#2.1
The basic processes of algebra
Knowledge and use of basic skills in manipulative algebra including use of the associative, commutative and distributive laws, are expected
#2.2
Definition of a function
Notation \(f(x)\) will be used, e.g. \(f(x) = x^2 - 9 \)
#2.3
Domain and range of a function
Domain may be expressed as, for example, \(x > 2\), or "for all x, except \(x = 0\)" and range may be expressed as \(f(x) > -1 \)
#2.4
Composite functions
The result of two or more functions, say \(f\) and \(g\), acting in succession.
\(fg(x)\) is \(g\) followed by \(f\)
#2.5
Inverse functions
The inverse function of \(f\) is written \(f^{-1}\)
Domains will be chosen for \(f\) to make \(f\) one-one
#2.6
Expanding brackets and collecting like terms
Expand and simplify
\((y^2 - 2y + 3) (2y - 1) - 2(y^3 - 3y^2 + 4y - 2)\)
#2.7
Expand \((a+b)^n\) for positive integer \(n\)
Expand and simplify \((5x + 2)^3 \)
Use Pascal's triangle to work out the coefficient of \(x^3\) in the expansion of \((3 + 2x)^5 \)
#2.8
Factorising
Factorise fully \((2x + 3)^2 - (2x - 5)^2 \)
Factorise \(15x^2 - 34xy - 16y^2 \)
Factorise fully \(x^4 - 25x^2 \)
#2.9
Manipulation of rational expressions:
Use of + – × ÷ for algebraic fractions with denominators being numeric, linear or quadratic
Simplify \(\dfrac{5}{x+2} - \dfrac{3}{2x-1} \)
Simplify \(\dfrac{x^3 + 2x^2 + x}{x^2 + x} \)
Simplify \(\dfrac{5x^2 - 14x - 3}{4x^2 - 25} ÷ \dfrac{x-3}{4x^2 + 10x} \)
#2.10
Use and manipulation of formulae and expressions
Rearrange \(\dfrac{1}{f} = \dfrac{1}{u} + \dfrac{1}{v}\) to make \(v\) the subject
#2.11
Use of the factor theorem for rational values of the variable for polynomials
Factorise \(x^3 - 2x^2 - 5x + 6 \)
Show that \(2x - 3 \) is a factor of \(2x^3 - x^2 - 7x + 6 \)
Solve \(x^3 + x^2 - 10x + 8 = 0 \)
Show that \(x - 7 \) is a factor of \(x^5 - 7x^4 - x + 7 \)
#2.12
Completing the square
Work out the values of \(a\), \(b\) and \(c\) such that
\(2x^2 + 6x + 7 ≡ a(x + b)^2 + c \)
#2.13
Drawing and sketching of functions
Interpretation of graphs
Graphs could be linear, quadratic, exponential and restricted to no more than 3 domains
Exponential graphs will be of the form \(y = ab^x \) and \(y = ab^{-x} \), where \(a\) and \(b\) are rational numbers
Sketch the graph of \(y = x^2 - 5x + 6 \)
Label clearly any points of the intersection with the axes
A function \(f\) is defined as
\( f(x) = x^2 \quad 0 ⩽ x < 1 \)
\(\qquad = 1 \quad 1 ⩽ x < 2 \)
\(\qquad = 3 - x \quad 2 ⩽ x < 3 \)
Draw the graph of \(y = f(x) \) on the grid below for values of \(x\) from 0 to 3
Given a sketch of \(y = ab^{-x} \), and two points, work out the values of \(a\) and \(b\)
#2.14
Solution of linear and quadratic equations
Solutions of quadratics to include solution by factorisation, by graph, by completing the square or by formula
Problems will be set in a variety of contexts, which result in the solution of linear or quadratic equations
#2.15
Algebraic and graphical solution of simultaneous equations in two unknowns, where the equations could both be linear or one linear and one second order
Solve \(4x - 3y = 0 \) and \(6x + 15y = 13 \)
Solve \(y = x + 2 \) and \(y^2 = 4x + 5 \)
Solve \(y = x^2 \) and \(y - 5x = 6 \)
Solve \(xy = 8 \) and \(x + y = 6 \)
#2.16
Algebraic solution of linear equations in three unknowns
Solve
\( 2x - 5y + 4z = 2 \)
\( 2x + y + 2z = 4 \)
\(x - y - 6z = -4 \)
#2.17
Solution of linear and quadratic inequalities
Solve \(5(x – 7) > 2(x + 1) \)
Solve \(x^2 < 9 \)
Solve \(2x^2 + 5x ⩽ 3 \)
#2.18
Index laws, including fractional and negative indices and the solution of equations
Express as a single power of \(x\)
\(\sqrt{x^{\frac{1}{2}} × x^{\frac{7}{2}}} \)
\(\sqrt{\dfrac{x^{\frac{3}{2}} × x^{\frac{7}{2}}}{x^2}} \)
Solve \(x^{-\frac{1}{2}} = 3 \)
Solve \(\sqrt{x} - \dfrac{10}{\sqrt{x}} = 3 \quad x > 0 \)
#2.19
Algebraic proof
Prove \( (n + 5)^2 - (n + 3)^2 \) is divisible by 4 for any integer value of \(n\)
#2.20
Using \(n\)th terms of sequences
Limiting value of a sequence as \(n → ∞ \)
Work out the difference between the 16th and 6th terms of the sequence with \(n\)th term \(\dfrac{2n}{n+4} \)
Write down the limiting value of \(\dfrac{2n}{n+4} \) as \(n → ∞ \)
#2.21
\(n\)th terms of linear sequences
A linear sequence starts 180 176 172 ...
By using the nth term, work out which term has value –1000
Work out the \(n\)th term of the linear sequence
\(r + s \quad r + 3s \quad r + 5s \quad ... \)
#2.22
\(n\)th terms of quadratic sequences
Work out the \(n\)th term of the quadratic sequence
10 16 18 16 ...
Which term has the value 0?