Algebra
KS4MA-KS4-D002
Using algebraic notation and manipulation, solving equations and inequalities, working with sequences, functions and graphs, including quadratics, simultaneous equations, and proof.
National Curriculum context
Algebra at KS4 extends the symbolic reasoning introduced in KS3 to encompass quadratic expressions and equations, simultaneous equations, algebraic fractions and formal proof. Pupils develop fluency with expanding and factorising, including completing the square and using the quadratic formula, and apply algebraic methods to model and solve problems in geometry, ratio and statistics contexts. The statutory curriculum requires pupils to interpret and construct graphs of linear, quadratic, cubic, reciprocal and exponential functions, and to understand transformations of graphs. Higher tier pupils extend this further to include algebraic proof, manipulation of algebraic fractions, functions and their inverses, gradient of curves and area under curves using calculus ideas informally.
8
Concepts
4
Clusters
4
Prerequisites
8
With difficulty levels
Lesson Clusters
Manipulate and factorise algebraic expressions including quadratics
introduction CuratedAlgebraic manipulation and factorisation (single brackets, double brackets, quadratics) is the foundational algebraic skill at GCSE upon which all equation and function work depends.
Solve linear and quadratic equations and simultaneous equations
practice CuratedLinear equations/inequalities, quadratic equations (factorising, formula, completing the square) and simultaneous equations are the core equation-solving cluster. C007 co-teaches with C008.
Work with sequences, linear functions and non-linear graphs
practice CuratedSequences/nth term, straight line graphs (y=mx+c) and non-linear graphs (quadratic, cubic, exponential, trigonometric) are the function and graph cluster at GCSE.
Understand function notation and apply graph transformations
practice CuratedFunctions (notation, composite, inverse) and graph transformations (f(x+a), f(ax), af(x)) are the most advanced algebra targets at GCSE and form a distinct conceptual cluster.
Prerequisites
Concepts from other domains that pupils should know before this domain.
Domain Vocabulary
89 terms across 8 concepts (89 domain-specific)(7 shared)
algebraic fraction(noun)
A fraction where the numerator, denominator, or both contain algebraic expressions.
amplitude(noun)
The maximum displacement from the centre line of a periodic function; half the distance between the peak and trough.
arithmetic(adjective)
A type of sequence where consecutive terms have a constant difference.
asymptote(noun)
A line that a curve approaches but never reaches, getting infinitely close without touching.
balance(noun)
A device for comparing the mass of objects, or the state when both sides are equal.
bracket(noun)
Symbols ( ) used to group terms in expressions, indicating these should be calculated first.
coefficient(noun)
The number placed in front of a variable in an algebraic term that shows how many of that variable there are.
Shared by 2 concepts
collinear(adjective)
Points that all lie on the same straight line.
common difference(noun)
The fixed amount added or subtracted to get from one term to the next in a linear sequence.
common factor(noun)
A number that divides exactly into two or more other numbers with no remainder.
common ratio(noun)
The constant multiplier between consecutive terms in a geometric sequence.
completing the square(noun)
A method for rewriting a quadratic expression in the form (x + p)² + q, useful for finding the vertex and solving equations.
Shared by 2 concepts
composite function(noun)
A function created by applying one function to the result of another, written as fg(x) or f(g(x)).
consistent(adjective)
Showing little variation; reliable and predictable results.
cubic(adjective)
A polynomial of degree 3, with the highest power of x being x³; the graph has up to two turning points.
difference of two squares(noun)
An algebraic identity: a² - b² = (a + b)(a - b), used for factorising expressions with two squared terms.
discriminant(noun)
The expression b² - 4ac from the quadratic formula, which determines the number of real roots.
domain(noun)
The set of all possible input values (x-values) for a function.
elimination(noun)
A method for solving simultaneous equations by adding or subtracting equations to remove one variable.
equation(noun)
A mathematical sentence with an equals sign showing that two sides have the same value.
equation of a line(noun)
An algebraic rule describing a straight line, usually in the form y = mx + c where m is the gradient and c is the y-intercept.
expand(verb)
To multiply out brackets in an algebraic expression.
explicit formula(noun)
A formula that directly calculates the nth term of a sequence from n, without needing previous terms.
exponential(adjective)
Growing or decaying by a constant multiplier; involving powers or indices.
