Preview
IB DP Mathematics: Applications and Interpretation | SL & HL | Unit 2: Functions
📋 Learning Intentions
- Calculate the gradient of a straight line from two coordinate points using the gradient formula.
- Write the equation of a straight line in gradient-intercept form \( y = mx + c \) and general form \( ax + by + d = 0 \).
- Find the equation of a line passing through a given point with a specified gradient, or through two given points.
- Identify whether two lines are parallel, perpendicular, or neither, using their gradients.
- Determine the \(x\)-intercept and \(y\)-intercept of a line from its equation and interpret these values in context.
- Use the GDC to graph lines, find intersections, and verify algebraic results.
Real-World Introduction
A road engineer designing a mountain highway must calculate the steepness of each section of road. Too steep, and heavy vehicles cannot climb; too shallow, and drainage fails. That steepness — rise divided by run — is exactly the gradient of a straight line.
Straight-line models appear throughout IB Mathematics AI: as regression lines fitted to data, as linear cost and revenue functions in financial modelling, and as the tangent lines that approximate curves in calculus. Mastering the equation of a straight line is therefore foundational to almost every other topic in the course.
🌍 Why This Matters In linear regression (Topic 4), the least-squares regression line \(\hat{y} = mx + c\) is itself the equation of a straight line — the gradient \(m\) is the rate of change of the response variable per unit increase in the explanatory variable, and the \(y\)-intercept \(c\) is the predicted value when the explanatory variable is zero.
Core Concepts & Definitions
Consider two points on a line. Moving from one to the other, the line rises (or falls) by a certain vertical amount for every horizontal unit travelled. This ratio of vertical change to horizontal change is constant for any two points on the same straight line — it is the defining property of linearity.
📐 Key Definition — Gradient The gradient (slope) of a straight line passing through points \( (x_1, y_1) \) and \( (x_2, y_2) \) is: \[ m = \frac{y_2 - y_1}{x_2 - x_1}, \quad x_1 \neq x_2 \] A positive gradient means the line rises left to right. A negative gradient means it falls. A zero gradient means the line is horizontal. An undefined gradient means the line is vertical.
The gradient is a rate of change. In a