IGCSE, Edexcel, IB and AP Science Courses | Dr Dabour Hub Academy

Overview

Define the quadratic function, find its vertex in both forms and sketch its graph.
Use the maximum or minimum value in real-life optimisation.
Find the zeros of polynomial functions by factoring, completing the square or reading the graph.
Determine the domain of a rational function, simplify it, and perform the four operations with their restrictions.

Quadratic, Polynomial and Rational Functions
├── 2-1 The quadratic function
│   ├── definition, values, points on the curve
│   ├── vertex form y = a(x - h)² + k
│   ├── equation from the vertex and one point
│   ├── sketch and vertex from the general form
│   └── maximum and minimum applications
├── 2-2 Polynomial functions
│   ├── degree and leading term
│   └── zeros: factoring, completing the square, from the graph
├── 2-3 Rational algebraic functions
│   ├── domain, simplifying
│   └── zeros of a rational function
├── 2-4 Common domain of two functions and equality of functions
└── 2-5 Operations on functions
    ├── addition and subtraction
    ├── multiplication and division
    ├── additive and multiplicative inverses
    └── properties of the operations

Unit 2: The Quadratic, Polynomial and Rational Functions and Their Operations · 1 / 13

2-1 The quadratic function: definition, values and points

A quadratic function is f(x) = ax² + bx + c with a, b, c real and a ≠ 0; its graph is a parabola.
The function value at x = k is f(k) substitute and compute — a point (k, m) lies on the curve exactly when f(k) = m.
To find an unknown coefficient from a point on the curve, substitute the point and solve the resulting linear equation.
The y-intercept is (0, c); the x-intercepts are the zeros of f.

Unit 2: The Quadratic, Polynomial and Rational Functions and Their Operations · 2 / 13

2-2 The vertex form y = a(x − h)² + k

In vertex form the vertex is (h, k) and the axis of symmetry is the line x = h.
a > 0 the parabola opens upward and k is the minimum value; a < 0 it opens downward and k is the maximum value.
Watch the sign inside the bracket y = (x − 3)² + 2 has vertex (3, 2), while y = (x + 3)² − 2 has vertex (−3, −2).

The Quadratic, Polynomial and Rational: a basketball flying along a parabola shows the quadratic function: vertex (2, 4) and zeros 0 and 4.

Equation from the vertex and one point

Write y = a(x − h)² + k with the given vertex, then substitute the other point to obtain a.
Vertex (1, −4) through (3, 0) 0 = a(3 − 1)² − 4 → 4a = 4 → a = 1, so y = (x − 1)² − 4 = x² − 2x − 3.

Unit 2: The Quadratic, Polynomial and Rational Functions and Their Operations · 3 / 13

2-3 The vertex from the general form

For f(x) = ax² + bx + ch = −b/(2a) and k = f(h) = (4ac − b²)/(4a).

Completing the square converts between the two formsx² + 6x + 5 = (x + 3)² − 4, vertex (−3, −4).

Zeros from the general formsolve ax² + bx + c = 0; the axis x = h is exactly halfway between the two zeros.

Unit 2: The Quadratic, Polynomial and Rational Functions and Their Operations · 4 / 13

2-4 Sketching the parabola

Comparison aspect	Main idea	Keywords
Plot, in order	the vertex the y-intercept (0 the reflection of the y-intercept across the axis (2h and the x-intercepts when they exist	vertexinterceptreflectionacross
Δ = b² − 4ac tells how many x-intercepts	two if Δ > 0 one (the vertex touches the axis) if Δ = 0 none if Δ < 0	vertextouchesaxisnone
•	Draw a smooth symmetric curve never a “V” — the parabola is curved everywhere	Drawsmoothsymmetriccurve
Translations	y = a(x − h)² + k is the image of y = ax² shifted h right and k up	imageshiftedright

Unit 2: The Quadratic, Polynomial and Rational Functions and Their Operations · 5 / 13

2-5 Maximum and minimum applications

Comparison aspect	Main idea	Keywords
•	A quantity modelled by a quadratic reaches its extremum at the vertex i.e. at x = −b/(2a) the extremum value is k	quantitymodelledquadraticreaches
Fenced area	with a fixed perimeter the area A(x) is quadratic in x and maximal at the vertex — the answer is always a square	fixedperimeterquadraticmaximal
Profit	revenue minus cost is quadratic the maximum profit is the vertex value and break-even points are its zeros	revenueminuscostquadratic
Projectile height	h(t) = −16t² + v₀t + h₀ opens downward so the vertex gives the greatest height and its time	16t²opensdownwardvertex
•	Reject any value of the variable that the situation forbids (negative time negative price)	Rejectvaluevariablesituation

Unit 2: The Quadratic, Polynomial and Rational Functions and Their Operations · 6 / 13