expression(noun)
A combination of numbers, variables, and operations that represents a value, but does not contain an equals sign.
factorise(verb)
To write an expression as a product of its factors; the reverse of expanding.
factorising(noun)
The process of writing an expression as a product of its factors, the reverse of expanding.
fibonacci(noun)
A sequence where each term is the sum of the two preceding terms: 1, 1, 2, 3, 5, 8, 13, 21...
function(noun)
A rule that assigns exactly one output to each input value.
geometric(adjective)
A type of sequence where consecutive terms have a constant ratio (multiplier).
gradient(noun)
The steepness of a line; calculated as the change in y divided by the change in x (rise over run).
highest common factor(noun)
The largest whole number that divides exactly into two or more given numbers with no remainder.
identity(noun)
An equation that is true for all values of the variable, such as 2(x + 3) ≡ 2x + 6.
inconsistent(adjective)
A system of equations that has no solution because the equations contradict each other.
inequality(noun)
A mathematical statement showing that two values are not equal, using symbols such as <, >, ≤, or ≥.
intercept(noun)
The point where a line or curve crosses an axis; the y-intercept is where x = 0.
Shared by 2 concepts
intersection(noun)
The point where two lines or curves cross; in sets, the elements that belong to both sets.
inverse function(noun)
A function that reverses the effect of the original function; written as f⁻¹(x).
linear(adjective)
Forming a straight line when graphed; involving variables to the power of 1 only.
Shared by 4 concepts
mapping(noun)
A rule that assigns each input value to exactly one output value; another word for function.
maximum(noun)
The greatest value; the highest point on a graph or the largest number in a data set.
minimum(noun)
The smallest value; the lowest point on a graph or the least number in a data set.
notation f(x)(noun)
Function notation where f(x) represents the output of function f when the input is x.
nth term(noun)
A formula giving the value of any term in a sequence based on its position number n.
number line(noun)
A straight line marked with numbers at equal intervals, used for counting, adding, and subtracting.
one-to-one(adjective)
A function where each input has a unique output and each output comes from a unique input; necessary for an inverse to exist.
parabola(noun)
The U-shaped curve that is the graph of a quadratic function y = ax² + bx + c.
parallel(adjective)
Two lines that are always the same distance apart and never meet, no matter how far they are extended.
Shared by 2 concepts
period(noun)
The horizontal length of one complete cycle of a periodic function like sine or cosine.
perpendicular(adjective)
Two lines that meet at exactly 90 degrees (a right angle).
position(noun)
Where something is located, described using words like above, below, left, right.
quadratic(adjective)
An expression or equation where the highest power of the variable is 2.
Shared by 5 concepts
quadratic formula(noun)
The formula x = (-b ± √(b²-4ac)) / 2a for solving any quadratic equation ax² + bx + c = 0.
range(noun)
The difference between the largest and smallest values in a data set, showing how spread out the data is.
rate of change(noun)
How quickly one quantity changes relative to another; shown by the gradient on a graph.
real roots(noun)
Solutions to an equation that are real numbers (not complex); where a graph crosses the x-axis.
rearrange(verb)
To change the order of numbers or operations, often to make a calculation easier.
reciprocal(noun)
The number you get when you flip a fraction (swap numerator and denominator); a number multiplied by its reciprocal equals 1.
recursive(adjective)
A formula that defines each term of a sequence using the previous term(s).
reflection(noun)
The mirror image of a shape produced by flipping it over a line of symmetry.
root(noun)
A value of x that makes an equation equal to zero; also the inverse of a power (e.g. square root).
roots(noun)
The values of x that satisfy an equation, making it equal to zero; also called solutions.
sequence(noun)
An ordered list of numbers that follows a rule or pattern.
simultaneous(adjective)
Occurring at the same time; in maths, equations that must both be satisfied by the same values.
sketch(noun)
A rough but recognisable drawing of a graph showing key features like intercepts, turning points, and shape.
slope(noun)
Another word for gradient; the steepness and direction of a line.
solution(noun)
The value or set of values that make an equation or problem true.
