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A Levels / Maths

Algebra and functions

Master the core A-Level Mathematics algebra and functions curriculum. Learn the laws of indices, methods to simplify surds and rationalise denominators, how to evaluate the quadratic discriminant, solve quadratic simultaneous equations, graph inequalities, apply the factor theorem, sketch modulus curves, decompose partial fractions, and perform graph transformations.

The laws of indices state that when multiplying terms with the same base you add powers (xa×xb=xa+bx^a \times x^b = x^{a+b}), when dividing you subtract powers (xa/xb=xa−bx^a / x^b = x^{a-b}), and fractional exponents represent roots where xmn=xmnx^{\frac{m}{n}} = \sqrt[n]{x^m}.
Negative indices represent reciprocals where x−n=1xnx^{-n} = \frac{1}{x^n}, allowing algebraic fractions with powers in the denominator to be converted into standard index form.
Surds rules allow expressions to be simplified using ab=a×b\sqrt{ab} = \sqrt{a} \times \sqrt{b} and ab=ab\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}, while terms with identical surd parts can be collected as like terms.
Rationalising the denominator of a fraction containing a binomial surd like a+ba + \sqrt{b} requires multiplying both the numerator and denominator by its conjugate pair a−ba - \sqrt{b}.
The discriminant of a quadratic equation ax2+bx+c=0ax^2 + bx + c = 0 is b2−4acb^2 - 4ac, where b2−4ac>0b^2 - 4ac > 0 indicates two distinct real roots, b2−4ac=0b^2 - 4ac = 0 indicates one repeated real root, and b2−4ac<0b^2 - 4ac < 0 indicates no real roots.
Completing the square transforms a quadratic into the form a(x+p)2+qa(x + p)^2 + q, immediately identifying the coordinates of the turning point at (−p,q)(-p, q) and the line of symmetry x=−px = -p.
Solving quadratic simultaneous equations involving one linear and one quadratic equation is achieved by rearranging the linear equation to isolate one variable and substituting it into the quadratic equation.
Solving quadratic inequalities requires finding the critical values, sketching the parabola, and stating the solution region using correct set notation or inequality symbols.
The factor theorem states that if substituting x=ax = a into a polynomial gives f(a)=0f(a) = 0, then (x−a)(x - a) is an exact factor of f(x)f(x), allowing the polynomial to be fully factorised via algebraic division.
Simplifying algebraic fractions requires completely factorising both the numerator and denominator into linear or quadratic factors and cancelling common factors.
Curve sketching for rational functions like y=axy = \frac{a}{x} and y=ax2y = \frac{a}{x^2} involves determining coordinate intercepts and identifying vertical and horizontal asymptotes.
The modulus function y=∣f(x)∣y = \vert{}f(x)\vert{} reflects any portion of the graph of y=f(x)y = f(x) that lies below the xx-axis into the positive yy-axis region.
Composite functions fg(x)fg(x) are formed by substituting the output of g(x)g(x) into f(x)f(x), while the inverse function f−1(x)f^{-1}(x) exists only for one-to-one functions and reflects y=f(x)y = f(x) across the line y=xy = x.
Graph transformations follow specific rules: y=f(x+a)y = f(x + a) shifts the graph horizontally by −a-a, y=f(x)+ay = f(x) + a shifts it vertically by +a+a, y=af(x)y = af(x) stretches it vertically by scale factor aa, and y=f(ax)y = f(ax) stretches it horizontally by scale factor 1a\frac{1}{a}.
Partial fractions decompose complex rational expressions into simpler components with distinct linear denominators Aax+b\frac{A}{ax + b} or repeated linear denominators B(ax+b)2\frac{B}{(ax + b)^2}, which is essential for algebraic integration and mathematical modelling.
A-Level Maths Algebra & Functions Revision Notes | TSL