2-6 Polynomial functions and their degree

Comparison aspect	Main idea	Keywords
•	A polynomial function is f(x) = aₙxⁿ + … + a₁x + a₀ with aₙ ≠ 0 n is the degree and aₙ the leading coefficient	polynomialfunctionaₙxⁿdegree
Names by degree	1 linear 2 quadratic 3 cubic constants have degree 0	linearquadraticcubicconstants
The end behaviour follows the leading term	even degree with a positive coefficient rises on both sides odd degree with a positive coefficient falls to the left and rises to the right	evendegreepositivecoefficient

Unit 2: The Quadratic, Polynomial and Rational Functions and Their Operations · 7 / 13

2-7 Zeros of a polynomial function

By factoring

Set f(x) = 0 and factor completely
each factor gives a zero. x³ − 4x = x(x − 2)(x + 2) → zeros 0
2
−2.

x² − 6x + 4 = 0 → (x − 3)² = 5 → x = 3 ± √5.

A zero is an x-intercept. If the curve crosses the axis the multiplicity is odd; if it only touches and turns the multiplicity is even.
A polynomial of degree n has at most n real zeros, and the sum of the multiplicities equals n.

Unit 2: The Quadratic, Polynomial and Rational Functions and Their Operations · 8 / 13

2-8 The rational algebraic function, its domain and simplifying

Comparison aspect	Main idea	Keywords
•	A rational function is f(x) = P(x)/Q(x) Q ≠ 0 where P and Q are polynomials	rationalfunctionpolynomials
Domain	all reals except the values that make Q(x) = 0. Example: Q(x) = x² − 9 excludes x = ±3 so the domain is ℝ − {−3	realsexceptvaluesmake
Simplifying	factor numerator and denominator cancel only common factors and keep the excluded values as restrictions. (x² − 1)/(x² + x) = (x − 1)(x + 1)/[x(x + 1)] = (x − 1)/x x ≠ 0 ±1… restrictions x ≠ 0 x ≠ −1	factornumeratordenominatorcancel
Zeros of a rational function	the zeros of the numerator that belong to the domain — a value that also vanishes the denominator is not a zero	numeratorbelongdomainvalue

Unit 2: The Quadratic, Polynomial and Rational Functions and Their Operations · 9 / 13

2-9 Common domain of two functions and equality of functions

The common domain of two or more functions is the intersection of their domains; any operation between them is defined only there.
Two functions are equal when they have the same domain and the same value for every element of it — identical simplified formulas on different domains are not the same function.
When asked for the domain of a sum, product or quotient, start from the intersection and then remove the zeros of any denominator that appears.

Unit 2: The Quadratic, Polynomial and Rational Functions and Their Operations · 10 / 13

2-10 Addition, subtraction, multiplication and division

Comparison aspect	Main idea	Keywords
Addition and subtraction	use the lowest common denominator rewrite each fraction over it then add or subtract the numerators only keeping the common denominator	lowestcommondenominatorrewrite
Multiplication	factor everything cancel common factors across the numerator and denominator then multiply straight across	factoreverythingcancelcommon
Division	multiply by the reciprocal of the divisor and add to the restrictions the zeros of the new numerator that came from flipping	multiplyreciprocaldivisorrestrictions
•	State the final answer in simplest form together with its domain restrictions	Statefinalanswersimplest

Unit 2: The Quadratic, Polynomial and Rational Functions and Their Operations · 11 / 13

2-11 Additive and multiplicative inverses, and properties

The additive inverse of f is −f; their sum is the zero function, the additive identity.
The multiplicative inverse (reciprocal) of f = P/Q is Q/P, defined only where P ≠ 0 and Q ≠ 0; their product is 1.
Properties on the common domain closure, commutativity and associativity of addition and multiplication, the distributive law of multiplication over addition, and the identities 0 (additive) and 1 (multiplicative).
Subtraction and division are not commutative and not associative — do not extend those properties to them.

Unit 2: The Quadratic, Polynomial and Rational Functions and Their Operations · 12 / 13

Exam traps and key takeaways

💡 High-yield traps

The vertex of y = a(x − h)² + k is (h, k) — the sign of h is opposite to what appears in the bracket.
k is a maximum only when a < 0; read the sign of a before answering.
For the general form use h = −b/(2a); forgetting the minus sign puts the vertex on the wrong side.
The definition of a quadratic requires a ≠ 0.
Zeros from the graph touching means even multiplicity, crossing means odd.
Cancel factors, never terms — (x + 2)/x is already simplest.
Simplifying does not enlarge the domain; the old restrictions must be restated.
A zero of a rational function comes from the numerator only, and must not vanish the denominator.
Division of fractions doubles the restrictions those of the divisor and those created by flipping it.
Two functions are equal only with the same domain and the same rule.

Unit 2: The Quadratic, Polynomial and Rational Functions and Their Operations · 13 / 13

Curriculum

Mathematics — Third Preparatory

Unit 1

Unit 2

Unit 3

Unit 4

Unit 2: The Quadratic, Polynomial and Rational Functions and Their Operations