Shared by 2 concepts
solution set(noun)
The complete set of all values that satisfy an equation or inequality.
solutions(noun)
The values of unknowns that make an equation or system of equations true.
solve(verb)
To find the value(s) of unknown variable(s) that make an equation true.
stretch(noun)
A transformation that changes the size of a shape in one direction only, either horizontally or vertically.
strict inequality(noun)
An inequality using < or > (not ≤ or ≥), where the boundary value is not included.
substitute(verb)
To replace a variable in an expression or equation with a specific numerical value.
substitution(noun)
Replacing a variable with a specific numerical value to evaluate an expression or solve an equation.
symmetry(noun)
A property of a shape where one half is a mirror image of the other when divided by a line.
system of equations(noun)
Two or more equations that share the same unknowns and must be solved together.
term(noun)
A number in a sequence or pattern, identified by its position (e.g. 1st term, 2nd term).
transformation(noun)
A change in the position, size, or orientation of a shape — includes reflection, rotation, and translation.
translation(noun)
A transformation that slides a shape to a new position without rotating or flipping it; every point moves the same distance in the same direction.
trigonometric(adjective)
Relating to the study of relationships between angles and sides of triangles using sine, cosine, and tangent.
turning point(noun)
A point on a curve where the direction changes from increasing to decreasing (maximum) or vice versa (minimum).
unknown(noun)
A number that has not been found yet; a missing value in a number sentence.
variable(noun)
A letter or symbol that represents a quantity which can change or take different values.
verify(verb)
To check that an answer is correct by using a different method or the inverse operation.
vertex(noun)
A point where two or more lines or edges meet; a corner of a shape.
x-intercept(noun)
The point where a graph crosses the x-axis; where y = 0.
y = mx + c(noun)
The standard form of a straight-line equation where m is the gradient and c is the y-intercept.
y-intercept(noun)
The point where a graph crosses the y-axis; the value of y when x = 0.
zero(noun)
The number 0; the starting point on a number line, representing nothing or no quantity.
Drag to rearrange. Click a term or concept for details. Scroll to zoom.
Concepts (8)
Algebraic Manipulation and Factorisation
process AI FacilitatedMA-KS4-C005
Expanding and factorising algebraic expressions, including single brackets, double brackets, and quadratic expressions; simplifying algebraic fractions.
Teaching guidance
Use area models (the grid method) for expanding double brackets before the FOIL algorithm — this gives a visual foundation. Teach factorising as the reverse of expanding and always ask pupils to check by re-expanding. Difference of two squares is a special case worth isolating. Higher pupils should work with algebraic fractions alongside numeric fractions to emphasise structural similarity.
Vocabulary (12 terms)
Common misconceptions
When expanding (a+b)², many pupils write a²+b² rather than a²+2ab+b² — the middle term is persistently dropped. Some pupils sign-strip when factorising, forgetting that factorising out -1 changes all signs inside. Cancelling in algebraic fractions is frequently misapplied (pupils cancel terms rather than factors).
Difficulty levels
Can expand a single bracket and collect like terms to simplify expressions.
Example task
Expand and simplify: 3(2x + 1) - 2(x - 4).
Model response: 6x + 3 - 2x + 8 = 4x + 11.
Expands double brackets and factorises simple quadratic expressions of the form x² + bx + c.
Example task
Factorise x² + 5x + 6.
Model response: Find two numbers that multiply to 6 and add to 5: 2 and 3. x² + 5x + 6 = (x + 2)(x + 3).
Factorises difference of two squares, expressions where the coefficient of x² is not 1, and simplifies algebraic fractions.
Example task
Factorise 4x² - 9 and simplify (4x² - 9)/(2x + 3).
Model response: 4x² - 9 = (2x)² - 3² = (2x + 3)(2x - 3). Simplify: (2x + 3)(2x - 3)/(2x + 3) = 2x - 3 (provided 2x + 3 ≠ 0).
Manipulates algebraic fractions (adding, subtracting, multiplying, dividing), factorises where the coefficient of x² is not 1, and uses completing the square. (Higher tier)
Example task
Simplify: 2/(x+1) + 3/(x-2).
Model response: Common denominator (x+1)(x-2): [2(x-2) + 3(x+1)] / [(x+1)(x-2)] = (2x - 4 + 3x + 3) / [(x+1)(x-2)] = (5x - 1)/[(x+1)(x-2)].
Delivery rationale
Secondary maths process concept — problem-solving benefits from structured AI delivery with facilitator for extended reasoning.
