-intercept (not b as you may have seen in other courses). Always match the notation your exam uses — it avoids confusion and keeps the examiner happy.

SECTION 5

Understanding Different Gradients

The sign and size of the gradient dramatically change what the linear model means in context. A positive gradient signals growth or increase, a negative gradient signals decline, and a zero gradient means no change at all. The diagram below compares four key cases side by side.

Four panels comparing gradient types: positive (upward slope), negative (downward slope), zero (horizontal), and steep vs. shallow (comparing magnitudes).

Gradient (m)	Direction of Line	Real-World Example
m > 0 (positive)	Rises from left to right	Earning $15/hour: total pay rises with hours worked
m < 0 (negative)	Falls from left to right	A phone battery losing 10 % per hour
m = 0	Horizontal line	A flat monthly subscription fee regardless of usage
Large \|m\|	Steep line	Rapid temperature drop during a cold front
Small \|m\|	Shallow line	Slow, steady erosion of a hillside over years

How gradient value translates to graph behaviour and real-world meaning

SECTION 6

Worked Example — Water Tank Model

A water tank is being filled at a constant rate. At time t = 0 minutes the tank already holds 20 litres of water. After 5 minutes, it holds 45 litres. Find the linear model that describes the volume V (in litres) as a function of time t (in minutes), and predict the volume after 12 minutes.

Finding the Linear Model for a Water Tank

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Step 1 — Identify Given Values

We have two data points: (t₁, V₁) = (0, 20) and (t₂, V₂) = (5, 45). Because t₁ = 0, we already know the y-intercept: c = 20.

c = 20
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Step 2 — Calculate the Gradient

Use the gradient formula: m = (V₂ − V₁) / (t₂ − t₁) = (45 − 20) / (5 − 0) = 25 / 5.

m = 5 litres per minute
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Step 3 — Write the Linear Model

Substitute m and c into y = mx + c. Using context-appropriate variable names:

V = 5t + 20
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Step 4 — Interpret in Context

The gradient m = 5 means the tank fills at a rate of 5 litres per minute. The y-intercept c = 20 means there were already 20 litres in the tank when timing began.
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Step 5 — Predict Volume at t = 12

Substitute t = 12 into the model: V = 5(12) + 20 = 60 + 20.

V = 80 litres

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Interpretation Matters

In IB exams, you often earn marks for interpreting the gradient and intercept in context, not just calculating them. Always write a sentence explaining what m and c mean in the real-world scenario.

SECTION 7

Strengths & Limitations of Linear Models

Linear models are beautifully simple, but simplicity is a double-edged sword. It's important to recognise when a straight line is a good fit for the data and when a more complex model (quadratic, exponential, etc.) would be more appropriate.

Strengths	Limitations
Easy to create — only two data points are needed to define the line	Assumes a constant rate of change, which rarely holds over large ranges
Simple to interpret — the gradient and intercept have clear, intuitive meanings	Cannot model curves, acceleration, or saturation effects
Great for interpolation (predicting within the data range)	Extrapolation can be unreliable if the trend changes outside the observed range
Foundational — many advanced models (regression, calculus) build on linearity	Real-world constraints (e.g., capacity limits) may make the model unrealistic

When to trust — and when to question — a linear model

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KEY TAKEAWAY

A linear model is like a weather forecast for the next hour: highly reliable in the short term but increasingly uncertain the further ahead you project. Always ask yourself, 'Is the rate of change likely to stay constant over this range?' If the answer is no, consider a different model.

SECTION 8

Connection to Advanced Models

Linear functions are the first step on a ladder of increasingly powerful models you'll encounter in IB Maths. Understanding how linear relates to what comes next helps you see the bigger picture and prepares you for SL 2.5 (modelling) and later HL topics.

Feature	Linear Model (SL 2.2)	Beyond Linear (SL 2.5+)
Equation form	y = mx + c	y = ax² + bx + c (quadratic), y = abˣ (exponential), etc.
Rate of change	Constant — same everywhere	Variable — changes depending on x
Graph shape	Straight line	Parabola, curve, exponential growth/decay
Typical use	Constant-speed motion, fixed-rate billing	Projectile motion, population growth, radioactive decay
Number of parameters	2 (m and c)	3 or more, allowing more flexible curve shapes

Linear vs. non-linear modelling at a glance

In calculus, you'll learn that the gradient of any smooth curve at a single point is found by zooming in until the curve looks like a straight line — a tangent line. So even when the world isn't linear, linear thinking is still the tool you use to analyse it locally. Mastering SL 2.2 now gives you a head start on those more advanced ideas.

