Best for coefficients, the y-intercept, discriminant and many parameter questions.
IB MATHEMATICS AA · TOPIC 2.2 · PREMIUM
Quadratic Functions & Equations
Quadratics are not just equations to solve. They connect algebra, graphs, parameters, inequalities and modelling. This chapter teaches students how to decide which form of a quadratic is useful, how the discriminant predicts a graph, how Vieta's formulae describe roots without solving, and how GDC methods support rather than replace the mathematics.
THE BIG IDEA
One parabola can tell several different stories.
The same quadratic can be written in expanded, factorised or vertex form. The mathematics has not changed, but the information that is visible has. Expanded form highlights coefficients and the y-intercept. Factorised form exposes the roots. Vertex form reveals the turning point, axis of symmetry and range. Good problem solving begins by asking: which form makes the information I need easiest to see?
Best for roots, sign analysis and constructing a quadratic from its x-intercepts.
Best for turning points, range, transformations and optimisation.
GRAPH BEFORE FORMULA
The discriminant predicts how the parabola meets the x-axis.
For \(ax^2+bx+c=0\), the discriminant is \(\Delta=b^2-4ac\). If \(\Delta>0\), there are two distinct real roots and the parabola cuts the x-axis twice. If \(\Delta=0\), there is one repeated root and the parabola is tangent to the x-axis. If \(\Delta<0\), there are no real roots and the graph does not meet the x-axis.
\(\Delta>0\)2 real roots\(\Delta=0\)1 repeated root\(\Delta<0\)no real roots
This link between algebra and graph shape is the basis of many IB parameter questions: instead of finding the roots, we choose the values of a parameter that make the discriminant positive, zero or negative.
VIETA'S FORMULAE
Sometimes we can reason about the roots without calculating them.
If \(\alpha\) and \(\beta\) are the roots of \(ax^2+bx+c=0\), then \(\alpha+\beta=-\frac{b}{a}\) and \(\alpha\beta=\frac{c}{a}\). These relationships are powerful when a question asks about the sum, product, squares, reciprocals or transformed roots. The notes show how to build new quadratics from root information and how Vieta's formulae combine with discriminants in harder parameter problems.
\(\alpha+\beta=-\frac{b}{a}\)
Use this when the problem gives or asks about the total of the two roots.
\(\alpha\beta=\frac{c}{a}\)
The sign of the product is especially useful when deciding whether roots have the same or opposite signs.
\(\alpha^2+\beta^2\)
Rewrite as \((\alpha+\beta)^2-2\alpha\beta\) so that Vieta's formulae can be used immediately.
\(x^2-Sx+P=0\)
If new roots have sum \(S\) and product \(P\), this monic quadratic reconstructs the equation.
GDC METHODS
Use the calculator to confirm structure, not to hide it.
The chapter gives explicit routes for both calculators. On the Casio fx-CG50, students learn how to graph a quadratic, find ROOT values and identify intersections. On the TI-Nspire CX II, the notes show the Graphs page and Analyze Graph tools for Zero, Minimum/Maximum and Intersection. Quadratic inequalities are handled by locating the critical x-values and then reading the sign of the graph over the required intervals.
MINI WORKED EXAMPLE
When does a line touch a parabola?
Consider \(y=x^2+4x-1\) and \(y=2x+c\). At an intersection, the y-values are equal:
\[x^2+4x-1=2x+c\;\Longrightarrow\;x^2+2x-(1+c)=0\]
For tangency there must be exactly one \(x\)-value, so \(\Delta=0\):
\[2^2-4(1)(-1-c)=0\;\Longrightarrow\;8+4c=0\;\Longrightarrow\;c=-2\]
Notice the strategy: turn the geometry into one quadratic equation, then translate “touches once” into \(\Delta=0\).
INSIDE THE COMPLETE PACK
A chapter designed for learning first and exam practice second.
16 pages
Story-led explanations, clean graphs and tables, worked examples, GDC routes, Vieta's formulae, modelling and boxed HL material.
35 questions
A curated set rather than repetitive drill: parameters, intersections, inequalities, optimisation, Vieta and long Section B-style problems.
9 pages
Full reasoning with method selection, exact algebra, interpretation and links between earlier and later subparts.
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The next complete pack treats functions as machines, develops composition and domains, then builds inverse functions through graph reflection, one-to-one restrictions and difficult mixed IB-style problems.