Solving Linear Equations and Inequalities
process AI FacilitatedMA-KS4-C006
Forming and solving linear equations in one variable, including equations with unknowns on both sides; solving and graphing linear inequalities in one or two variables.
Teaching guidance
Develop the balance model early (doing the same operation to both sides maintains equality) using physical balance imagery. Progress from one-step to multi-step equations systematically. Inequalities require careful attention to the direction reversal when multiplying or dividing by a negative — use examples where pupils test both directions to discover this rule.
Vocabulary (14 terms)
Common misconceptions
Pupils frequently forget to reverse the inequality sign when multiplying or dividing by a negative number. When equations have unknowns on both sides, many pupils only subtract the smaller unknown (e.g. 5x = 3x + 8 becomes 2x = 8, correct, but 3x = 5x + 8 often becomes 2x = 8 rather than -2x = 8). Brackets are often expanded with the wrong sign applied to terms beyond the first.
Difficulty levels
Solves one- and two-step linear equations with positive coefficients.
Example task
Solve 4x + 3 = 19.
Model response: 4x = 16. x = 4.
Solves linear equations with the unknown on both sides and equations involving brackets.
Example task
Solve 3(2x - 1) = 5x + 7.
Model response: 6x - 3 = 5x + 7. x = 10.
Solves linear equations with fractional coefficients and solves linear inequalities, representing solutions on a number line.
Example task
Solve (3x + 2)/4 = 5 and solve 2x - 7 < 3x + 1.
Model response: Equation: 3x + 2 = 20, 3x = 18, x = 6. Inequality: -7 - 1 < 3x - 2x, -8 < x, so x > -8.
Solves equations involving algebraic fractions and forms/solves equations from complex real-world and geometric contexts.
Example task
Solve 5/(x-1) - 2/(x+3) = 1.
Model response: Common denominator (x-1)(x+3): 5(x+3) - 2(x-1) = (x-1)(x+3). 5x+15-2x+2 = x²+2x-3. 3x+17 = x²+2x-3. x²-x-20 = 0. (x-5)(x+4) = 0. x = 5 or x = -4. Check both: x = 5: 5/4 - 2/8 = 5/4 - 1/4 = 1 ✓. x = -4: 5/-5 - 2/-1 = -1+2 = 1 ✓.
Delivery rationale
Secondary maths process concept — problem-solving benefits from structured AI delivery with facilitator for extended reasoning.
Quadratic Equations
process AI FacilitatedMA-KS4-C007
Solving quadratic equations by factorising, completing the square, and the quadratic formula; understanding discriminant and nature of roots.
Teaching guidance
Teach factorisation first as an entry point, but explicitly introduce the quadratic formula as the general-purpose method that always works. Completing the square should be linked to the vertex form of a parabola — connecting algebra to graph interpretation. The discriminant (b²-4ac) is a powerful discriminator: use it to predict the number of solutions before solving.
Vocabulary (12 terms)
Common misconceptions
Many pupils apply factorisation only when the leading coefficient is 1. When completing the square, pupils frequently forget to adjust the constant term after moving the half-coefficient inside. In the quadratic formula, pupils often substitute incorrectly when b or c are negative, missing the double negation.
Difficulty levels
Recognises quadratic expressions and can solve quadratic equations when they are already factorised.
Example task
Solve (x - 3)(x + 5) = 0.
Model response: x - 3 = 0 or x + 5 = 0. x = 3 or x = -5.
Solves quadratic equations by factorising, including those requiring rearrangement first.
Example task
Solve x² + 2x - 15 = 0.
Model response: Find two numbers that multiply to -15 and add to 2: 5 and -3. (x + 5)(x - 3) = 0. x = -5 or x = 3.
Solves quadratic equations using the quadratic formula and by completing the square, and interprets solutions graphically.
Example task
Solve 2x² + 3x - 7 = 0 using the quadratic formula.
Model response: a=2, b=3, c=-7. x = (-3 ± √(9+56))/4 = (-3 ± √65)/4. x = (-3+√65)/4 ≈ 1.27 or x = (-3-√65)/4 ≈ -2.77.
Uses the discriminant to determine the nature of roots, completes the square to find the vertex form, and solves quadratic equations in applied contexts. (Higher tier)
Example task
For what values of k does kx² + 6x + 3 = 0 have no real roots?
