Knowledge organisers

Functions: IB Maths AA HL knowledge organiser

Everything to know about functions on one page: key definitions, the formulas, a worked example, the mistakes to avoid and a checklist of what you should be able to do.

Download the A4 sheet (PDF)

Key definitions

Factor theorem: (x − a) is a factor of a polynomial p(x) exactly when p(a) = 0.
Vertical asymptote: A line x = a that the graph approaches but never meets, often where a denominator is zero.
Odd and even functions: Even: f(−x) = f(x), symmetric in the y-axis. Odd: f(−x) = −f(x), rotational symmetry about the origin.
Sum and product of roots: For a polynomial of degree n, the sum and product of its roots can be read from its coefficients.

Key formulas

Formulas marked with a label are given in the exam (we only say so where our formula sheet confirms it). Learn the rest.

LinesIn the formula booklet	\(m=\frac{y_2-y_1}{x_2-x_1},\) \(y=mx+c,\) \(ax+by+d=0,\) \(y-y_1=m(x-x_1)\)
Perpendicular	\(m_1m_2=-1\)
QuadraticsIn the formula booklet	\(x=-\frac b{2a},\) \(x=\frac{-b\pm\sqrt{b^2-4ac}}{2a},\) \(\Delta=b^2-4ac\)
\(\Delta>0\): two real roots; \(\Delta=0\): repeated root; \(\Delta<0\): complex conjugate roots
Sum, product of roots of \(\sum_0^na_rx^r\)In the formula booklet	\(-\frac{a_{n-1}}{a_n};\) \(\frac{(-1)^na_0}{a_n}\)
Factor & remainder theorems	\(p(a)=0\iff(x-a)\text{ is a factor};\) \(\text{remainder of }p(x)\div(x-a)\text{ is }p(a)\)
Odd; even	\(f(-x)=-f(x);\) \(f(-x)=f(x)\)
Composite; inverse	\((f\circ g)(x)=f(g(x));\) \(f\big(f^{-1}(x)\big)=x\)
Transformations: \(f(x)+b\) up \(b\); \(f(x-a)\) right \(a\); \(pf(x)\) vertical ×\(p\); \(f(qx)\) horizontal ×\(\tfrac1q\); \(-f(x)\), \(f(-x)\) reflect in \(x\)-, \(y\)-axis
Graphs: \(\|f(x)\|\) reflects negative parts up; \(f(\|x\|)\) mirrors \(x\ge0\); \(\frac1{f(x)}\) has asymptotes at zeros of \(f\)
\(y=\frac{ax+b}{cx+d}\) asymptotes	\(x=-\frac dc,\) \(y=\frac ac\)

Worked example

Show that (x − 1) is a factor of p(x) = x³ − 2x² − 5x + 6 and factorise p(x) fully.

p(1) = 1 − 2 − 5 + 6 = 0, so (x − 1) is a factor
Divide: p(x) = (x − 1)(x² − x − 6)
x² − x − 6 = (x + 2)(x − 3)

Answer: p(x) = (x − 1)(x + 2)(x − 3)

Common mistakes

Graphs and transformations: describing a rational function, and graphs of [f(x)]²
Complex numbers and polynomials: conjugate roots, all the roots, general solutions
Mixing up the perpendicular gradient rule (m₁·m₂ = −1)
Sign errors completing the square

More on what examiners see students get wrong: Examiner Insights.

You should be able to…

Find the gradient and midpoint of two points and write the equation of a line in gradient-intercept, general and point-gradient forms, such as y − 3 = 2(x − 1).
Solve quadratics with the formula, use the discriminant to count real roots and find unknown constants, and sketch parabolas showing intercepts and the vertex.
Sketch y = f(x) + a, y = f(x + a), y = af(x) and y = f(ax) from a given graph, describing each transformation and tracking key points.
Sketch rational functions with a quadratic numerator or denominator, finding vertical, horizontal and oblique asymptotes and the intercepts.
Use the factor and remainder theorems to find factors and remainders of polynomials and unknown coefficients, such as k when (x − 2) is a factor.
Solve inequalities such as |2x − 1| > x + 3 and rational inequalities both graphically and analytically, giving the solution set clearly.

The printable sheet

Functions knowledge organiser for IB Maths AA HL: one A4 page of key definitions, formulas, a worked example and common mistakes — Functions knowledge organiser (IB Maths AA HL), A4. Download the PDF.

Revise it next

Other IB Maths AA HL topics: Number and algebra · Geometry and trigonometry · Statistics and probability · Calculus · All IB Maths AA HL organisers