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

A friend says, 'The y-intercept of a linear function is the point where the line is steepest.' Is this correct? Explain your reasoning.

Show answer

This is incorrect. A linear function has a constant gradient, meaning the 'steepness' is exactly the same at every point on the line — it does not change from one location to another. The y-intercept is simply the point where the line crosses the y-axis (where x = 0). It tells you the starting value, not anything about how steep the line is. The steepness is determined by the gradient m, which is independent of where the line crosses the axis.

Question 2: BASIC CALCULATION

Find the equation of the line that passes through the points (2, 7) and (6, 19). Write your answer in the form y = mx + c.

Show answer

First, calculate the gradient: m = (19 − 7) / (6 − 2) = 12 / 4 = 3. Next, substitute m = 3 and one point, say (2, 7), into y = mx + c: 7 = 3(2) + c, so 7 = 6 + c, giving c = 1. The equation is y = 3x + 1. You can verify with the second point: 3(6) + 1 = 19 ✓.

Question 3: INTERMEDIATE

A taxi company charges a flat fee of $4.50 plus $2.80 per kilometre. Write a linear model for the total cost C in terms of distance d (km). How far can you travel for $25?

Show answer

The linear model is C = 2.80d + 4.50, where the gradient 2.80 represents the cost per kilometre and the y-intercept 4.50 is the flat fee. To find the distance for $25, set C = 25: 25 = 2.80d + 4.50 → 20.50 = 2.80d → d = 20.50 / 2.80 ≈ 7.32 km. In context, you could travel approximately 7.3 km for $25. Note that this assumes the linear rate holds for all distances and ignores rounding to the nearest tenth of a kilometre.

Question 4: APPLIED

A scientist records the temperature of a cooling liquid. At t = 3 minutes the temperature is 68 °C, and at t = 10 minutes it is 40 °C. Assuming a linear model is appropriate over this interval, find the rate of cooling and estimate the temperature at t = 15 minutes. Comment on whether the linear model is likely to remain valid.

Show answer

The gradient (rate of cooling) is m = (40 − 68) / (10 − 3) = −28 / 7 = −4 °C per minute. Using point (3, 68): 68 = −4(3) + c → c = 80. The model is T = −4t + 80. At t = 15: T = −4(15) + 80 = −60 + 80 = 20 °C. However, in reality, the rate of cooling typically slows as the liquid approaches room temperature (Newton's Law of Cooling is exponential, not linear). Therefore, the linear model may underestimate the temperature at t = 15 because the actual cooling rate would be slower than −4 °C/min by that point.

Question 5: CRITICAL THINKING

Two linear functions are defined: f(x) = 3x − 2 and g(x) = −0.5x + 7. Find the point where they intersect. Then explain what the intersection point means if f models revenue (in thousands of dollars) and g models cost (in thousands of dollars) as functions of units sold x (in hundreds).

Show answer

Set f(x) = g(x): 3x − 2 = −0.5x + 7 → 3.5x = 9 → x = 9 / 3.5 ≈ 2.571. Then f(2.571) = 3(2.571) − 2 ≈ 5.71. The intersection is approximately (2.57, 5.71). In the business context, this is the break-even point: when about 257 units are sold, both revenue and cost equal roughly $5,710. For fewer than 257 units, cost exceeds revenue (a loss), and for more than 257 units, revenue exceeds cost (a profit). This analysis assumes both revenue and cost change at constant rates, which is a simplification — in practice, economies of scale or market saturation could alter the gradients.

SUMMARY

Lesson Summary

Summary

A linear function has the form y = mx + c, where m (the gradient) represents the constant rate of change and c (the y-intercept) is the output value when x = 0. The gradient is calculated using m = (y₂ − y₁) / (x₂ − x₁) and can be positive (line rises), negative (line falls), or zero (horizontal line).

Linear models are powerful for situations with constant rates — such as fixed-speed travel, hourly wages, or steady filling of a tank — but have limitations when the rate of change is not truly constant. Always interpret m and c in context for full IB marks, and question whether extrapolation beyond the given data is reasonable. Mastering these ideas prepares you for quadratic, exponential, and other non-linear models later in the course.