Model response: Discriminant < 0: b² - 4ac < 0. 36 - 12k < 0. 12k > 36. k > 3. So for k > 3, the equation has no real roots.
Delivery rationale
Secondary maths process concept — problem-solving benefits from structured AI delivery with facilitator for extended reasoning.
Simultaneous Equations
process AI FacilitatedMA-KS4-C008
Solving pairs of simultaneous linear equations by elimination, substitution, and graphically; solving linear-quadratic simultaneous equation systems (Higher).
Teaching guidance
Begin with graphical interpretation so pupils understand that the solution is the intersection point — this builds conceptual understanding before procedural methods. Teach elimination and substitution as complementary methods: elimination is efficient for symmetric coefficient pairs, substitution is generalisable. For linear-quadratic systems, always check both solutions by substitution into the original equations.
Vocabulary (11 terms)
Common misconceptions
During elimination, pupils often make sign errors when subtracting equations, particularly when one equation has a negative term. Many pupils find only one solution of a linear-quadratic system, not realising two solutions are possible. Graphical solutions are imprecise — pupils must learn to interpret approximate graphical solutions vs exact algebraic solutions.
Difficulty levels
Understands that simultaneous equations are two equations that must both be true at the same time, and can solve them graphically.
Example task
From the graph, read the solution of y = 2x + 1 and y = -x + 7.
Model response: The lines cross at (2, 5). So x = 2, y = 5.
Solves pairs of linear simultaneous equations by elimination or substitution.
Example task
Solve: 3x + 2y = 12 and x - y = 1.
Model response: From the second equation: x = y + 1. Substitute into the first: 3(y+1) + 2y = 12. 3y + 3 + 2y = 12. 5y = 9. y = 9/5 = 1.8. x = 2.8.
Solves simultaneous equations requiring multiplication of one or both equations before elimination.
Example task
Solve: 2x + 3y = 7 and 5x + 4y = 13.
Model response: ×5: 10x + 15y = 35. ×2: 10x + 8y = 26. Subtract: 7y = 9. y = 9/7. x = (7 - 3(9/7))/2 = (49/7 - 27/7)/2 = (22/7)/2 = 11/7.
Solves linear-quadratic simultaneous equation systems and interprets the solutions geometrically. (Higher tier)
Example task
Solve: y = x² - 2x + 1 and y = 2x - 3.
Model response: x² - 2x + 1 = 2x - 3. x² - 4x + 4 = 0. (x - 2)² = 0. x = 2 (repeated). y = 2(2) - 3 = 1. The line is tangent to the parabola at (2, 1) — they touch at exactly one point.
Delivery rationale
Secondary maths process concept — problem-solving benefits from structured AI delivery with facilitator for extended reasoning.
Sequences and nth Term
knowledge AI DirectMA-KS4-C009
Identifying term-to-term and position-to-term rules for arithmetic and geometric sequences; finding the nth term formula for linear and quadratic sequences.
Teaching guidance
Use tabular representations consistently (position, term, difference, second difference) to make the pattern visible before formulating rules. Arithmetic sequences have constant first differences (leading to linear nth term), quadratic sequences have constant second differences (leading to quadratic nth term). Fibonacci-type sequences model non-algebraic recursive thinking and are worth exploring computationally.
Vocabulary (13 terms)
Common misconceptions
Many pupils confuse the term-to-term rule with the position-to-term (nth term) formula, substituting n=1,2,3 to generate terms rather than using the nth term algebraically. When identifying the nth term of an arithmetic sequence, pupils often write the common difference as the coefficient of n but forget to adjust the constant term.
Difficulty levels
Can continue a sequence given a term-to-term rule and identify whether a sequence is arithmetic (constant difference) or geometric (constant ratio).
Example task
Identify the type of sequence and find the next two terms: 5, 8, 11, 14, ...
Model response: Arithmetic sequence with common difference 3. Next terms: 17, 20.
Finds the nth term of a linear (arithmetic) sequence and uses it to determine whether a given number is in the sequence.
Example task
Find the nth term of 7, 11, 15, 19, ... Is 99 in this sequence?
Model response: Common difference = 4. nth term = 4n + 3. Is 99 a term? 4n + 3 = 99, 4n = 96, n = 24. Yes, 99 is the 24th term.
