Learn this topic properly. Interactive diagrams, worked examples, check questions and exam-style practice with mark schemes. Open “Functions” in FortyFive →
Rise ÷ run: how much y changes when x increases by 1.
y-intercept c
Where the line crosses the y-axis.
Parallel lines
Same gradient.
Perpendicular lines
Gradients multiply to −1: m₂ = −1 ÷ m₁.
Gradient-intercept form \(y=mx+c\); general form \(ax+by+d=0\) (usually with integer coefficients); point-gradient form \(y-y_1=m(x-x_1)\).
The full notes, worked example and check questions for this subtopic are in the app.
Exam tip: “General form” usually means integer coefficients with everything on one side. Read which form the question asks for.
2.2Functions: domain, range and inverse
Key terms
Function
Each input x gives exactly one output f(x).
Domain
The set of allowed inputs (x-values).
Range
The set of outputs (y-values) the function actually produces.
Inverse function f⁻¹
Undoes f: f⁻¹(f(x)) = x. Its graph is the reflection of f in y = x.
Function notation: \(f(3)\) is the output when \(x=3\); \(f(x)=10\) is an equation to solve for \(x\).
The full notes, worked example and check questions for this subtopic are in the app.
Exam tip: When asked for a range, use the correct variable: write f(x) ≥ 2 or y ≥ 2, not x ≥ 2.
2.3Graphs of functions and using technology
Key terms
Sketch
Shows the general shape and key features, labelled, but not to scale.
Draw
Accurate, to scale, using a ruler and plotted points.
Viewing window
The x and y range shown on the GDC screen.
Key features
Intercepts, maximum and minimum points, asymptotes.
A sketch must show the correct shape, the domain, and labelled key features (intercepts, turning points, asymptotes) with their coordinates.
The full notes, worked example and check questions for this subtopic are in the app.
Exam tip: Label points with coordinates directly on the sketch. An unlabelled curve rarely gets full marks.
2.4Key features of graphs and intersections
Key terms
Zero (root)
An x-value where f(x) = 0: an x-intercept.
Vertex
The turning point of a parabola.
Axis of symmetry
For y = ax² + bx + c, the line x = −b ÷ (2a).
Asymptote
A line the graph approaches but does not reach.
Point of intersection
Solves f(x) = g(x).
Quadratic \(y=ax^{2}+bx+c\): opens up if \(a\gt0\), down if \(a\lt0\); vertex on \(x=-\frac{b}{2a}\); \(y\)-intercept \(c\).
The full notes, worked example, interactive diagram and check questions for this subtopic are in the app.
Exam tip: Give coordinates as pairs (x, y). If the question asks for x only, don't waste time finding y.
2.5Modelling with functions
Key terms
Linear model
Constant rate of change: f(x) = mx + c.
Exponential model
Constant percentage change: f(x) = ka^x + c.
Direct variation
y = kxⁿ with n > 0: y increases as x increases.
Inverse variation
y = k ÷ xⁿ: y decreases as x increases.
Sinusoidal model
f(x) = a sin(bx) + d for repeating patterns such as tides.
Linear \(f(x)=mx+c\); quadratic \(f(x)=ax^{2}+bx+c\); cubic \(f(x)=ax^{3}+bx^{2}+cx+d\).
The full notes, worked example, interactive diagram and check questions for this subtopic are in the app.
Exam tip: Check your sinusoidal model at a known point (a maximum or minimum) before you use it.
2.6Modelling skills
Key terms
Develop
Choose a model type from the shape of the data or the context.
Fit
Find the parameters, by hand or with GDC regression.
Test
Compare predictions with data you did not use.
Reflect
Discuss assumptions, limitations and the domain where the model is valid.
Extrapolation
Predicting outside the range of the data: unreliable.
The modelling cycle: develop → fit → test → reflect → use, then improve.
The full notes, worked example and check questions for this subtopic are in the app.
Exam tip: “Comment on the model” questions want a specific reason linked to the context, not “it might be wrong”.
2.7Composite and inverse functions HL
Key terms
Composite function
(f ∘ g)(x) = f(g(x)): apply g first, then f.
One-to-one
Each output comes from exactly one input; only these have inverses.
Self-inverse
f⁻¹ = f, e.g. f(x) = 1/x.
In general \(f\circ g\ne g\circ f\).
The full notes, worked example and check questions for this subtopic are in the app.
Exam tip: Always state the domain of an inverse. It is the range of the original function.
2.8Transformations of graphs HL
Key terms
Translation
y = f(x − h) + k moves the graph h right and k up.
Vertical stretch
y = p f(x): scale factor p, parallel to the y-axis.
Horizontal stretch
y = f(qx): scale factor 1/q, parallel to the x-axis.
Reflections
y = −f(x) in the x-axis; y = f(−x) in the y-axis.
Changes outside the function affect \(y\) and behave as expected; changes inside affect \(x\) and behave the opposite way: \(f(x-3)\) moves right 3, \(f(2x)\) halves the \(x\)-values.
The full notes, worked example and check questions for this subtopic are in the app.
Exam tip: Use the exact wording: “vertical stretch, scale factor 2” and “translation by the vector (3, 1)”. Vague words like “moves” or “bigger” lose marks.
2.9Exponential, logarithmic, logistic and piecewise models HL
Key terms
Natural exponential model
f(x) = ke^(rx) + c: r > 0 growth, r < 0 decay.
Logarithmic model
f(x) = a + b ln x: grows ever more slowly.
Logistic model
f(x) = L ÷ (1 + Ce^(−kx)): growth that levels off.
Carrying capacity L
The upper limit a logistic model approaches.
Piecewise model
Different formulas on different parts of the domain.
A logistic curve starts like exponential growth, has its fastest growth at \(y=\frac L2\), then levels off towards \(y=L\).
The full notes, worked example, interactive diagram and check questions for this subtopic are in the app.
Exam tip: For long-term behaviour, say what happens to the exponential term (→ 0) and then state the limit.
2.10Logarithmic scales and linearising data HL
Key terms
Log scale
Equal steps multiply by 10 (e.g. Richter, decibels, pH).
Semi-log graph
log y against x: a straight line means an exponential model.
Log-log graph
log y against log x: a straight line means a power model.
Plot transformed data; if it is a straight line, fit a linear regression, then transform back.
The full notes, worked example and check questions for this subtopic are in the app.
Exam tip: Keep track of which log you used (log or ln). Transforming back with the wrong base is the most common error.