FortyFive › Revision notes › IB Maths: Analysis & Approaches › IB Maths: Analysis & Approaches: Functions
Domain, range, inverses, transformations, quadratics, exponentials. 16 subtopics.
In this topic: 2.1 · 2.2 · 2.3 · 2.4 · 2.5 · 2.6 · 2.7 · 2.8 · 2.9 · 2.10 · 2.11 · 2.12 · 2.13 · 2.14 · 2.15 · 2.16
Parallel lines have equal gradients.
The full notes, worked example and check questions for this subtopic are in the app.
A function maps each input in its domain to exactly one output.
The full notes, worked example and check questions for this subtopic are in the app.
A sketch shows the general shape and key features (intercepts, turning points, asymptotes), labelled, but not to scale.
The full notes, worked example and check questions for this subtopic are in the app.
Look for maximum and minimum points, intercepts, symmetry, the vertex, zeros, and vertical and horizontal asymptotes.
The full notes, worked example and check questions for this subtopic are in the app.
To find \(f^{-1}\): write \(y=f(x)\), swap \(x\) and \(y\), and make \(y\) the subject.
The full notes, worked example and check questions for this subtopic are in the app.
\(a\gt0\): opens upward (minimum). \(a\lt0\): opens downward (maximum).
The full notes, worked example, interactive diagram and check questions for this subtopic are in the app.
\(\Delta\gt0\): two distinct real roots. \(\Delta=0\): one repeated root. \(\Delta\lt0\): no real roots.
The full notes, worked example, interactive diagram and check questions for this subtopic are in the app.
Also find the intercepts: \(x=-\frac ba\) and \(y=\frac bd\).
The full notes, worked example and check questions for this subtopic are in the app.
\(y=a^{x}\ (a\gt0)\) passes through \((0,1)\) and has horizontal asymptote \(y=0\).
The full notes, worked example and check questions for this subtopic are in the app.
Solve analytically when possible: factorise, use log laws, substitute.
The full notes, worked example and check questions for this subtopic are in the app.
Changes inside the bracket affect \(x\) and act the “opposite” way to what you might expect.
The full notes, worked example and check questions for this subtopic are in the app.
Factor theorem: if \(f(a)=0\) then \((x-a)\) is a factor of \(f(x)\).
The full notes, worked example and check questions for this subtopic are in the app.
For \(\dfrac{ax+b}{cx^{2}+dx+e}\): vertical asymptotes at the real roots of the denominator; horizontal asymptote \(y=0\).
The full notes, worked example and check questions for this subtopic are in the app.
Even: \(f(-x)=f(x)\), symmetric about the \(y\)-axis.
The full notes, worked example and check questions for this subtopic are in the app.
Bring everything to one side, factorise, and use a sign diagram or a graph.
The full notes, worked example and check questions for this subtopic are in the app.
\(y=|f(x)|\): reflect any part below the \(x\)-axis upward.
The full notes, worked example and check questions for this subtopic are in the app.