Finds the nth term of quadratic sequences (constant second differences) and generates terms from geometric sequence formulae.
Example task
Find the nth term of: 2, 6, 12, 20, 30, ...
Model response: First differences: 4, 6, 8, 10. Second differences: 2 (constant). Since second difference = 2, coefficient of n² is 2/2 = 1. Sequence - n²: 2-1, 6-4, 12-9, 20-16, 30-25 = 1, 2, 3, 4, 5 = n. So nth term = n² + n = n(n+1).
Works with geometric sequences including sum to infinity, and solves problems requiring sequences in unfamiliar contexts. (Higher tier)
Example task
A geometric sequence has first term 6 and common ratio 1/3. Find the sum to infinity.
Model response: S∞ = a/(1-r) = 6/(1-1/3) = 6/(2/3) = 9. This works because |r| < 1, so the terms converge to 0 and the sum converges.
Delivery rationale
Secondary maths concept — abstract, procedural, and objectively assessable.
Straight Line Graphs and Linear Functions
Keystone knowledge AI DirectMA-KS4-C010
Understanding and using y = mx + c; finding equations of lines; identifying parallel and perpendicular gradients; interpreting gradient and intercept in context.
Teaching guidance
Build gradient from its physical interpretation (steepness, rate of change) before introducing the formula. The connection between gradient and rate of change should be explicit and persistent — revisit in ratio, distance-time graphs, and statistics scatter graphs. Perpendicular gradients (product = -1) require careful derivation using rotation of slope triangles.
Vocabulary (12 terms)
Common misconceptions
Pupils frequently confuse x-intercept and y-intercept, or read gradient as rise alone rather than rise over run. Many pupils state parallel lines have 'the same gradient' but cannot identify perpendicular gradients. When finding the equation of a line through two points, sign errors in computing (y₂-y₁)/(x₂-x₁) are common.
Difficulty levels
Can plot a straight line from a table of values and identify whether a line slopes upward or downward.
Example task
Plot y = 2x - 1 for x = 0, 1, 2, 3, 4.
Model response: Points: (0,-1), (1,1), (2,3), (3,5), (4,7). The line slopes upward.
Identifies gradient and y-intercept from y = mx + c, and finds the equation of a line from a graph.
Example task
A line passes through (0, 3) and (4, 11). Find its equation.
Model response: Gradient = (11-3)/(4-0) = 8/4 = 2. y-intercept = 3. Equation: y = 2x + 3.
Finds equations of lines from two points or from a point and gradient, identifies parallel lines (same gradient) and perpendicular lines (gradient product = -1).
Example task
Find the equation of the line perpendicular to y = 3x - 2 through (6, 1).
Model response: Perpendicular gradient = -1/3. y - 1 = -1/3(x - 6). y = -x/3 + 2 + 1 = -x/3 + 3.
Interprets gradient as rate of change in context (speed, cost per unit), uses distance-time and velocity-time graphs, and derives line equations in unfamiliar coordinate geometry problems.
Example task
A distance-time graph shows a car travelling at constant speed: after 2 hours it has covered 150 km, after 5 hours it has covered 375 km. Find the speed and the equation relating distance d and time t.
Model response: Speed = gradient = (375-150)/(5-2) = 225/3 = 75 km/h. Using (2, 150): d - 150 = 75(t - 2). d = 75t. This means the car started from the origin (d = 0 when t = 0), consistent with constant speed from rest.
Delivery rationale
Secondary maths concept — abstract, procedural, and objectively assessable.
Non-linear Graphs
knowledge AI DirectMA-KS4-C011
Recognising, sketching and interpreting graphs of quadratic, cubic, reciprocal, exponential and trigonometric functions; understanding key features including turning points, asymptotes, and roots.
Teaching guidance
Use graphing technology (Desmos or GeoGebra) to build intuition for graph shapes before working analytically. Focus on key features: roots (where f(x)=0), y-intercept (x=0), turning points, and asymptotes. Transformations of graphs are best understood experientially — pupils should discover how f(x+a) and f(x)+a shift the graph in opposite/same directions respectively before memorising the rules.
Vocabulary (15 terms)
Common misconceptions
Pupils commonly confuse y = kˣ (exponential) with y = xᵏ (power), not recognising the qualitative difference in growth. The direction of horizontal translations is counterintuitive: f(x+2) shifts left, not right. Many pupils draw reciprocal graphs touching the axes rather than having asymptotes.
