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LearnEdexcel9MA0First teaching 2017Paper 2 · Pure Mathematics 2Algebra and functions

Paper 2 · Module

Algebra and functions

16 subtopics≈ 645 min in total

Start the module: Completing the square and the vertex Open graphing calculator

Completing the square and the vertex
Rewrite $ax^2+bx+c$ as $a(x+p)^2+q$ , read off the vertex, line of symmetry and least or greatest value, and use it to sketch parabolas and solve real problems about maximum height or area.
≈ 45 min
Solving quadratic equations
Three ways to solve $ax^2+bx+c=0$ (factorising, completing the square and the formula), choosing the quickest, giving exact answers, and spotting quadratics in disguise such as $x^4-5x^2+4=0$ .
≈ 45 min
The discriminant and the nature of roots
How $b^2-4ac$ tells you whether a quadratic has two, one or no real roots without solving it, how to find unknown constants from that, and how to show an expression is always positive.
≈ 40 min
Linear and quadratic inequalities
Solve linear inequalities, then quadratic ones using critical values and a sketch, write the answer correctly (and, or, set notation), and combine two conditions.
≈ 35 min
Simultaneous equations and lines meeting curves
Solve one linear and one quadratic equation together by substitution, find the points where a line meets a curve, and use the discriminant to decide whether a line crosses, touches or misses a curve.
≈ 40 min
Indices and surds
$a^{\frac mn}$ means "take the $n$ th root, then raise to the power $m$ ", and $a^{-n}=\dfrac1{a^n}$ . Surds are roots left exact: simplify them by pulling out square factors, and remove them from denominators by multiplying by the conjugate.
≈ 30 min
Polynomials: division, factor and remainder theorems
Long division for polynomials, why the remainder on dividing by $(x-a)$ is just $p(a)$ , and how the factor theorem lets you factorise cubics and find unknown coefficients.
≈ 45 min
Partial fractions
Splitting a complicated fraction into simpler ones, which you need for integrating and for binomial expansions: the right form for each kind of denominator, and quick ways to find the constants.
≈ 45 min
Modulus graphs and equations
What $|x|$ means, how to sketch $y=|ax+b|$ by reflecting the negative part, and two reliable ways to solve modulus equations: squaring both sides, or splitting into cases and checking.
≈ 40 min
Modulus inequalities
Solving inequalities with moduli: the "between" and "outside" patterns, squaring when both sides are moduli, and using a sketch when a modulus is compared with a line.
≈ 40 min
Curve sketching and transformations
A factorised cubic crosses the $x$ -axis at each factor's root and touches it at a repeated root. Reciprocal graphs have asymptotes. Changes outside the bracket act vertically and do what you expect; changes inside act horizontally and do the opposite.
≈ 40 min
Functions, domain and range
What makes a rule a function, function notation, the largest possible domain, and how to find the range of linear and quadratic functions on any domain.
≈ 40 min
Composite functions
Applying one function after another: what $fg(x)$ means, finding and solving with composite expressions, and when a composite function exists.
≈ 35 min
One-one and inverse functions
Which functions can be undone, how to find $f^{-1}(x)$ , why the domain of the inverse is the range of the function, and how to invert a quadratic on a restricted domain.
≈ 45 min
Graphs of a function and its inverse
Why the graph of $f^{-1}$ is the reflection of $f$ in the line $y=x$ , how points, intercepts and asymptotes swap over, and where a function meets its own inverse.
≈ 30 min
Transformations of graphs
Translating, stretching and reflecting $y=f(x)$ : what each of $f(x)+a$ , $f(x+a)$ , $af(x)$ and $f(ax)$ does, how to describe it in exam language, and how to combine transformations in the right order.
≈ 50 min