Difficulty levels
Recognises the shapes of basic non-linear graphs: quadratic (U/n-shape), cubic (S-shape), reciprocal (two branches).
Example task
Match each equation to its graph shape: y = x², y = x³, y = 1/x.
Model response: y = x²: U-shaped parabola. y = x³: S-shaped curve through the origin. y = 1/x: two separate curves in opposite corners.
Sketches quadratic and cubic graphs, identifying roots, y-intercept and turning points.
Example task
Sketch y = (x-1)(x-3). Label the roots and y-intercept.
Model response: Roots at x = 1 and x = 3. y-intercept: (0)(−3) × ... wait, y(0) = (-1)(-3) = 3. Vertex at x = 2: y = (1)(-1) = -1. So minimum at (2, -1). U-shaped parabola crossing at (1,0) and (3,0), y-intercept at (0,3).
Recognises, sketches and interprets exponential (y = kˣ), trigonometric (y = sin x, y = cos x) and reciprocal graphs, identifying key features.
Example task
Sketch y = 2ˣ for -3 ≤ x ≤ 3, labelling key features.
Model response: Points: (-3, 1/8), (-2, 1/4), (-1, 1/2), (0, 1), (1, 2), (2, 4), (3, 8). The graph passes through (0,1), increases exponentially for positive x, and approaches 0 (but never reaches it) for negative x. Horizontal asymptote at y = 0.
Sketches transformed graphs and identifies the effect of parameters on graph features. Applies graphical analysis to solve equations and model real-world phenomena. (Higher tier)
Example task
Given the graph of y = f(x), sketch y = -f(x + 2) and describe the transformations.
Model response: First, f(x + 2) translates the graph 2 units to the left. Then -f(x + 2) reflects the result in the x-axis. If f had a maximum at (3, 5), the transformed graph has a minimum at (1, -5).
Delivery rationale
Secondary maths concept — abstract, procedural, and objectively assessable.
Functions and Graph Transformations
knowledge AI DirectMA-KS4-C012
Using function notation; finding composite and inverse functions; applying and describing graph transformations f(x+a), f(x)+a, af(x), f(ax).
Teaching guidance
Introduce function notation as a generalisation of the input-output machine model used at KS3. Inverse functions undo the original function — link to solving equations by rearranging. Transformations should be taught with reference to any general function y=f(x), then applied to specific cases. The domain and range of functions should be made explicit, particularly for inverse functions.
Vocabulary (12 terms)
Common misconceptions
Pupils frequently confuse f(2x) with 2f(x) — horizontal stretch vs vertical stretch. Many pupils believe every function has an inverse, not recognising that non-one-to-one functions do not have inverses over their full domain. The notation f⁻¹(x) is confused with [f(x)]⁻¹ = 1/f(x).
Difficulty levels
Understands function notation f(x) as an instruction: input x, apply the rule, get the output.
Example task
If f(x) = 3x + 2, find f(5).
Model response: f(5) = 3(5) + 2 = 17.
Evaluates composite functions and understands that the order of composition matters.
Example task
f(x) = 2x + 1 and g(x) = x². Find fg(3) and gf(3).
Model response: fg(3) = f(g(3)) = f(9) = 19. gf(3) = g(f(3)) = g(7) = 49. Note: fg(3) ≠ gf(3).
Finds inverse functions algebraically and understands that the inverse undoes the original function.
Example task
Find f⁻¹(x) for f(x) = (2x - 3)/5.
Model response: Let y = (2x-3)/5. Rearrange for x: 5y = 2x - 3. 2x = 5y + 3. x = (5y+3)/2. So f⁻¹(x) = (5x+3)/2.
Applies graph transformations using function notation (translations, stretches, reflections) and finds composite and inverse functions for complex expressions. (Higher tier)
Example task
f(x) = x² + 4x. Express f(x) in the form (x + a)² + b. Hence state the coordinates of the minimum point and find the range of f.
Model response: f(x) = x² + 4x = (x+2)² - 4. Minimum point at (-2, -4). Range: f(x) ≥ -4. Completing the square converts to vertex form, immediately revealing the turning point.
Delivery rationale
Secondary maths concept — abstract, procedural, and objectively assessable.